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1
+ arXiv:2301.13439v1 [hep-ph] 31 Jan 2023
2
+ USTC-ICTS/PCFT-23-04
3
+ January 2023
4
+ Rare W-boson decays into a vector meson and lepton pair
5
+ Dao-Neng Gao†
6
+ Interdisciplinary Center for Theoretical Study, University of Science and Technology of China,
7
+ Hefei, Anhui 230026 China
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+ Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026 China
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+ Abstract
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+ We have presented a theoretical study of exclusive rare W-boson decays, W β†’ V β„“Β―Ξ½β„“
11
+ with V denoting a neutral vector meson and β„“ = e or Β΅, in the standard model.
12
+ The leading-order contributions to these processes are given by W β†’ Ξ³βˆ—β„“Β―Ξ½β„“ with the
13
+ subsequent Ξ³βˆ— β†’ V transition. Branching fractions of these decay modes, for V = ρ,
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+ Ο‰, Ο†, and J/Ξ¨, respectively, have been calculated and predicted around 10βˆ’6 ∼ 10βˆ’7,
15
+ which are surprisingly larger than those of two-body hadronic radiative decays W Β± β†’
16
+ MΒ±Ξ³ with M denoting a pseudoscalar or vector meson. Thus it is expected that rare
17
+ W decays into a neutral vector meson plus lepton pair may be the promising channels
18
+ in future experimental facilities with a large number of W-boson events produced.
19
+ † E-mail address: [email protected]
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+
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+ Exclusive rare W-boson decays, which contain hadronic final states, could provide inter-
22
+ esting probes to increase our understanding of the properties of the fundamental weak gauge
23
+ boson as well as offer some deep insights into quantum chromodynamics [1, 2, 3, 4, 5]. Experi-
24
+ mentally, no such processes have been observed so far, and only upper limits on the branching
25
+ fractions of three exclusive modes: B(W Β± β†’ DΒ±
26
+ s Ξ³) < 1.3Γ—10βˆ’3, B(W Β± β†’ π±γ) < 7Γ—10βˆ’6,
27
+ and B(W Β± β†’ Ο€+Ο€βˆ’Ο€Β±) < 1.01 Γ— 10βˆ’6, were set at 95% confidence level [6]. On the other
28
+ hand, a huge number of W events, about O(1011), will be expectedly accumulated in the
29
+ high-luminosity Large Hadron Collider (LHC) [3]. This may significantly facilitate the ex-
30
+ perimental studies of rare W-boson decay channels, which can be very helpful both to test
31
+ the standard model (SM) and to search for new physics beyond the SM.
32
+ Our main focus in the present paper is on another types of rare W-boson decays: W β†’
33
+ V β„“Β―Ξ½β„“ with V denoting the neutral vector particle including heavy quarkonium J/Ξ¨ or light
34
+ mesons ρ, Ο‰, and Ο† etc.
35
+ β„“ is the lepton with β„“ = e or Β΅.
36
+ The leading-order Feynman
37
+ diagrams contributing to these processes in the SM have been shown in Figure 1, in which
38
+ the transitions can proceed through W β†’ Ξ³βˆ—β„“Β―Ξ½β„“, followed by Ξ³βˆ— β†’ Β―qq β†’ V . This is similar
39
+ to the case of Z β†’ V β„“+β„“+ decays, which have been studied in Refs. [7, 8, 9].
40
+ First let us go into the decay amplitude of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“. Using the standard vertices
41
+ Wβ„“Β―Ξ½β„“, Ξ³qΒ―q, and Ξ³WW, one can carry out the direct calculation for Figure 1, which gives
42
+ M(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“)
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+ =
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+ βˆ’e2gQV fV
45
+ 2
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+ √
47
+ 2mV
48
+ Η«Β΅(p)Η«βˆ—
49
+ Ξ½(q)Β―u(k1)
50
+ οΏ½2kΞ½
51
+ 1Ξ³Β΅ + Ξ³Ξ½q/Ξ³Β΅
52
+ q2 + 2k1 Β· q
53
+ βˆ’(2k + q)Ξ½Ξ³Β΅ + 2q¡γν βˆ’ 2q/g¡ν
54
+ q2 + 2k Β· q
55
+ οΏ½
56
+ (1 βˆ’ Ξ³5)v(k2),
57
+ (1)
58
+ where p, q, k1, and k2 represent the momenta of W βˆ’ and the final particles including V , β„“βˆ’,
59
+ and Β―Ξ½β„“, respectively. k = k1 + k2 denotes the momentum sum of lepton pair. e is the QED
60
+ coupling constant and g is the weak SU(2)L coupling constant. fV is the decay constant of
61
+ the vector meson, which is defined by
62
+ ⟨V (p, Η«)|Β―qΞ³Ξ½q|0⟩ = fV mV Η«βˆ—
63
+ Ξ½.
64
+ (2)
65
+ Here Η«βˆ—
66
+ Ξ½ is polarization vector of V , and the value of fV can be extracted from the measured
67
+ V β†’ e+eβˆ’ width. As shown in Ref. [3], it has been already given that, fρ = 216.3 Β± 1.3
68
+ MeV, fω = 194.2 ± 2.1 MeV, fφ = 223.0 ± 1.4 MeV, and fJ/Ψ = 403.3 ± 5.1 MeV. QV is the
69
+ quantity related to the electric charge of the quark inside V with Qρ = 1/
70
+ √
71
+ 2, Qω = 1/3
72
+ √
73
+ 2,
74
+ QΟ† = βˆ’1/3, and QJ/Ξ¨ = 2/3. Note that the use of the relation (2) in deriving eq. (1) also
75
+ fulfills the hadronization of the electromagnetic current ¯qγνq into the final state particle V .
76
+ Next, by squaring the decay amplitude (1), summing or averaging the polarizations of
77
+ final or initial particles, the differential decay rate of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ can be expressed as
78
+ dΞ“
79
+ dsV dsβ„“
80
+ = mW
81
+ 256Ο€3
82
+ 1
83
+ 3
84
+ οΏ½
85
+ spins
86
+ |M(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“)|2.
87
+ (3)
88
+ Consequently, we get
89
+ dΞ“
90
+ dsV dsβ„“
91
+ = Ξ±2
92
+ emQ2
93
+ V g2f 2
94
+ V
95
+ 384Ο€mWr2
96
+ V
97
+ IV ,
98
+ (4)
99
+ 1
100
+
101
+ *
102
+ *
103
+ *
104
+ (a)
105
+ (b)
106
+ W
107
+ W
108
+ W
109
+ V
110
+ V
111
+ οΏ½
112
+ οΏ½
113
+ οΏ½
114
+ οΏ½
115
+ οΏ½οΏ½
116
+ οΏ½
117
+ οΏ½
118
+ _
119
+ _
120
+ Figure 1: The lowest-order Feynman diagrams for W β†’ V β„“Β―Ξ½β„“ decays.
121
+ where Ξ±em = e2/4Ο€, rV = mV /mW, and the lepton mass has been neglected in the calcula-
122
+ tion. The explicit expression of the dimensionless quantity IV is a little tedious, which will
123
+ be shown in the Appendix. The Lorentz invariant dimensionless kinematical variables are
124
+ defined as
125
+ sV ≑ (p βˆ’ q)2/m2
126
+ W,
127
+ sβ„“ ≑ (p βˆ’ k1)2/m2
128
+ W,
129
+ (5)
130
+ and the phase space can be given by
131
+ 0 ≀ sV ≀ (1 βˆ’ sβ„“)(1 βˆ’ r2
132
+ V /sβ„“),
133
+ r2
134
+ V ≀ sβ„“ ≀ 1.
135
+ (6)
136
+ Meanwhile, it is easy to compute the leading-order contribution to the width of pure
137
+ leptonic W-boson decay for β„“ = e or Β΅, which reads
138
+ Ξ“(W βˆ’ β†’ β„“βˆ’Β―Ξ½β„“) = g2mW
139
+ 48Ο€
140
+ = GFm3
141
+ W
142
+ 6
143
+ √
144
+ 2Ο€ ≑ Ξ“0,
145
+ (7)
146
+ where GF is the Fermi constant given by GF/
147
+ √
148
+ 2 = g2/8m2
149
+ W.
150
+ Then one can choose to
151
+ normalize the decay rate of W βˆ’ β†’ V β„“βˆ’β„“Β―Ξ½β„“ to Ξ“0, which leads to
152
+ 1
153
+ Ξ“0
154
+ dΞ“
155
+ dsV dsβ„“
156
+ = Ξ±2
157
+ emQ2
158
+ V f 2
159
+ V
160
+ 8m2
161
+ V
162
+ IV .
163
+ (8)
164
+ By further defining
165
+ YV ≑
166
+ οΏ½
167
+ IV dsV dsβ„“
168
+ (9)
169
+ with the integral bound is given in eq. (6), one can get
170
+ Ξ“(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“)
171
+ Ξ“0
172
+ = Ξ±2
173
+ emQ2
174
+ V f 2
175
+ V
176
+ 8m2
177
+ V
178
+ YV .
179
+ (10)
180
+ As mentioned above, the decay constants (fV ) of the neutral vector mesons have been
181
+ extracted by the authors of Ref. [3] from the experimental data, and
182
+ Ξ“(V β†’ e+eβˆ’) = 4Ο€Q2
183
+ V f 2
184
+ V
185
+ 3mV
186
+ Ξ±2
187
+ em(mV )
188
+ (11)
189
+ 2
190
+
191
+ V
192
+ mV (GeV)
193
+ Ξ“(V β†’ e+eβˆ’)(keV)
194
+ YV
195
+ Ξ“(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“)/Ξ“0
196
+ ρ
197
+ 0.775
198
+ 7.04 Β± 0.06
199
+ 194.91
200
+ (5.28 Β± 0.04) Γ— 10βˆ’5
201
+ Ο‰
202
+ 0.782
203
+ 0.60 Β± 0.02
204
+ 193.94
205
+ (4.44 Β± 0.15) Γ— 10βˆ’6
206
+ Ο†
207
+ 1.019
208
+ 1.27 Β± 0.04
209
+ 166.32
210
+ (6.18 Β± 0.19) Γ— 10βˆ’6
211
+ J/Ξ¨
212
+ 3.097
213
+ 5.53 Β± 0.10
214
+ 74.53
215
+ (3.97 Β± 0.07) Γ— 10βˆ’6
216
+ Table 1: Decay rates of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ normalized to Ξ“(W βˆ’ β†’ β„“βˆ’Β―Ξ½β„“) for β„“ = e or Β΅. The
217
+ values of Ξ“(V β†’ e+eβˆ’) are taken from Ref. [6].
218
+ has been used. Therefore, after integrating over IV in eq. (9) to get YV , one can easily
219
+ predict the decay rates of W β†’ V β„“Β―Ξ½β„“ for V = ρ, Ο‰, Ο†, and J/Ξ¨, respectively.
220
+ On the other hand, note that the scale of the electromagnetic coupling Ξ±em in eq. (8)
221
+ should also be at mV since, in Figure 1, this electromagnetic transition is via Ξ³βˆ— β†’ V .
222
+ Therefore, combing eq. (10) with eq. (11), one will obtain
223
+ Ξ“(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“)
224
+ Ξ“0
225
+ =
226
+ 3YV
227
+ 32Ο€mV
228
+ Ξ“(V β†’ e+eβˆ’),
229
+ (12)
230
+ which means that we can get Ξ“(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“)/Ξ“0 using the experimental data of Ξ“(V β†’
231
+ e+eβˆ’) given by Particle Data Group [6] directly. Numerical results have been listed in Table
232
+ 1, and the errors of the predictions in the fifth column are due to the uncertainties of the
233
+ measured widths of Ξ“(V β†’ e+eβˆ’) only. To transform them into the branching fractions of
234
+ W β†’ V β„“Β―Ξ½β„“, one may use the experimental data of B(W β†’ β„“Β―Ξ½β„“), which can be found in Ref.
235
+ [6] that
236
+ B(W βˆ’ β†’ eβˆ’Β―Ξ½e) = (10.71 Β± 0.16)%,
237
+ B(W βˆ’ β†’ Β΅βˆ’Β―Ξ½Β΅) = (10.63 Β± 0.15)%.
238
+ (13)
239
+ For our numerical analysis, we take
240
+ B(W βˆ’ β†’ β„“βˆ’Β―Ξ½β„“) = (10.67 Β± 0.16)%
241
+ (14)
242
+ by simply averaging over the electron and muon modes. Thus, it is straightforward to obtain
243
+ the branching fractions of rare W-boson decays into a vector meson and lepton pair, for β„“ = e
244
+ or Β΅, which read
245
+ B(W βˆ’ β†’ Οβ„“βˆ’Β―Ξ½β„“) = (5.64 Β± 0.10) Γ— 10βˆ’6,
246
+ (15)
247
+ B(W βˆ’ β†’ Ο‰β„“βˆ’Β―Ξ½β„“) = (4.74 Β± 0.17) Γ— 10βˆ’7,
248
+ (16)
249
+ B(W βˆ’ β†’ Ο†β„“βˆ’Β―Ξ½β„“) = (6.60 Β± 0.23) Γ— 10βˆ’7,
250
+ (17)
251
+ B(W βˆ’ β†’ J/Ξ¨β„“βˆ’Β―Ξ½β„“) = (4.24 Β± 0.10) Γ— 10βˆ’7.
252
+ (18)
253
+ Here the quoted errors of our theoretical results show the uncertainties from the experimental
254
+ values of Ξ“(V β†’ e+eβˆ’) in the third column of Table 1, and also B(W βˆ’ β†’ β„“βˆ’Β―Ξ½β„“) in eq. (14).
255
+ It is found that branching ratios of W β†’ V β„“Β―Ξ½β„“ decays obtained in the present work are
256
+ quite larger than those of the hadronic radiative decays W Β± β†’ MΒ±Ξ³ (M is a pseudoscalar
257
+ 3
258
+
259
+ W
260
+ W
261
+ J/οΏ½
262
+ οΏ½
263
+ οΏ½
264
+ οΏ½
265
+ _
266
+ c
267
+ s
268
+ c
269
+ _
270
+ *
271
+ Figure 2: The Feynman diagram contributing to W β†’ J/Ξ¨β„“Β―Ξ½β„“ decays via W β†’ J/Ξ¨W βˆ—
272
+ transition.
273
+ or vector meson such as Ο€, K, ρ, Kβˆ—, and Ds etc), which are maximally around 10βˆ’8 or even
274
+ smaller, predicted by the authors of Ref. [3]. Naively, one may expect that Ξ“(W β†’ V β„“Β―Ξ½β„“)
275
+ should be smaller than Ξ“(W Β± β†’ MΒ±Ξ³) since the former rate is suppressed by a power of
276
+ Ξ±em compared to the latter rate. However, careful observation can tell us this expectation is
277
+ not correct. As given in Ref. [3], we know
278
+ Ξ“(W Β± β†’ MΒ±Ξ³) ∼ Ξ±emf 2
279
+ M
280
+ 192mW
281
+ .
282
+ (19)
283
+ Comparing with eq. (4), one will find a relevant factor m2
284
+ W/m2
285
+ V in the formula of Ξ“(W βˆ’ β†’
286
+ V β„“βˆ’Β―Ξ½β„“), which could significantly counteract the suppression of Ξ±em if the mass of vector
287
+ meson is very small relative to the W mass. Obviously, the appearance of this factor is
288
+ actually due to the virtual photon propagator of Ξ³βˆ— β†’ V transition in Figure 1.
289
+ Similar situation also occurs in rare Z-boson decays. In particular, it has been shown in
290
+ Ref. [8] that the dominant contribution to Z β†’ V β„“+β„“βˆ’ comes from Z β†’ Ξ³βˆ—β„“+β„“βˆ’ with the
291
+ subsequent transition Ξ³βˆ— β†’ V , since, in comparison, the radiative decays Z β†’ V Ξ³ are quite
292
+ suppressed. One can thus neglect the contribution from Z β†’ V Ξ³βˆ— β†’ V β„“+β„“βˆ’ although it is
293
+ of the same order of Ξ±em as the dominant part.
294
+ Analogous to Z β†’ V Ξ³βˆ— β†’ V β„“+β„“βˆ’, the rare charged weak gauge boson decays considered
295
+ in the present paper could happen through W β†’ V W βˆ— β†’ V β„“Β―Ξ½β„“. The Feynman diagram
296
+ has been displayed in Figure 2, and we take V = J/Ξ¨ as an explicit example. As a good
297
+ approximation for the leading order calculation, the momenta of the quark (c) and anti-
298
+ quark (Β―c) are taken to be one half of J/Ξ¨ momentum q, so the strange quark propagator in
299
+ this diagram is proportional to 1/(k + q
300
+ 2)2, which is of order 1/m2
301
+ W. By contrast, the virtual
302
+ photon propagator in the diagrams of Figure 1 is of order 1/m2
303
+ J/Ξ¨. This means that the
304
+ contribution from Figure 2, relative to that from Figure 1, is strongly suppressed, which can
305
+ be safely neglected.
306
+ Furthermore, recall that the differential decay rate of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ has been given in
307
+ eq. (4). Now one can rewrite
308
+ sV = 1 + r2
309
+ V βˆ’ 2EV /mW,
310
+ sβ„“ = 1 βˆ’ 2Eβ„“/mW,
311
+ (20)
312
+ where EV is the vector meson energy and Eβ„“ is the lepton energy in the rest frame of W
313
+ 4
314
+
315
+ 0
316
+ 10
317
+ 20
318
+ 30
319
+ 40
320
+ 0.00
321
+ 0.01
322
+ 0.02
323
+ 0.03
324
+ 0.04
325
+ 0.05
326
+ 0.06
327
+ EJ (GeV)
328
+ 1/
329
+ οΏ½
330
+ d
331
+ οΏ½
332
+ /dEJ (GeV-1)
333
+ 0
334
+ 10
335
+ 20
336
+ 30
337
+ 40
338
+ 0.00
339
+ 0.01
340
+ 0.02
341
+ 0.03
342
+ 0.04
343
+ 0.05
344
+ 0.06
345
+ Eβ„“
346
+ (GeV)
347
+ 1/Ξ“ dΞ“/dEβ„“ (GeV-1)
348
+ Figure 3: The normalized energy spectrum of W β†’ J/Ξ¨β„“βˆ’Β―Ξ½β„“ decays with respect to J/Ξ¨
349
+ energy EJ (left plot), and with respect to the lepton energy Eβ„“ (right plot).
350
+ boson. In terms of EV and Eβ„“, we have
351
+ dΞ“
352
+ dEV dEβ„“
353
+ = Ξ±2
354
+ emQ2
355
+ V g2f 2
356
+ V
357
+ 96Ο€m3
358
+ Wr2
359
+ V
360
+ IV .
361
+ (21)
362
+ Thus the energy spectrum of the rare decays can be obtained by integrating over EV or Eβ„“.
363
+ The normalized energy distributions of W βˆ’ β†’ J/Ξ¨β„“βˆ’Β―Ξ½β„“ with respect to EJ and Eβ„“ have
364
+ been plotted in Figure 3, respectively. The peak of the distribution is corresponding to the
365
+ small J/ψ energy or large lepton energy region. Since we have neglected the lepton mass in
366
+ the calculation, the spectrum in left plot does not go to zero for EJ at ∼ mW/2. We are not
367
+ going to display the plots for the differential rate of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ decays when V is the light
368
+ vector meson (ρ, Ο‰, and Ο†) because it is believed that one will achieve the similar behavior
369
+ as above.
370
+ To summarize, we have presented the analysis of exclusive rare W-boson decays into a
371
+ vector meson and lepton pair. In the SM, the leading order contributions to these processes
372
+ come from W β†’ Ξ³βˆ—β„“Β―Ξ½β„“, followed by Ξ³βˆ— β†’ V . Using the measured widths of Ξ“(V β†’ e+eβˆ’)
373
+ given in [6], we have determined the branching fractions of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ for V = ρ, Ο‰,
374
+ Ο†, and J/Ξ¨, respectively, as shown in eqs.
375
+ (15) – (18).
376
+ It is surprising that branching
377
+ fractions of these three-body decays, although they are suppressed by a power of Ξ±em, are
378
+ quite larger than those of two-body hadronic radiative decays W Β± β†’ MΒ±Ξ³, which have been
379
+ predicted by the authors of Ref. [3] already. Furthermore, note that the Ξ³WW vertex, as
380
+ shown in Figure 1(b), is involved in the transition, thus both experimental and theoretical
381
+ investigations of W β†’ V β„“Β―Ξ½β„“ decays may be also helpful to test triple gauge couplings.
382
+ Our experimentalists have been trying to search for exclusive rare W-boson processes
383
+ containing hadronic final states. Unfortunately, so far no such decays have been observed.
384
+ Theoretical predictions on branching fractions of W β†’ V β„“Β―Ξ½β„“ in the present paper are around
385
+ 10βˆ’6 ∼ 10βˆ’7. Experimentally, the heavy quarkonium J/ψ is in general reconstructed via
386
+ leptonic decays with their rates: B(J/Ξ¨ β†’ β„“+β„“βˆ’) = (5.971Β±0.032)% [6]; while for light vector
387
+ mesons, ρ decays almost exclusively to Ο€+Ο€βˆ’, Ο‰ and Ο† have a large rate into Ο€+Ο€βˆ’Ο€βˆ’ and
388
+ K+Kβˆ’, respectively, in the event construction. Therefore, our analysis seems to indicate that
389
+ 5
390
+
391
+ these exclusive rare W decay modes could be the promising candidates in future experimental
392
+ machines, for instance, in the high-luminosity LHC, where large amount of W bosons about
393
+ O(1011) events will be produced. We eagerly await some dedicated searches for such decays
394
+ at these facilities.
395
+ Acknowledgments
396
+ This work was supported in part by the National Natural Science Foundation of China
397
+ under Grants No. 11575175, No. 12047502, and No. 12247103, and by National Research
398
+ and Development Program of China under Contract No. 2020YFA0406400.
399
+ Appendix: Explicit expression of IV
400
+ After squaring the W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ decay amplitude and summing/averaging spins of all par-
401
+ ticles, we get the differential decay rate of eq. (4) as
402
+ dΞ“
403
+ dsV dsβ„“
404
+ = Ξ±2
405
+ emQ2
406
+ V g2f 2
407
+ V
408
+ 384Ο€mWr2
409
+ V
410
+ IV .
411
+ The full expression of IV , in terms of dot product of the relevant four-momenta, can be given
412
+ by
413
+ IV = I1 + I2 + I3,
414
+ (A1)
415
+ where
416
+ I1 =
417
+ 8
418
+ (q2 + 2k1 Β· q)2
419
+ οΏ½
420
+ (2k1 Β· qk2 Β· q βˆ’ q2k1 Β· k2) + 2p Β· k2
421
+ m2
422
+ W
423
+ (2k1 Β· qp Β· q βˆ’ q2k1 Β· p)
424
+ οΏ½
425
+ ,
426
+ (A2)
427
+ I2 =
428
+ 8
429
+ (q2 + 2k1 Β· q)(q2 + 2k Β· q)
430
+ οΏ½
431
+ 2k1 Β· k2(4k1 Β· k2 + 2k Β· q + 4q2) βˆ’ 4k1 Β· qk2 Β· q
432
+ βˆ’ 1
433
+ m2
434
+ W
435
+ [(2k1 Β· q + 4k Β· q + 3q2)(2k1 Β· qk2 Β· q βˆ’ q2k1 Β· k2)
436
+ βˆ’4p Β· k2(q2 + 2k2 Β· q)k1 Β· k2 + 4p Β· q(q2 + 2k1 Β· q)k1 Β· k2]
437
+ οΏ½
438
+ ,
439
+ (A3)
440
+ and
441
+ I3 =
442
+ 8
443
+ (q2 + 2k Β· q)2
444
+ οΏ½
445
+ 12k1 Β· qk2 Β· q βˆ’ ((2k + q)2 + 10q2)k1 Β· k2
446
+ βˆ’
447
+ 1
448
+ 2m2
449
+ W
450
+ (2k + q)2(2k1 Β· qk2 Β· q βˆ’ q2k1 Β· k2) + 4(p Β· q)2
451
+ m2
452
+ W
453
+ οΏ½
454
+ .
455
+ (A4)
456
+ On the other hand, one can easily get
457
+ k1 Β· k2 = m2
458
+ W
459
+ 2 sV ,
460
+ p Β· q = m2
461
+ W
462
+ 2 (1 + r2
463
+ V βˆ’ sV ),
464
+ 6
465
+
466
+ k1 Β· q = m2
467
+ W
468
+ 2 (1 βˆ’ sV βˆ’ sβ„“),
469
+ k2 Β· q = m2
470
+ W
471
+ 2 (sβ„“ βˆ’ r2
472
+ V ),
473
+ k1 Β· p = m2
474
+ W
475
+ 2 (1 βˆ’ sβ„“),
476
+ k2 Β· p = m2
477
+ W
478
+ 2 (sV + sβ„“ βˆ’ r2
479
+ V ).
480
+ For on-shell initial and final states particles, we could take p2 = m2
481
+ W, q2 = m2
482
+ V , and k2
483
+ 1 =
484
+ k2
485
+ 2 = 0 (lepton masses are set to be zero already). This shows that IV can be in terms of the
486
+ kinematical variables sV and sβ„“ completely.
487
+ References
488
+ [1] L. Arnellos, W. J. Marciano, and Z. Parsa, Nucl. Phys. B196, 378 (1982).
489
+ [2] M. Mangano and T. Melia, Eur. Phys. J. C 75, 258 (2015), arXiv:1410.7475 [hep-ph].
490
+ [3] Y. Grossman, M. K¨onig, and M. Neubert, J. High Energy Phys. 04 (2015) 101,
491
+ arXiv:1501.06569 [hep-ph].
492
+ [4] Y.Y. Keum and X.Y. Pham, Mod. Phys. Lett. A 9, 1545 (1994), hep-ph/9303300.
493
+ [5] S. Ishaq, Y. Jia, X. Xiong, and D.-S. Yang, Phys. Rev. D 100, 054027 (2019),
494
+ arXiv:1903.12627 [hep-ph]; F. Feng, Y. Jia, and W.-L. Sang, arXiv:1902.11288 [hep-
495
+ ph].
496
+ [6] R.L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01
497
+ (2022).
498
+ [7] L. Bergstr¨om and R.W. Robinett, Phys. Lett. B 245, 249 (1990).
499
+ [8] S. Fleming, Phys. Rev. D 48, R1914 (1993), hep-ph/9304270; S. Fleming, Phys. Rev.
500
+ D 50, 5808 (1994), hep-ph/9403396.
501
+ [9] CMS Collaboration, A.M. Sirunyan et al., Phys. Rev. Lett. 121, 141801 (2018),
502
+ arXiv:1806.04213 [hep-ex].
503
+ 7
504
+
-NFQT4oBgHgl3EQf6jZ7/content/tmp_files/load_file.txt ADDED
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1
+ filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf,len=260
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+ page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
3
+ page_content='13439v1 [hep-ph] 31 Jan 2023 USTC-ICTS/PCFT-23-04 January 2023 Rare W-boson decays into a vector meson and lepton pair Dao-Neng Gao† Interdisciplinary Center for Theoretical Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
4
+ page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
5
+ page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
6
+ page_content=' Anhui 230026 China Peng Huanwu Center for Fundamental Theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
7
+ page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
8
+ page_content=' Anhui 230026 China Abstract We have presented a theoretical study of exclusive rare W-boson decays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
9
+ page_content=' W β†’ V β„“Β―Ξ½β„“ with V denoting a neutral vector meson and β„“ = e or Β΅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
10
+ page_content=' in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
11
+ page_content=' The leading-order contributions to these processes are given by W β†’ Ξ³βˆ—β„“Β―Ξ½β„“ with the subsequent Ξ³βˆ— β†’ V transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
12
+ page_content=' Branching fractions of these decay modes, for V = ρ, Ο‰, Ο†, and J/Ξ¨, respectively, have been calculated and predicted around 10βˆ’6 ∼ 10βˆ’7, which are surprisingly larger than those of two-body hadronic radiative decays W Β± β†’ MΒ±Ξ³ with M denoting a pseudoscalar or vector meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
13
+ page_content=' Thus it is expected that rare W decays into a neutral vector meson plus lepton pair may be the promising channels in future experimental facilities with a large number of W-boson events produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
14
+ page_content=' † E-mail address: gaodn@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
15
+ page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
16
+ page_content='cn Exclusive rare W-boson decays, which contain hadronic final states, could provide inter- esting probes to increase our understanding of the properties of the fundamental weak gauge boson as well as offer some deep insights into quantum chromodynamics [1, 2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
17
+ page_content=' Experi- mentally, no such processes have been observed so far, and only upper limits on the branching fractions of three exclusive modes: B(W Β± β†’ DΒ± s Ξ³) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
18
+ page_content='3Γ—10βˆ’3, B(W Β± β†’ π±γ) < 7Γ—10βˆ’6, and B(W Β± β†’ Ο€+Ο€βˆ’Ο€Β±) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
19
+ page_content='01 Γ— 10βˆ’6, were set at 95% confidence level [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
20
+ page_content=' On the other hand, a huge number of W events, about O(1011), will be expectedly accumulated in the high-luminosity Large Hadron Collider (LHC) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
21
+ page_content=' This may significantly facilitate the ex- perimental studies of rare W-boson decay channels, which can be very helpful both to test the standard model (SM) and to search for new physics beyond the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
22
+ page_content=' Our main focus in the present paper is on another types of rare W-boson decays: W β†’ V β„“Β―Ξ½β„“ with V denoting the neutral vector particle including heavy quarkonium J/Ξ¨ or light mesons ρ, Ο‰, and Ο† etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
23
+ page_content=' β„“ is the lepton with β„“ = e or Β΅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
24
+ page_content=' The leading-order Feynman diagrams contributing to these processes in the SM have been shown in Figure 1, in which the transitions can proceed through W β†’ Ξ³βˆ—β„“Β―Ξ½β„“, followed by Ξ³βˆ— β†’ Β―qq β†’ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
25
+ page_content=' This is similar to the case of Z β†’ V β„“+β„“+ decays, which have been studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
26
+ page_content=' [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
27
+ page_content=' First let us go into the decay amplitude of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
28
+ page_content=' Using the standard vertices Wβ„“Β―Ξ½β„“, Ξ³qΒ―q, and Ξ³WW, one can carry out the direct calculation for Figure 1, which gives M(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“) = βˆ’e2gQV fV 2 √ 2mV Η«Β΅(p)Η«βˆ— Ξ½(q)Β―u(k1) οΏ½2kΞ½ 1Ξ³Β΅ + Ξ³Ξ½q/Ξ³Β΅ q2 + 2k1 Β· q βˆ’(2k + q)Ξ½Ξ³Β΅ + 2q¡γν βˆ’ 2q/g¡ν q2 + 2k Β· q οΏ½ (1 βˆ’ Ξ³5)v(k2), (1) where p, q, k1, and k2 represent the momenta of W βˆ’ and the final particles including V , β„“βˆ’, and Β―Ξ½β„“, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
29
+ page_content=' k = k1 + k2 denotes the momentum sum of lepton pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
30
+ page_content=' e is the QED coupling constant and g is the weak SU(2)L coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
31
+ page_content=' fV is the decay constant of the vector meson, which is defined by ⟨V (p, Η«)|Β―qΞ³Ξ½q|0⟩ = fV mV Η«βˆ— Ξ½.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
32
+ page_content=' (2) Here Η«βˆ— Ξ½ is polarization vector of V , and the value of fV can be extracted from the measured V β†’ e+eβˆ’ width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
33
+ page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
34
+ page_content=' [3], it has been already given that, fρ = 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
35
+ page_content='3 Β± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
36
+ page_content='3 MeV, fω = 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
37
+ page_content='2 Β± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
38
+ page_content='1 MeV, fφ = 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
39
+ page_content='0 Β± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
40
+ page_content='4 MeV, and fJ/Ξ¨ = 403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
41
+ page_content='3 Β± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
42
+ page_content='1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
43
+ page_content=' QV is the quantity related to the electric charge of the quark inside V with Qρ = 1/ √ 2, QΟ‰ = 1/3 √ 2, QΟ† = βˆ’1/3, and QJ/Ξ¨ = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
44
+ page_content=' Note that the use of the relation (2) in deriving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
45
+ page_content=' (1) also fulfills the hadronization of the electromagnetic current ¯qγνq into the final state particle V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
46
+ page_content=' Next, by squaring the decay amplitude (1), summing or averaging the polarizations of final or initial particles, the differential decay rate of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ can be expressed as dΞ“ dsV dsβ„“ = mW 256Ο€3 1 3 οΏ½ spins |M(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
47
+ page_content=' (3) Consequently, we get dΞ“ dsV dsβ„“ = Ξ±2 emQ2 V g2f 2 V 384Ο€mWr2 V IV , (4) 1 (a) (b) W W W V V οΏ½ οΏ½ οΏ½ οΏ½ οΏ½οΏ½ οΏ½ οΏ½ _ _ Figure 1: The lowest-order Feynman diagrams for W β†’ V β„“Β―Ξ½β„“ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
48
+ page_content=' where Ξ±em = e2/4Ο€, rV = mV /mW, and the lepton mass has been neglected in the calcula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
49
+ page_content=' The explicit expression of the dimensionless quantity IV is a little tedious, which will be shown in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
50
+ page_content=' The Lorentz invariant dimensionless kinematical variables are defined as sV ≑ (p βˆ’ q)2/m2 W, sβ„“ ≑ (p βˆ’ k1)2/m2 W, (5) and the phase space can be given by 0 ≀ sV ≀ (1 βˆ’ sβ„“)(1 βˆ’ r2 V /sβ„“), r2 V ≀ sβ„“ ≀ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
51
+ page_content=' (6) Meanwhile, it is easy to compute the leading-order contribution to the width of pure leptonic W-boson decay for β„“ = e or Β΅, which reads Ξ“(W βˆ’ β†’ β„“βˆ’Β―Ξ½β„“) = g2mW 48Ο€ = GFm3 W 6 √ 2Ο€ ≑ Ξ“0, (7) where GF is the Fermi constant given by GF/ √ 2 = g2/8m2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
52
+ page_content=' Then one can choose to normalize the decay rate of W βˆ’ β†’ V β„“βˆ’β„“Β―Ξ½β„“ to Ξ“0, which leads to 1 Ξ“0 dΞ“ dsV dsβ„“ = Ξ±2 emQ2 V f 2 V 8m2 V IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
53
+ page_content=' (8) By further defining YV ≑ οΏ½ IV dsV dsβ„“ (9) with the integral bound is given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
54
+ page_content=' (6), one can get Ξ“(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“) Ξ“0 = Ξ±2 emQ2 V f 2 V 8m2 V YV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
55
+ page_content=' (10) As mentioned above, the decay constants (fV ) of the neutral vector mesons have been extracted by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
56
+ page_content=' [3] from the experimental data, and Ξ“(V β†’ e+eβˆ’) = 4Ο€Q2 V f 2 V 3mV Ξ±2 em(mV ) (11) 2 V mV (GeV) Ξ“(V β†’ e+eβˆ’)(keV) YV Ξ“(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“)/Ξ“0 ρ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
57
+ page_content='775 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
58
+ page_content='04 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
59
+ page_content='06 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
60
+ page_content='91 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
61
+ page_content='28 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
62
+ page_content='04) Γ— 10βˆ’5 Ο‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
63
+ page_content='782 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
64
+ page_content='60 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
65
+ page_content='02 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
66
+ page_content='94 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
67
+ page_content='44 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
68
+ page_content='15) Γ— 10βˆ’6 Ο† 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
69
+ page_content='019 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
70
+ page_content='27 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
71
+ page_content='04 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
72
+ page_content='32 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
73
+ page_content='18 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
74
+ page_content='19) Γ— 10βˆ’6 J/Ξ¨ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
75
+ page_content='097 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
76
+ page_content='53 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
77
+ page_content='10 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
78
+ page_content='53 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
79
+ page_content='97 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
80
+ page_content='07) Γ— 10βˆ’6 Table 1: Decay rates of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ normalized to Ξ“(W βˆ’ β†’ β„“βˆ’Β―Ξ½β„“) for β„“ = e or Β΅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
81
+ page_content=' The values of Ξ“(V β†’ e+eβˆ’) are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
82
+ page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
83
+ page_content=' has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
84
+ page_content=' Therefore, after integrating over IV in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
85
+ page_content=' (9) to get YV , one can easily predict the decay rates of W β†’ V β„“Β―Ξ½β„“ for V = ρ, Ο‰, Ο†, and J/Ξ¨, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
86
+ page_content=' On the other hand, note that the scale of the electromagnetic coupling Ξ±em in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
87
+ page_content=' (8) should also be at mV since, in Figure 1, this electromagnetic transition is via Ξ³βˆ— β†’ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
88
+ page_content=' Therefore, combing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
89
+ page_content=' (10) with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
90
+ page_content=' (11), one will obtain Ξ“(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“) Ξ“0 = 3YV 32Ο€mV Ξ“(V β†’ e+eβˆ’), (12) which means that we can get Ξ“(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“)/Ξ“0 using the experimental data of Ξ“(V β†’ e+eβˆ’) given by Particle Data Group [6] directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
91
+ page_content=' Numerical results have been listed in Table 1, and the errors of the predictions in the fifth column are due to the uncertainties of the measured widths of Ξ“(V β†’ e+eβˆ’) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
92
+ page_content=' To transform them into the branching fractions of W β†’ V β„“Β―Ξ½β„“, one may use the experimental data of B(W β†’ β„“Β―Ξ½β„“), which can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
93
+ page_content=' [6] that B(W βˆ’ β†’ eβˆ’Β―Ξ½e) = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
94
+ page_content='71 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
95
+ page_content='16)%, B(W βˆ’ β†’ Β΅βˆ’Β―Ξ½Β΅) = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
96
+ page_content='63 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
97
+ page_content='15)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
98
+ page_content=' (13) For our numerical analysis, we take B(W βˆ’ β†’ β„“βˆ’Β―Ξ½β„“) = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
99
+ page_content='67 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
100
+ page_content='16)% (14) by simply averaging over the electron and muon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
101
+ page_content=' Thus, it is straightforward to obtain the branching fractions of rare W-boson decays into a vector meson and lepton pair, for β„“ = e or Β΅, which read B(W βˆ’ β†’ Οβ„“βˆ’Β―Ξ½β„“) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
102
+ page_content='64 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
103
+ page_content='10) Γ— 10βˆ’6, (15) B(W βˆ’ β†’ Ο‰β„“βˆ’Β―Ξ½β„“) = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
104
+ page_content='74 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
105
+ page_content='17) Γ— 10βˆ’7, (16) B(W βˆ’ β†’ Ο†β„“βˆ’Β―Ξ½β„“) = (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
106
+ page_content='60 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
107
+ page_content='23) Γ— 10βˆ’7, (17) B(W βˆ’ β†’ J/Ξ¨β„“βˆ’Β―Ξ½β„“) = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
108
+ page_content='24 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
109
+ page_content='10) Γ— 10βˆ’7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
110
+ page_content=' (18) Here the quoted errors of our theoretical results show the uncertainties from the experimental values of Ξ“(V β†’ e+eβˆ’) in the third column of Table 1, and also B(W βˆ’ β†’ β„“βˆ’Β―Ξ½β„“) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
111
+ page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
112
+ page_content=' It is found that branching ratios of W β†’ V β„“Β―Ξ½β„“ decays obtained in the present work are quite larger than those of the hadronic radiative decays W Β± β†’ MΒ±Ξ³ (M is a pseudoscalar 3 W W J/οΏ½ οΏ½ οΏ½ οΏ½ _ c s c _ Figure 2: The Feynman diagram contributing to W β†’ J/Ξ¨β„“Β―Ξ½β„“ decays via W β†’ J/Ξ¨W βˆ— transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
113
+ page_content=' or vector meson such as Ο€, K, ρ, Kβˆ—, and Ds etc), which are maximally around 10βˆ’8 or even smaller, predicted by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
114
+ page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
115
+ page_content=' Naively, one may expect that Ξ“(W β†’ V β„“Β―Ξ½β„“) should be smaller than Ξ“(W Β± β†’ MΒ±Ξ³) since the former rate is suppressed by a power of Ξ±em compared to the latter rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
116
+ page_content=' However, careful observation can tell us this expectation is not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
117
+ page_content=' As given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
118
+ page_content=' [3], we know Ξ“(W Β± β†’ MΒ±Ξ³) ∼ Ξ±emf 2 M 192mW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
119
+ page_content=' (19) Comparing with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
120
+ page_content=' (4), one will find a relevant factor m2 W/m2 V in the formula of Ξ“(W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“), which could significantly counteract the suppression of Ξ±em if the mass of vector meson is very small relative to the W mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
121
+ page_content=' Obviously, the appearance of this factor is actually due to the virtual photon propagator of Ξ³βˆ— β†’ V transition in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
122
+ page_content=' Similar situation also occurs in rare Z-boson decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
123
+ page_content=' In particular, it has been shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
124
+ page_content=' [8] that the dominant contribution to Z β†’ V β„“+β„“βˆ’ comes from Z β†’ Ξ³βˆ—β„“+β„“βˆ’ with the subsequent transition Ξ³βˆ— β†’ V , since, in comparison, the radiative decays Z β†’ V Ξ³ are quite suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
125
+ page_content=' One can thus neglect the contribution from Z β†’ V Ξ³βˆ— β†’ V β„“+β„“βˆ’ although it is of the same order of Ξ±em as the dominant part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
126
+ page_content=' Analogous to Z β†’ V Ξ³βˆ— β†’ V β„“+β„“βˆ’, the rare charged weak gauge boson decays considered in the present paper could happen through W β†’ V W βˆ— β†’ V β„“Β―Ξ½β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
127
+ page_content=' The Feynman diagram has been displayed in Figure 2, and we take V = J/Ξ¨ as an explicit example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
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+ page_content=' As a good approximation for the leading order calculation, the momenta of the quark (c) and anti- quark (Β―c) are taken to be one half of J/Ξ¨ momentum q, so the strange quark propagator in this diagram is proportional to 1/(k + q 2)2, which is of order 1/m2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
129
+ page_content=' By contrast, the virtual photon propagator in the diagrams of Figure 1 is of order 1/m2 J/Ξ¨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
130
+ page_content=' This means that the contribution from Figure 2, relative to that from Figure 1, is strongly suppressed, which can be safely neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
131
+ page_content=' Furthermore, recall that the differential decay rate of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ has been given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
132
+ page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
133
+ page_content=' Now one can rewrite sV = 1 + r2 V βˆ’ 2EV /mW, sβ„“ = 1 βˆ’ 2Eβ„“/mW, (20) where EV is the vector meson energy and Eβ„“ is the lepton energy in the rest frame of W 4 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
134
+ page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
135
+ page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
136
+ page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
137
+ page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
138
+ page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
139
+ page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
140
+ page_content='06 EJ (GeV) 1/ οΏ½ d οΏ½ /dEJ (GeV-1) 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
141
+ page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
142
+ page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
143
+ page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
144
+ page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
145
+ page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
146
+ page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
147
+ page_content='06 Eβ„“ (GeV) 1/Ξ“ dΞ“/dEβ„“ (GeV-1) Figure 3: The normalized energy spectrum of W β†’ J/Ξ¨β„“βˆ’Β―Ξ½β„“ decays with respect to J/Ξ¨ energy EJ (left plot), and with respect to the lepton energy Eβ„“ (right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
148
+ page_content=' boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
149
+ page_content=' In terms of EV and Eβ„“, we have dΞ“ dEV dEβ„“ = Ξ±2 emQ2 V g2f 2 V 96Ο€m3 Wr2 V IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
150
+ page_content=' (21) Thus the energy spectrum of the rare decays can be obtained by integrating over EV or Eβ„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
151
+ page_content=' The normalized energy distributions of W βˆ’ β†’ J/Ξ¨β„“βˆ’Β―Ξ½β„“ with respect to EJ and Eβ„“ have been plotted in Figure 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
152
+ page_content=' The peak of the distribution is corresponding to the small J/ψ energy or large lepton energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
153
+ page_content=' Since we have neglected the lepton mass in the calculation, the spectrum in left plot does not go to zero for EJ at ∼ mW/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
154
+ page_content=' We are not going to display the plots for the differential rate of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ decays when V is the light vector meson (ρ, Ο‰, and Ο†) because it is believed that one will achieve the similar behavior as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
155
+ page_content=' To summarize, we have presented the analysis of exclusive rare W-boson decays into a vector meson and lepton pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
156
+ page_content=' In the SM, the leading order contributions to these processes come from W β†’ Ξ³βˆ—β„“Β―Ξ½β„“, followed by Ξ³βˆ— β†’ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
157
+ page_content=' Using the measured widths of Ξ“(V β†’ e+eβˆ’) given in [6], we have determined the branching fractions of W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ for V = ρ, Ο‰, Ο†, and J/Ξ¨, respectively, as shown in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
158
+ page_content=' (15) – (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
159
+ page_content=' It is surprising that branching fractions of these three-body decays, although they are suppressed by a power of Ξ±em, are quite larger than those of two-body hadronic radiative decays W Β± β†’ MΒ±Ξ³, which have been predicted by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
160
+ page_content=' [3] already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
161
+ page_content=' Furthermore, note that the Ξ³WW vertex, as shown in Figure 1(b), is involved in the transition, thus both experimental and theoretical investigations of W β†’ V β„“Β―Ξ½β„“ decays may be also helpful to test triple gauge couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
162
+ page_content=' Our experimentalists have been trying to search for exclusive rare W-boson processes containing hadronic final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
163
+ page_content=' Unfortunately, so far no such decays have been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
164
+ page_content=' Theoretical predictions on branching fractions of W β†’ V β„“Β―Ξ½β„“ in the present paper are around 10βˆ’6 ∼ 10βˆ’7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
165
+ page_content=' Experimentally, the heavy quarkonium J/ψ is in general reconstructed via leptonic decays with their rates: B(J/Ξ¨ β†’ β„“+β„“βˆ’) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
166
+ page_content='971Β±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
167
+ page_content='032)% [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
168
+ page_content=' while for light vector mesons, ρ decays almost exclusively to Ο€+Ο€βˆ’, Ο‰ and Ο† have a large rate into Ο€+Ο€βˆ’Ο€βˆ’ and K+Kβˆ’, respectively, in the event construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
169
+ page_content=' Therefore, our analysis seems to indicate that 5 these exclusive rare W decay modes could be the promising candidates in future experimental machines, for instance, in the high-luminosity LHC, where large amount of W bosons about O(1011) events will be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
170
+ page_content=' We eagerly await some dedicated searches for such decays at these facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
171
+ page_content=' Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
172
+ page_content=' 11575175, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
173
+ page_content=' 12047502, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
174
+ page_content=' 12247103, and by National Research and Development Program of China under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
175
+ page_content=' 2020YFA0406400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
176
+ page_content=' Appendix: Explicit expression of IV After squaring the W βˆ’ β†’ V β„“βˆ’Β―Ξ½β„“ decay amplitude and summing/averaging spins of all par- ticles, we get the differential decay rate of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
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+ page_content=' (4) as dΞ“ dsV dsβ„“ = Ξ±2 emQ2 V g2f 2 V 384Ο€mWr2 V IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
178
+ page_content=' The full expression of IV ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
179
+ page_content=' in terms of dot product of the relevant four-momenta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
180
+ page_content=' can be given by IV = I1 + I2 + I3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
181
+ page_content=' (A1) where I1 = 8 (q2 + 2k1 Β· q)2 οΏ½ (2k1 Β· qk2 Β· q βˆ’ q2k1 Β· k2) + 2p Β· k2 m2 W (2k1 Β· qp Β· q βˆ’ q2k1 Β· p) οΏ½ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
182
+ page_content=' (A2) I2 = 8 (q2 + 2k1 Β· q)(q2 + 2k Β· q) οΏ½ 2k1 Β· k2(4k1 Β· k2 + 2k Β· q + 4q2) βˆ’ 4k1 Β· qk2 Β· q βˆ’ 1 m2 W [(2k1 Β· q + 4k Β· q + 3q2)(2k1 Β· qk2 Β· q βˆ’ q2k1 Β· k2) βˆ’4p Β· k2(q2 + 2k2 Β· q)k1 Β· k2 + 4p Β· q(q2 + 2k1 Β· q)k1 Β· k2] οΏ½ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
183
+ page_content=' (A3) and I3 = 8 (q2 + 2k Β· q)2 οΏ½ 12k1 Β· qk2 Β· q βˆ’ ((2k + q)2 + 10q2)k1 Β· k2 βˆ’ 1 2m2 W (2k + q)2(2k1 Β· qk2 Β· q βˆ’ q2k1 Β· k2) + 4(p Β· q)2 m2 W οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
184
+ page_content=' (A4) On the other hand, one can easily get k1 Β· k2 = m2 W 2 sV , p Β· q = m2 W 2 (1 + r2 V βˆ’ sV ), 6 k1 Β· q = m2 W 2 (1 βˆ’ sV βˆ’ sβ„“), k2 Β· q = m2 W 2 (sβ„“ βˆ’ r2 V ), k1 Β· p = m2 W 2 (1 βˆ’ sβ„“), k2 Β· p = m2 W 2 (sV + sβ„“ βˆ’ r2 V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
185
+ page_content=' For on-shell initial and final states particles, we could take p2 = m2 W, q2 = m2 V , and k2 1 = k2 2 = 0 (lepton masses are set to be zero already).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
186
+ page_content=' This shows that IV can be in terms of the kinematical variables sV and sβ„“ completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
187
+ page_content=' References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
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189
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190
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191
+ page_content=' Parsa, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
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195
+ page_content=' Mangano and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
196
+ page_content=' Melia, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
197
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198
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199
+ page_content=' C 75, 258 (2015), arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
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201
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202
+ page_content=' Grossman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
203
+ page_content=' K¨onig, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
204
+ page_content=' Neubert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
205
+ page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
206
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209
+ page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
210
+ page_content=' Keum and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
211
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212
+ page_content=' Pham, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
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215
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216
+ page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
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1
+ arXiv:2301.00743v1 [cs.SC] 2 Jan 2023
2
+ 1
3
+ Computing square roots in quaternion algebras
4
+ PrzemysΕ‚aw Koprowski
5
+ Institute of Mathematics
6
+ University of Silesia in Katowice
7
+ ul. Bankowa 14, 40-007 Katowice, Poland
8
9
+ Abstract. We present an explicit algorithmic method for computing square roots in quaternion
10
+ algebras over global fields of characteristic different from 2.
11
+ Keywords:
12
+ square root computation, quaternion algebra, number fields, global fields
13
+ 1.
14
+ Introduction
15
+ The computation of square roots is one of the most basic operations in mathematics. Effective methods
16
+ for computing square roots are among the oldest algorithms in the realm of computational mathemat-
17
+ ics. In fact, Heron’s method for a numerical approximation of a square root of a real number is two
18
+ thousand years old and preceded by the Euclidean algorithm (wildly believed to be the oldest mathe-
19
+ matical algorithm) by only about three to four centuries (for an in-depth discussion on the chronology
20
+ see [1]). Although numerous methods for computing square roots in various algebraic structures are
21
+ known nowadays, some important omissions prevail. Among them are general quaternion algebras.
22
+ Computation of square roots in the algebra of Hamilton quaternions H =
23
+ οΏ½ βˆ’1,βˆ’1
24
+ R
25
+ οΏ½
26
+ is well-known (see
27
+ [2]) and very simple as for every quaternion
28
+ ∈ H there is a subfield K ∼= C of H containing , and
29
+ so the computation the square root in H can be reduced to the computation of the square root in C. It
30
+ is no longer so in a general quaternion algebra Q =
31
+ οΏ½ Ξ±,Ξ²
32
+ K
33
+ οΏ½
34
+ for an arbitrary field K and two elements
35
+ Ξ±, Ξ² ∈ KΓ—. To the best of our knowledge, no algorithm for computing quaternionic square roots ex-
36
+ ists in the literature. One possible explanation for this (quite surprising) fact is that in the commutative
37
+ Address for correspondence: Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland
38
+
39
+ 2
40
+ P. Koprowski / Computing square roots in quaternion algebras
41
+ case when one considers a field extension L/K, a typical way to compute a square root of an element
42
+ a ∈ L is to factor the polynomial x2 βˆ’ a in L[x]. However, for quaternion algebras, there are no
43
+ known polynomial factorization algorithms.
44
+ The sole purpose of this paper is to correct this evident omission and present an explicit algorithm
45
+ for computing square roots in quaternion algebras over arbitrary global fields of characteristic different
46
+ from 2.
47
+ 2.
48
+ Notation
49
+ Throughout this paper, K will denote an arbitrary global field of characteristic char K ̸= 2. Hence, K
50
+ is either a number field, i.e. a finite extension of Q (then its characteristic is just 0) or a global function
51
+ field, that is, a finite extension of a rational function field over a finite field Fq, where q is a power of
52
+ an odd prime. The set of nonzero elements of K is denoted KΓ—.
53
+ Recall that a quaternion algebra Q =
54
+ οΏ½ Ξ±,Ξ²
55
+ K
56
+ οΏ½
57
+ over K is a 4-dimensional K-algebra with a basis
58
+ {1, i, j, k} and a multiplication gathered by the rules:
59
+ i2 = Ξ±,
60
+ j2 = Ξ²,
61
+ ij = k = βˆ’ji.
62
+ As usual, we shall identify the field K with the subfield K · 1 of Q, which is known to coincide with
63
+ the center Z(Q) of Q. We refer the reader to [3, 4] for a comprehensive presentation of the theory of
64
+ quaternion algebras.
65
+ A quaternion
66
+ is called pure (see e.g., [4, Definition 5.2.1]) if
67
+ ∈ spanK{i, j, k}. Every quater-
68
+ nion
69
+ ∈ Q can be uniquely expressed as a sum
70
+ = a +
71
+ 0 of a scalar a ∈ K and a pure quaternion
72
+ 0. We write
73
+ := a βˆ’
74
+ 0 for the conjugate of . The map that sends a quaternion to its conjugate is an
75
+ involution.
76
+ If x is an element of either a quadratic field extension L = K
77
+ �√α
78
+ οΏ½
79
+ of K or a quaternion algebra
80
+ Q =
81
+ οΏ½ Ξ±,Ξ²
82
+ K
83
+ οΏ½
84
+ over K, we write N (x) := xx and call it the norm of x. If the domain is not clear from
85
+ the context, we write NL/K or NQ/K.
86
+ Remark 2.1. When Q is a quaternion algebra, the norm of in the above sense should not be confused
87
+ with the determinant of the endomorphism of Q defined by the multiplication by , which is often also
88
+ called the norm. For this reason, in [3, 4] the map
89
+ οΏ½β†’
90
+ is called the reduced norm and denoted nrd.
91
+ In that manner, our terminology in the present paper agrees with the one used by Lam in [5] but not
92
+ with the one used by Vigneras in [3] and Voight in [4].
93
+ Equivalency classes of valuations on K are called places. Throughout this paper, places are de-
94
+ noted using fraktur letters p, q, r. Every place of a global field is either archimedean, when it extends
95
+ the standard absolute value on Q (then the field K is necessarily a number field) or non-archimedean.
96
+ Over a global function field, every place is non-archimedean. To avoid monotonous repetitions, non-
97
+ archimedean places will also be called primes (or finite primes when we want to emphasize the fact
98
+ that they are non-archimedean). The completion of K with respect to a place p is denoted Kp. If p is
99
+ a finite prime, we write ordp : K β†’ Z to denote the corresponding (normalized) discrete valuation
100
+
101
+ P. Koprowski / Computing square roots in quaternion algebras
102
+ 3
103
+ on K. The prime p is called dyadic if ordp 2 ΜΈ= 0. The map ordp induces a natural map KΓ—/KΓ—2 β†’ Z/2Z
104
+ on the group of square classes of K that is again denoted ordp.
105
+ If p is an archimedean place, then the completion Kp is isomorphic either to C or to R. The places
106
+ of the second kind are called real. We write sgnr a for the sign of a ∈ K with respect to for the unique
107
+ ordering of K induced by a real place r.
108
+ Given some nonzero elements a1, . . . , an ∈ K we denote by ⟨a1, . . . , an⟩ the quadratic form
109
+ a1x2
110
+ 1 + Β· Β· Β· + anx2
111
+ n. Further, if p is a place and a, b ∈ KΓ— we write (a, b)p for the Hilber symbol of a
112
+ and b at p, that is
113
+ (a, b)p :=
114
+ οΏ½
115
+ 1
116
+ if
117
+ οΏ½a,b
118
+ K
119
+ � ∼= M2Kp,
120
+ βˆ’1
121
+ otherwise.
122
+ For a quadratic form ξ = ⟨a1, . . . , an⟩ we define its Hasse invariant spξ at p by the formula (see e.g.,
123
+ [5, Definition V.3.17]):
124
+ spΞΎ :=
125
+ οΏ½
126
+ i<j
127
+ (ai, aj)p.
128
+ Finally, abusing the notation harmlessly, by log-1 we will denote the (unique) isomorphism from the
129
+ multiplicative group {Β±1} to the additive group {0, 1} with addition modulo 2.
130
+ 3.
131
+ Square roots of non-central elements
132
+ Let us begin by writing down the explicit formula for a square in quaternion algebra so that we can
133
+ easily reference it in the discussion that follows.
134
+ Observation 3.1. If
135
+ = q0 + q1i + q2j + q3k ∈ Q is a quaternion, then
136
+ 2 = (q2
137
+ 0 + q2
138
+ 1Ξ± + q2
139
+ 2Ξ² βˆ’ q2
140
+ 3Ξ±Ξ²) + 2q0q1i + 2q0q2j + 2q0q3k
141
+ = (2q2
142
+ 0 βˆ’ N( )) + 2q0 Β· (q1i + q2j + q3k).
143
+ (1)
144
+ An immediate consequence of the previous observation is the following rather well-known fact.
145
+ Corollary 3.2. If
146
+ ∈ Q is a pure quaternion, then
147
+ 2 ∈ Z(Q) = K.
148
+ Another direct consequence of Eq. (1) is the following observation that may be treated as a partial
149
+ converse of Corollary 3.2.
150
+ Observation 3.3. Let
151
+ ∈ Q be a square root of some element a ∈ K. Then
152
+ is either pure or
153
+ ∈ K.
154
+ Proof:
155
+ Let
156
+ = q0 + q1i + q2j + q3k. If
157
+ 2 = a ∈ K then by Eq. (1) we have
158
+ 2q0q1 = 2q0q2 = 2q0q3 = 0.
159
+ Therefore, if
160
+ is not pure, that is if q0 ΜΈ= 0, then q1 = q2 = q3 = 0, hence
161
+ ∈ K.
162
+ βŠ“βŠ”
163
+
164
+ 4
165
+ P. Koprowski / Computing square roots in quaternion algebras
166
+ Combining Corollary 3.2 with Observation 3.3 we see that for computing the square roots in
167
+ quaternion algebras it is crucial to distinguish between the case when one computes a quaternionic
168
+ square root of an element in K (i.e., in the center of Q) and the case when the argument comes from
169
+ Q \Z(Q). It turns out that the latter case is, in fact, trivial and requires nothing more than high-school
170
+ mathematics.
171
+ Algorithm 1. Let Q =
172
+ οΏ½ Ξ±,Ξ²
173
+ K
174
+ οΏ½
175
+ be a quaternion algebra over a field K of characteristic char K ̸= 2.
176
+ Given a quaternion
177
+ = q0 + q1i + q2j + q3k ∈ Q \ Z(Q), this algorithm outputs its square root or
178
+ reports a failure when
179
+ is not a square.
180
+ 1. Check if the norm N( ) of
181
+ is a square in K.
182
+ (a) If it is not, then report a failure and quit.
183
+ (b) If it is, let d be an element of K such that d2 = N( ).
184
+ 2. Check if any of the following two elements is a square in K:
185
+ a+ := q0 + d
186
+ 2
187
+ ,
188
+ aβˆ’ := q0 βˆ’ d
189
+ 2
190
+ .
191
+ 3. If neither of them is a square, then report a failure and quit.
192
+ 4. Otherwise, fix r0 such that either r2
193
+ 0 = a+ or r2
194
+ 0 = aβˆ’.
195
+ 5. Set
196
+ r1 := q1
197
+ 2r0
198
+ ,
199
+ r2 := q2
200
+ 2r0
201
+ ,
202
+ r3 := q3
203
+ 2r0
204
+ .
205
+ 6. Output r = r0 + r1i + r2j + r3k.
206
+ Proof of correctness:
207
+ Since the norm N : Q β†’ K is multiplicative, it is obvious that if N( ) /∈ K2, then
208
+ cannot be a
209
+ square in Q. This fact justifies the early exit in step (1a) of the algorithm. Assume that N( ) = d2 and
210
+ let r = r0 + r1i + r2j + r3k be the sought square root of , if it exists. By Eq. (1) we have
211
+ q1 = 2r0r1,
212
+ q2 = 2r0r2,
213
+ q3 = 2r0r3.
214
+ It is, thus, clear that it suffices to find r0. Again by Eq. (1) we may write
215
+ q0 = r2
216
+ 0 + r2
217
+ 1Ξ± + r2
218
+ 2Ξ² βˆ’ r2
219
+ 3Ξ±Ξ² = r2
220
+ 0 +
221
+ οΏ½ q1
222
+ 2r0
223
+ οΏ½2
224
+ Ξ± +
225
+ οΏ½ q2
226
+ 2r0
227
+ οΏ½2
228
+ Ξ² βˆ’
229
+ οΏ½ q3
230
+ 2r0
231
+ οΏ½2
232
+ Ξ±Ξ².
233
+ The above formula can be rewritten in the form of a bi-quadratic equation:
234
+ 4r4
235
+ 0 βˆ’ 4q0r2
236
+ 0 +
237
+ οΏ½
238
+ q2
239
+ 1Ξ± + q2
240
+ 2Ξ² βˆ’ q2
241
+ 3Ξ±Ξ²
242
+ οΏ½
243
+ = 0.
244
+
245
+ P. Koprowski / Computing square roots in quaternion algebras
246
+ 5
247
+ If we treat the left-hand-side as a quadratic equation in r2
248
+ 0, then its discriminant equals 16 Β· N( ) =
249
+ (4d)2, hence
250
+ r2
251
+ 0 = q0 Β± d
252
+ 2
253
+ = aΒ±.
254
+ It follows that the sought quaternion r exists if and only if either a+ or aβˆ’ is a square in K. This
255
+ proves the correctness of the algorithm.
256
+ βŠ“βŠ”
257
+ Remark 3.4. In the above proof, we constructed the square root r of a quaternion
258
+ ∈ Q \ Z(Q) by
259
+ solving a bi-quadratic equation. Such equations in general, may have four roots. Hence, one may
260
+ suspect that there are four distinct quaternions r such that r2 = . It is not the case. It is clear from
261
+ the above proof that
262
+ ∈ Q \ Z(Q) has only finitely moan square roots in Q. Now, if r2 = Q, then r
263
+ is a root of a quaternionic polynomial x2 βˆ’ . But [6, Theorem 5] asserts that a quadratic polynomial
264
+ over Q which has more than two zeros must have infinitely many of them. This way, we conclude
265
+ that
266
+ has just two square roots. Notice that for hamiltonian quaternions this fact has been observed
267
+ already 80 years ago by Niven in [2].
268
+ 4.
269
+ Square roots of central elements. Split case
270
+ It is evident from the preceding section that the only non-trivial case that must be considered is how
271
+ to compute a quaternionic square root of an element of the base field K, which is not a square in K.
272
+ In contrast to the previous case (cf. Remark 3.4), in general, an element a ∈ K = Z(Q) may have
273
+ infinitely many square roots in Q. Once again, for hamiltonian quaternions it has been observed
274
+ already by Niven.
275
+ First, we need, however, to introduce an auxiliary algorithm that is not specific to quaternions, as
276
+ it deals with an arbitrary quadratic form. Recall that a quadratic form is called isotropic (see e.g., [5,
277
+ Definition I.3.1]) if it represents zero non-trivially. It is well known (see, e.g., [5, Theorem I.3.4]) that
278
+ every isotropic form represents all elements of K.
279
+ Algorithm 2. Let ξ be an isotropic quadratic form of dimension n over a field K of characteristic
280
+ char K ̸= 2. Given an element a ∈ K and a vector V ∈ Kn such that ξ(V ) = 0, this algorithm
281
+ outputs a vector W ∈ Kn satisfying the condition ξ(W) = a.
282
+ 1. Find a vector U ∈ Kn such that U and V are linearly independent.
283
+ 2. Set b := ΞΎ(U) and c := 1/2 Β·
284
+ οΏ½
285
+ ΞΎ(U + V ) βˆ’ ΞΎ(U)
286
+ οΏ½
287
+ .
288
+ 3. Output
289
+ W := U + a βˆ’ b
290
+ 2c
291
+ Β· V.
292
+ Proof of correctness:
293
+
294
+ 6
295
+ P. Koprowski / Computing square roots in quaternion algebras
296
+ Just compute:
297
+ ΞΎ(W) = ΞΎ
298
+ οΏ½
299
+ U + a βˆ’ b
300
+ 2c
301
+ Β· V
302
+ οΏ½
303
+ = ΞΎ(U) + a βˆ’ b
304
+ 2c
305
+ Β·
306
+ οΏ½
307
+ ΞΎ(U + V ) βˆ’ ΞΎ(U) βˆ’ ΞΎ(V )
308
+ οΏ½
309
+ + (a βˆ’ b)2
310
+ 4c2
311
+ ΞΎ(V )
312
+ = b + a βˆ’ b
313
+ 2c
314
+ Β· 2c + 0 = a
315
+ βŠ“βŠ”
316
+ Recall that a quaternion algebra Q =
317
+ οΏ½ Ξ±,Ξ²
318
+ K
319
+ οΏ½
320
+ is said to split (see e.g., [4, Definition 5.4.5]) if Q
321
+ is isomorphic to the matrix ring M2K. It is well known (see e.g., [4, Theorem 5.4.4] or [5, Theo-
322
+ rem III.2.7]) that Q is split if and only if the quadratic form βŸ¨βˆ’Ξ±, βˆ’Ξ², αβ⟩ is isotropic. If it is the case,
323
+ the preceding algorithm combined with Eq. (1) lets us compute the quaternionic square root of any
324
+ element of the base field. In particular, when K is a global field, char K ̸= 2, then the computation
325
+ of the square root of a ∈ K in a split quaternion algebra boils down to solving a norm equation in
326
+ a quadratic extension of K. Algorithms for the latter task are well known. They can be found in
327
+ [7, 8, 9, 10, 11].
328
+ Algorithm 3. Let Q =
329
+ οΏ½Ξ±,Ξ²
330
+ K
331
+ οΏ½
332
+ be a split quaternion algebra over a global field K of characteristic
333
+ char K ̸= 2. Given a nonzero element a ∈ K, this algorithm outputs a pure quaternion
334
+ ∈ Q such
335
+ that
336
+ 2 = a.
337
+ 1. Check if Ξ± is a square in K. If there is c ∈ KΓ— such that c2 = Ξ±, then set V := (0, c, 1).
338
+ 2. Otherwise, if Ξ± /∈ KΓ—2, then:
339
+ (a) Construct a quadratic field extension L = K
340
+ �√α
341
+ οΏ½
342
+ of K.
343
+ (b) Solve the norm equation
344
+ NL/K (x) = βˆ’Ξ±
345
+ Ξ²
346
+ and denote the solution by λ = b + c√α.
347
+ (c) Set V := (1, b, c).
348
+ 3. Let ΞΎ := βŸ¨βˆ’Ξ±, βˆ’Ξ², αβ⟩ be the pure subform of the norm form of Q. Execute Algorithm 2 with
349
+ the input (βˆ’a, V, ΞΎ) to construct a vector W = (w1, w2, w3) such that ΞΎ(W) = βˆ’a.
350
+ 4. Output
351
+ = 0 + w1i + w2j + w3k.
352
+ Proof of correctness:
353
+ We claim that the vector V constructed either in step (1) or in step (2) of the algorithm is an isotropic
354
+ vector for ΞΎ. First, suppose that Ξ± is a square in K. Say Ξ± = c2 for some c ∈ KΓ—. Then
355
+ βˆ’Ξ± Β· 02 βˆ’ Ξ² Β· c2 + Ξ±Ξ² Β· 12 = 0.
356
+
357
+ P. Koprowski / Computing square roots in quaternion algebras
358
+ 7
359
+ Conversely, assume that Ξ± /∈ KΓ—2 and so L = K
360
+ �√α
361
+ οΏ½
362
+ is a proper extension of K. Let λ = b + c√α
363
+ be an element of L such that N (Ξ») = βˆ’Ξ±/Ξ². Then
364
+ βˆ’Ξ±
365
+ Ξ² = λλ = b2 βˆ’ Ξ±c2.
366
+ It follows that
367
+ βˆ’Ξ± Β· 12 βˆ’ Ξ² Β· b2 + Ξ±Ξ² Β· c2 = 0.
368
+ Hence, in both cases V is an isotropic vector of ΞΎ, as claimed. Consequently, executing Algorithm 2
369
+ in step (3) we obtain a vector W satisfying the condition ΞΎ(W) = βˆ’a. Now, by Eq. (1) the square of
370
+ the quaternion
371
+ outputted by the algorithm equals
372
+ 2 = βˆ’N( ) = βˆ’ΞΎ(W) = a.
373
+ Thus, to conclude the proof, we only need to show that the norm equation in step (2b) is solvable. But
374
+ this follows immediately from the fact that Q is split. Hence ΞΎ is isotropic. Indeed, if V = (v1, v2, v3)
375
+ is an isotropic vector of ΞΎ, then
376
+ βˆ’Ξ± Β· v2
377
+ 1 βˆ’ Ξ² Β· v2
378
+ 2 + Ξ±Ξ² Β· v2
379
+ 3 = 0.
380
+ Observe that v1 must be nonzero since otherwise, Ξ± would be a square. It follows that
381
+ βˆ’Ξ±
382
+ Ξ² =
383
+ οΏ½v2
384
+ v1
385
+ οΏ½2
386
+ βˆ’ Ξ±
387
+ οΏ½v3
388
+ v1
389
+ οΏ½2
390
+ = NL/K
391
+ οΏ½v2
392
+ v1
393
+ + v3
394
+ v1
395
+ √α
396
+ οΏ½
397
+ .
398
+ Therefore, the norm equation is solvable, as claimed.
399
+ βŠ“βŠ”
400
+ Remark 4.1. The construction of the isotropic vector V in steps (1–2) of Algorithm 3 is equivalent
401
+ to establishing an explicit isomorphism Q ∼= M2K. For details, see [5, Chapter III]. Of course, if
402
+ the quaternion algebra Q is fixed, the vector V should be computed only once and cached between
403
+ successive computations of square roots.
404
+ Remark 4.2. If the isomorphism Q ∼= M2K is a priori known explicitly, then the computation of the
405
+ quaternionic square root of any a ∈ KΓ— trivializes, as we have the identity
406
+ οΏ½
407
+ 0
408
+ a
409
+ 1
410
+ 0
411
+ οΏ½2
412
+ =
413
+ οΏ½
414
+ a
415
+ 0
416
+ 0
417
+ a
418
+ οΏ½
419
+ .
420
+ 5.
421
+ Square roots of central elements. Non-split case
422
+ Now the only case left to be dealt with is when a ∈ KΓ— but Q is not split. Here we have to solve
423
+ not one but two norm equations (see Algorithm 5 below). First, however, we need to introduce the
424
+ following auxiliary algorithm that constructs an element simultaneously represented by two binary
425
+
426
+ 8
427
+ P. Koprowski / Computing square roots in quaternion algebras
428
+ forms. Recall (see e.g., [5, Definition I.2.1]) that for a given quadratic form ξ of dimension d, we
429
+ denote the set of nonzero elements of K represented by ΞΎ by the symbol
430
+ DK(ΞΎ) :=
431
+ οΏ½
432
+ ξ(V ) | V ∈ Kd and ξ(V ) ̸= 0
433
+ οΏ½
434
+ .
435
+ Let P be any finite set of primes of K. Recall that an element a ∈ KΓ— is called P-singular if
436
+ ordp a ≑ 0 (mod 2) for all finite primes p /∈ P. The set of all P-singular elements forms a subgroup
437
+ of the group KΓ— containing KΓ—2. Thus, the notion of P-singularity generalizes naturally to the square
438
+ classes. Define the set
439
+ EP :=
440
+ οΏ½
441
+ aKΓ—2 | a is P-singular
442
+ οΏ½
443
+ of P-singular square classes. It is a subgroup of the group KΓ—/KΓ—2 of square classes of K, hence a
444
+ vector space over F2. It is known that the dimension of this vector space is finite. In fact it equals (see
445
+ e.g., [12, p. 607])
446
+ dimF2 EP = |P| + dimF2 CP/C2
447
+ P,
448
+ where CP is the P-class group of K. There is a number of know algorithms to construct a basis of
449
+ this vector space. For details see e.g., [13, 14, 15].
450
+ Algorithm 4. Let K be a global field of characteristic char K ̸= 2. Given two binary quadratic
451
+ forms ξ = ⟨x0, x1⟩ and ΢ = ⟨z0, z1⟩ over K with x0, x1, z0, z1 ̸= 0, this algorithm outputs a nonzero
452
+ element d ∈ KΓ— such that d ∈ DK(ΞΎ) ∩ DK(ΞΆ) or reports a failure if there is no such d.
453
+ 1. If βˆ’x0x1 is a square in K, then output z0 and quit.
454
+ 2. Likewise, if βˆ’z0z1 is a square in K, then output x0 and quit.
455
+ 3. Check (using e.g., [16, Algorithm 5]) whether the form
456
+ ΞΎ βŠ₯ (βˆ’ΞΆ) = ⟨x0, x1, βˆ’z0, βˆ’z1⟩
457
+ is isotropic. If it is not, then report a failure and quit.
458
+ 4. Construct a set P consisting of all dyadic places of K (if there are any) and of all these non-
459
+ dyadic primes of K where at least one of the elements x0, x1, z0, z1 has an odd valuation.
460
+ 5. If K is a formally real number field, then:
461
+ (a) Construct the set R of all the real places of K, where either ξ or ΢ is definite and denote
462
+ its cardinality by r, i.e.
463
+ R =
464
+ οΏ½
465
+ r | sgnr x0x1 = 1 or sgnr z0z1 = 1
466
+ οΏ½
467
+ ,
468
+ r = |R|.
469
+ (b) [Notation only] Let r1, . . . , rr be all the elements of R.
470
+ (c) Construct a vector W = (w1, . . . , wr) ∈ {0, 1}r setting
471
+ wi =
472
+ οΏ½
473
+ log-1 sgnri x0
474
+ if sgnri x0x1 = 1,
475
+ log-1 sgnri z0
476
+ if sgnri x0x1 = βˆ’1.
477
+
478
+ P. Koprowski / Computing square roots in quaternion algebras
479
+ 9
480
+ Otherwise, if the field K is non-real, set R := βˆ…, r = 0 and W := ().
481
+ 6. Repeat the following steps until the sought element d is found:
482
+ (a) [Notation only] Let p1, . . . , ps be all the elements of P.
483
+ (b) Construct a basis B = {Ξ²1, . . . , Ξ²k} of the group EP of P-singular square classes.
484
+ (c) Construct vectors U = (u1, . . . , us) and V = (v1, . . . , vs) setting
485
+ ui = log-1(x0, x1)pi
486
+ and
487
+ vi = log-1(z0, z1)pi.
488
+ (d) Construct matrices A = (aij) and B = (bij), with k = |B| columns and s = |P| rows,
489
+ setting
490
+ aij = log-1(βˆ’x0x1, Ξ²j)pi
491
+ and
492
+ bij = log-1(βˆ’z0z1, Ξ²j)pi.
493
+ (e) If R ΜΈ= βˆ… construct a matrix C = (cij) with k columns and r = |R| rows, setting
494
+ cij = log-1 sgnri Ξ²j.
495
+ Otherwise, when R = βˆ…, set C = ().
496
+ (f) Check if the following system of F2-linear equations has a solution
497
+ 
498
+ 
499
+ ο£­
500
+ A
501
+ B
502
+ C
503
+ ο£Ά
504
+ ο£·
505
+ ο£Έ Β·
506
+ 
507
+ 
508
+ 
509
+ ο£­
510
+ x1
511
+ ...
512
+ xk
513
+ ο£Ά
514
+ ο£·
515
+ ο£·
516
+ ο£Έ =
517
+ 
518
+ 
519
+ ο£­
520
+ U
521
+ V
522
+ W
523
+ ο£Ά
524
+ ο£·
525
+ ο£Έ
526
+ (Ϙ)
527
+ (g) If it does, denote the solution by (Ρ1, . . . , Ρk) ∈ {0, 1}k. Output d = βΡ1
528
+ 1 Β· Β· Β· Ξ²Ξ΅k
529
+ k and quit.
530
+ (h) If the system (Ϙ) has no solution, then append a new prime p to P (see Remark 5.1 below)
531
+ and reiterate the loop.
532
+ Proof of correctness:
533
+ First, suppose that βˆ’x0x1 is a square in K. This means that the form ΞΎ is isotropic (see, e.g., [5,
534
+ Theorem I.3.2]). Hence, by [5, Theorem I.3.4] it represents every element of K. In particular, it
535
+ represents z0. Since ΞΆ also represents z0 (trivially), step (1) of the algorithm outputs the correct result.
536
+ The same argument also applies to step (2), when it is the form ΞΆ that is isotropic. It is also clear that
537
+ the sets DK(ΞΎ) and DK(ΞΆ) of elements represented by ΞΎ and ΞΆ, intersect if and only if ΞΎ βŠ₯ (βˆ’ΞΆ) is
538
+ isotropic. This justifies the test in step (3). Therefore, without loss of generality, for the remainder of
539
+ the proof, we may assume that ΞΎ βŠ₯ (βˆ’ΞΆ) is isotropic while both forms ΞΎ and ΞΆ are anisotropic.
540
+ We will first show that the algorithm terminates. Let W = (w0, w1, w2, w3) ∈ K4 be an isotropic
541
+ vector of ΞΎ βŠ₯ (βˆ’ΞΆ). Denote e := ΞΎ(w0, w1) = ΞΆ(w2, w3). Further, let R and P be the sets of
542
+ places (real and non-archimedean, respectively) constructed in steps (4–5) of the algorithm. Now, [17,
543
+ Lemma 2.1] asserts that there exists a finite prime p0 of K and an element d ∈ KΓ— such that:
544
+ i. ordp d = 0 for every finite prime p /∈ P βˆͺ {p0};
545
+
546
+ 10
547
+ P. Koprowski / Computing square roots in quaternion algebras
548
+ ii. d ≑ e (mod p1+ordp 4) for every p ∈ P;
549
+ iii. ordp0 d = 1;
550
+ iv. sgnr d = sgnr e for every real places r of K.
551
+ Let B = {Ξ²1, . . . , Ξ²k} be a basis of the group EPβˆͺ{p0} of
552
+ οΏ½
553
+ P βˆͺ {p0}
554
+ οΏ½
555
+ -singular square classes.
556
+ The element d is
557
+ οΏ½
558
+ P βˆͺ {p0}
559
+ οΏ½
560
+ -singular, hence it can be expressed in the form
561
+ d = Ξ²Ξ΅1
562
+ 1 Β· Β· Β· Ξ²Ξ΅k
563
+ k ,
564
+ where Ρ1, . . . , Ρk ∈ F2 are the coordinates of d with respect to B.
565
+ Fix a real place ri ∈ R. First, suppose that sgnri x0x1 = 1, so the form ΞΎ βŠ— Kp is definite. Then
566
+ sgnri(βˆ’d) = sgnri(βˆ’e) = sgnri x0 since βŸ¨βˆ’e, x0, x1⟩ is isotropic. But this implies that
567
+ k
568
+ οΏ½
569
+ j=1
570
+ (βˆ’1)cijΞ΅j =
571
+ k
572
+ οΏ½
573
+ j=1
574
+ sgnri Ξ²Ξ΅j
575
+ j = sgnri d = sgnri x0 = (βˆ’1)wi.
576
+ Consequently
577
+ ci1Ξ΅1 + Β· Β· Β· + cikΞ΅k = wi.
578
+ (2)
579
+ Conversely, assume that ΞΎ βŠ— Kri is indefinete, hence ΞΆ βŠ— Kri must be definete. Applying the same
580
+ arguments to the form ΞΆ instead of ΞΎ, we show that Eq. (2) also holds in this case.
581
+ Now fix a finite prime pi ∈ P. Observe that by the local square theorem (see, e.g., [5, The-
582
+ orem VI.2.19]) condition (ii) implies that the local squares classes dKΓ—2
583
+ pi
584
+ and eKΓ—2
585
+ pi
586
+ coincide. It
587
+ follows that the form
588
+ βŸ¨βˆ’d, x0, x1⟩ βŠ— Kpi ∼= βŸ¨βˆ’e, x0, x1⟩ βŠ— Kpi
589
+ is isotropic. Now, [5, Proposition V.3.22] asserts that the Hasse invariant of βŸ¨βˆ’d, x0, x1⟩ βŠ— Kpi equals
590
+ spiβŸ¨βˆ’d, x0, x1⟩ = (βˆ’1, x0x1 Β· d)pi.
591
+ This can be rewritten as
592
+ (βˆ’x0x1, d)pi = (x0, x1)pi.
593
+ Substituting Ξ²Ξ΅1
594
+ 1 Β· Β· Β· Ξ²Ξ΅k
595
+ k for d we obtain
596
+ k
597
+ οΏ½
598
+ j=1
599
+ (βˆ’x0x1, Ξ²j)Ξ΅j
600
+ pi = (x0, x1)pi.
601
+ Now, (x0, x1)pi = (βˆ’1)ui and (βˆ’x0x1, Ξ²j)pi = (βˆ’1)aij, where ui, aij ∈ {0, 1} are the elements
602
+ constructed in steps (6c–6d). Therefore, the last condition can be expressed as a linear equation over
603
+ F2:
604
+ ai1Ξ΅1 + Β· Β· Β· + aikΞ΅k = ui.
605
+ (3)
606
+
607
+ P. Koprowski / Computing square roots in quaternion algebras
608
+ 11
609
+ Finally, we will show that the above equation also holds for the index i = 0, that is for the prime p0
610
+ appended to P. This fact follows from Hilbert reciprocity law (see, e.g., [5, Theorem VI.5.5]). We
611
+ already know that for every i ∈ {1, . . . , s} we have
612
+ (βˆ’x0x1, d)pi = (x0, x1)pi.
613
+ The same also holds for primes not in P. Indeed, if q /∈ P βˆͺ {p0} then q is non-dyadin and all three
614
+ elements x0, x1 and d have even valuations at q. Consequently, by [5, Corollary VI.2.5] one obtains
615
+ (βˆ’x0x1, d)q = (x0, x1)q = 1.
616
+ Now, by Hilbert reciprocity law, we can write
617
+ 1 =
618
+ οΏ½
619
+ p
620
+ (βˆ’x0x1, d)p Β·
621
+ οΏ½
622
+ p
623
+ (x0, x1)p
624
+ = (βˆ’x0x1, d)p0(x0, x1)p0 Β·
625
+ οΏ½
626
+ p∈P
627
+ οΏ½
628
+ (βˆ’x0x1, d)p(x0, x1)p
629
+ οΏ½
630
+ Β·
631
+ οΏ½
632
+ q/∈Pβˆͺ{p0}
633
+ οΏ½
634
+ (βˆ’x0x1, d)p(x0, x1)p
635
+ οΏ½
636
+ = (βˆ’x0x1, d)p0(x0, x1)p0.
637
+ Hence, in the same way as above, we show that Eq. (3) also holds for i = 0. Applying the same
638
+ arguments to the form ΞΆ, we obtain
639
+ bi1Ξ΅1 + Β· Β· Β· + bikΞ΅k = vi,
640
+ (4)
641
+ for all i ∈ {0, 1, . . . , s}.
642
+ All in all, we have proved that Eq. (Ϙ) has a solution in EPβˆͺ{p0}. Now, for every Pβ€² βŠ‡ P βˆͺ {p0}
643
+ we have EPβˆͺ{p0} βŠ† EPβ€², hence once the prime p0 is appended to P the algorithm terminates (see also
644
+ Remark 5.1 w below).
645
+ Now, when we have proved that the algorithm stops, we must show that it outputs a correct result.
646
+ To this end, we will show that the forms βŸ¨βˆ’d, x0, x1⟩ and βŸ¨βˆ’d, z0, z1⟩ are locally isotropic in every
647
+ completion of K. The assumptions are symmetric with respect to both forms, except in real places.
648
+ Hence it generally suffices to prove the isotropy of one of them.
649
+ Both forms are trivially isotropic in all complex completions of K (provided that there are any)
650
+ and in all real completions Kr for r /∈ R. Fix now a real place ri ∈ R. First, assume that the form
651
+ ⟨x0, x1⟩ βŠ— Kri is definite. From the preceding part we know that the element d = Ξ²Ξ΅1
652
+ 1 Β· Β· Β· Ξ²Ξ΅k
653
+ k , con-
654
+ structed by the algorithm, satisfies the condition sgnri d = sgnri x0. Therefore the form βŸ¨βˆ’d, x0, x1βŸ©βŠ—
655
+ Kri is isotropic. Now, the form ΞΎ βŠ₯ (βˆ’ΞΆ) is isotropic because otherwise, the execution of the algo-
656
+ rithm would have been interrupted already in step (3). Thus, either sgnri z0 = sgnri x0 = sgnri d or
657
+ sgnri z1 = sgnri x0 = sgnri d. In both cases, we have that the form βŸ¨βˆ’d, z0, z1⟩ βŠ— Kri is isotropic, as
658
+ well. Conversely, assume that ΞΎ βŠ— Kri is indefinite, and so it is ΞΆ βŠ— Kri that must be definite. Then,
659
+ βŸ¨βˆ’d, x0, x1βŸ©βŠ—Kri is trivially isotropic and to the form βŸ¨βˆ’d, z0, z1βŸ©βŠ—Kri we apply the some argument
660
+ as to the form βŸ¨βˆ’d, x0, x1⟩ βŠ— Kri in the previous case.
661
+ We may now concentrate on finite primes. Fix a prime p. Suppose p is not among the primes
662
+ constituting P (here, we allow P to have been already enlarged during the execution of the algorithm).
663
+
664
+ 12
665
+ P. Koprowski / Computing square roots in quaternion algebras
666
+ In that case, p is certainly non-dyadic, and all three elements x0, x1, and d have even valuations at p.
667
+ Hence, [5, Corollary VI.2.5] asserts that βŸ¨βˆ’d, x0, x1⟩ βŠ— Kp is isotropic. On the other hand, we know
668
+ from the first part of the proof that if p = pi ∈ P, then d satisfies the condition (βˆ’x0x1, d)p =
669
+ (x0, x1)p, which is equivalent to spβŸ¨βˆ’d, x0, x1⟩ = (βˆ’1, x0x1 Β· d)p. The later condition implies that
670
+ βŸ¨βˆ’d, x0, x1⟩ βŠ— Kp is isotropic, again by [5, Proposition V.3.22]. The very same arguments may be
671
+ applied to the form βŸ¨βˆ’d, z0, z1⟩ βŠ— Kp.
672
+ All in all, we have shown that the forms βŸ¨βˆ’d, x0, x1⟩ and βŸ¨βˆ’d, z0, z1⟩ are locally isotropic in every
673
+ completion of K. Thus, they are isotropic over K by the Hasse–Minkowski principle (see e.g., [5,
674
+ Theorem VI.3.1]). This means that the forms ΞΎ and ΞΆ represent d over K by [5, Corollary I.3.5].
675
+ βŠ“βŠ”
676
+ Remark 5.1. To rigorously prove that Algorithm 4 terminates, we must show that the new primes
677
+ may be appended to the set P in such a way that after finitely many iterations the set will contain the
678
+ prime p0 specified in the proof of correctness. If K is a number field, let p1, p2, p3, . . . = 2, 3, 5, . . .
679
+ be the (strictly increasing) sequence of all prime numbers. On the other hand, if K is a global function
680
+ field, i.e., a finite extension of a rational function field Fq(x), let p1, p2, p3, . . . be a sequence of all
681
+ the irreducible polynomials from Fq[x] ordered in such a way that deg pj ≀ deg pj+1 for every j.
682
+ Now, appending new primes to P, we can first use the places of K that extend p1, then the ones that
683
+ extend p2, then p3, and so on. It is clear that this is an exhaustive method, so eventually, we will
684
+ append p0 and consequently terminate the algorithm. This proves that the algorithm stops but does
685
+ not present a complete picture. The existence of the prime p0 follows from [17, Lemma 2.1], which in
686
+ turn uses Chebotarev’s density theorem. In particular, if P0 denotes the set of primes of K such that
687
+ appending any of them makes the algorithm stop, then the set P0 has positive density. This means that
688
+ in practice, one can just append primes to P at random with exponentially diminishing probability
689
+ that the system (Ϙ) fails to be solvable.
690
+ We are now in a position to present an algorithm that computes a square root of a scalar in a
691
+ non-split quaternion algebra.
692
+ Algorithm 5. Let Q =
693
+ οΏ½ Ξ±,Ξ²
694
+ K
695
+ οΏ½
696
+ be a non-split quaternion algebra over a global field of characteristic
697
+ char K ̸= 2. Given a nonzero element a ∈ K this algorithm outputs a quaternion
698
+ ∈ Q such that
699
+ 2 = a or reports a failure if a is not a square in Q.
700
+ 1. Check if a is a square in K. If there is c ∈ KΓ— such that a = c2, then output
701
+ = c+0i+0j+0k
702
+ and quit.
703
+ 2. Check if aΞ± is a square in K. If there is c ∈ KΓ— such that aΞ± = c2, then output
704
+ = 0+(c/Ξ±)i+
705
+ 0j + 0k and quit.
706
+ 3. Check if aΞ² is a square in K. If there is c ∈ KΓ— such that aΞ² = c2, then output
707
+ = 0 + 0i +
708
+ (c/Ξ²)j + 0k and quit.
709
+ 4. Execute Algorithm 4 with input ΞΎ = ⟨a, βˆ’Ξ±βŸ© and ΞΆ = ⟨β, βˆ’Ξ±Ξ²βŸ©. If it fails, then report a failure
710
+ and quit. Otherwise, let d ∈ KΓ— denote the outputted element represented by these two binary
711
+ forms.
712
+
713
+ P. Koprowski / Computing square roots in quaternion algebras
714
+ 13
715
+ 5. Construct two quadratic extensions of K:
716
+ L := K
717
+ �√α
718
+ οΏ½
719
+ and
720
+ M := K
721
+ �√aα
722
+ οΏ½
723
+ .
724
+ 6. Solve the following two norm equations:
725
+ d
726
+ Ξ² = NL/K (x)
727
+ and
728
+ d
729
+ a = NM/K (y) .
730
+ Denote the solutions by
731
+ Ξ» = l0 + l1
732
+ √α
733
+ and
734
+ Β΅ = m0 + m1
735
+ √aα,
736
+ respectively.
737
+ 7. Output
738
+ = 0 + a Β· m1
739
+ m0 i + l0
740
+ m0 j + l1
741
+ m0 k.
742
+ Proof of correctness:
743
+ The correctness of the results outputted in step (1) is obvious as is the correctness of output of steps
744
+ (2–3). Indeed, if aΞ± = c2 for some c ∈ KΓ— and
745
+ = (c/Ξ±)i, then
746
+ 2 = Ξ± Β· c2/Ξ±2 = a. In the remainder
747
+ of the proof, we can, thus, assume that neither a nor aΞ± is a square in K. Likewise, Ξ± is not a square,
748
+ either, since otherwise, the quaternion algebra Q would split. Therefore, L and M are proper quadratic
749
+ extensions of K. It follows from Eq. (1) that a is a square of some pure quaternion
750
+ = q1i+q2j+q3k
751
+ if and only if
752
+ a Β· 12 βˆ’ Ξ± Β· q2
753
+ 1 = Ξ² Β· q2
754
+ 2 βˆ’ Ξ±Ξ² Β· q2
755
+ 3.
756
+ This equality is equivalent to the condition that the sets of elements of K represented by the binary
757
+ forms ΞΎ = ⟨a, βˆ’Ξ±βŸ© and ΞΆ = ⟨β, βˆ’Ξ±Ξ²βŸ© have a non-empty intersection. This proves the correctness of
758
+ step (4). Now, assume that Algorithm 4 returned some element d ∈ DK(ξ) ∩ DK(΢). Then there are
759
+ l0, l1, m0, m1 ∈ K such that
760
+ οΏ½
761
+ d = am2
762
+ 0 βˆ’ Ξ±(am1)2 = a Β· NM/K (m0 + m1
763
+ √aα)
764
+ d = Ξ²l2
765
+ 0 βˆ’ Ξ±Ξ²l2
766
+ 1 = Ξ² Β· NL/K (l0 + l1
767
+ √α) .
768
+ Rearranging the terms we have
769
+ a = Ξ±
770
+ οΏ½am1
771
+ m0
772
+ οΏ½2
773
+ + Ξ²
774
+ οΏ½ l0
775
+ m0
776
+ οΏ½2
777
+ βˆ’ Ξ±Ξ²
778
+ οΏ½ l1
779
+ m0
780
+ οΏ½2
781
+ .
782
+ Now, the right-hand-side is nothing else but the square of the quaternion
783
+ constructed in step (7). This
784
+ proves that the algorithm is correct.
785
+ βŠ“βŠ”
786
+
787
+ 14
788
+ P. Koprowski / Computing square roots in quaternion algebras
789
+ References
790
+ [1] Heath T. A history of Greek mathematics. Vol. I. Dover Publications, Inc., New York, 1981. ISBN
791
+ 0-486-24073-8. From Thales to Euclid, Corrected reprint of the 1921 original.
792
+ [2] Niven I. The roots of a quaternion. Amer. Math. Monthly, 1942. 49:386–388. doi:10.2307/2303134. URL
793
+ https://doi.org/10.2307/2303134.
794
+ [3] VignΒ΄eras MF. ArithmΒ΄etique des alg`ebres de quaternions, volume 800 of Lecture Notes in Mathematics.
795
+ Springer, Berlin, 1980. ISBN 3-540-09983-2.
796
+ [4] Voight J.
797
+ Quaternion algebras, volume 288 of Graduate Texts in Mathematics.
798
+ Springer, Cham,
799
+ [2021] Β©2021. ISBN 978-3-030-56692-0; 978-3-030-56694-4. doi:10.1007/978-3-030-56694-4. URL
800
+ https://doi.org/10.1007/978-3-030-56694-4.
801
+ [5] Lam TY. Introduction to quadratic forms over fields, volume 67 of Graduate Studies in Mathematics.
802
+ American Mathematical Society, Providence, RI, 2005. ISBN 0-8218-1095-2.
803
+ [6] Gordon B, Motzkin TS. On the zeros of polynomials over division rings. Trans. Amer. Math. Soc., 1965.
804
+ 116:218–226. doi:10.2307/1994114. URL https://doi.org/10.2307/1994114.
805
+ [7] Cohen
806
+ H.
807
+ Advanced
808
+ topics
809
+ in
810
+ computational
811
+ number
812
+ theory,
813
+ volume
814
+ 193
815
+ of
816
+ Graduate
817
+ Texts in Mathematics.
818
+ Springer-Verlag,
819
+ New York,
820
+ 2000.
821
+ ISBN 0-387-98727-4.
822
+ doi:
823
+ 10.1007/978-1-4419-8489-0.
824
+ https://doi.org/10.1007/978-1-4419-8489-0,
825
+ URL
826
+ https://doi.org/10.1007/978-1-4419-8489-0.
827
+ [8] Fieker
828
+ C,
829
+ Jurk
830
+ A,
831
+ Pohst
832
+ M.
833
+ On
834
+ solving
835
+ relative
836
+ norm
837
+ equations
838
+ in
839
+ alge-
840
+ braic
841
+ number
842
+ fields.
843
+ Math.
844
+ Comp.,
845
+ 1997.
846
+ 66(217):399–410.
847
+ doi:10.1090/
848
+ S0025-5718-97-00761-8.
849
+ https://doi.org/10.1090/S0025-5718-97-00761-8,
850
+ URL
851
+ https://doi.org/10.1090/S0025-5718-97-00761-8.
852
+ [9] Fincke U, Pohst M. A procedure for determining algebraic integers of given norm. In: Computer al-
853
+ gebra (London, 1983), volume 162 of Lecture Notes in Comput. Sci., pp. 194–202. Springer, Berlin,
854
+ 1983.
855
+ doi:10.1007/3-540-12868-9\ 103.
856
+ https://doi.org/10.1007/3-540-12868-9_103, URL
857
+ https://doi.org/10.1007/3-540-12868-9_103.
858
+ [10] Garbanati
859
+ DA.
860
+ An
861
+ algorithm
862
+ for
863
+ finding
864
+ an
865
+ algebraic
866
+ number
867
+ whose
868
+ norm
869
+ is
870
+ a
871
+ given
872
+ rational
873
+ number.
874
+ J.
875
+ Reine
876
+ Angew.
877
+ Math.,
878
+ 1980.
879
+ 316:1–13.
880
+ doi:
881
+ 10.1515/crll.1980.316.1.
882
+ https://doi.org/10.1515/crll.1980.316.1,
883
+ URL
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+ https://doi.org/10.1515/crll.1980.316.1.
885
+ [11] Simon
886
+ D.
887
+ Solving
888
+ norm
889
+ equations
890
+ in
891
+ relative
892
+ number
893
+ fields
894
+ us-
895
+ ing
896
+ S-units.
897
+ Math.
898
+ Comp.,
899
+ 2002.
900
+ 71(239):1287–1305.
901
+ doi:10.1090/
902
+ S0025-5718-02-01309-1.
903
+ https://doi.org/10.1090/S0025-5718-02-01309-1,
904
+ URL
905
+ https://doi.org/10.1090/S0025-5718-02-01309-1.
906
+ [12] CzogaΕ‚a A.
907
+ Witt rings of Hasse domains of global fields.
908
+ J. Algebra, 2001. 244(2):604–630. doi:
909
+ 10.1006/jabr.2001.8918. URL https://doi.org/10.1006/jabr.2001.8918.
910
+ [13] Cannon J, Bosma W, Fieker C, (eds) AS. Handbook of Magma Functions, 2.26-4 edition, 2021.
911
+ [14] Koprowski P.
912
+ Computing singular elements modulo squares.
913
+ Fund. Inform., 2021.
914
+ 179(3):227–
915
+ 238.
916
+ doi:10.3233/fi-2021-2022.
917
+ https://doi.org/10.3233/fi-2021-2022,
918
+ URL
919
+ https://doi.org/10.3233/fi-2021-2022.
920
+
921
+ P. Koprowski / Computing square roots in quaternion algebras
922
+ 15
923
+ [15] Koprowski
924
+ P,
925
+ Rothkegel
926
+ B.
927
+ The
928
+ anisotropic
929
+ part
930
+ of
931
+ a
932
+ quadratic
933
+ form
934
+ over
935
+ a
936
+ number
937
+ field.
938
+ J.
939
+ Symbolic Comput.,
940
+ 2023.
941
+ 115:39–52.
942
+ doi:10.1016/j.jsc.2022.07.003.
943
+ URL
944
+ https://doi.org/10.1016/j.jsc.2022.07.003.
945
+ [16] Koprowski P, CzogaΕ‚a A. Computing with quadratic forms over number fields. J. Symbolic Comput., 2018.
946
+ 89:129–145. doi:10.1016/j.jsc.2017.11.009. URL https://doi.org/10.1016/j.jsc.2017.11.009.
947
+ [17] Leep D, Wadsworth A. The Hasse norm theorem mod squares. J. Number Theory, 1992. 42(3):337–348.
948
+ doi:10.1016/0022-314X(92)90098-A. URL https://doi.org/10.1016/0022-314X(92)90098-A.
949
+
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
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:2301.03299v1 [math.NA] 9 Jan 2023
2
+ ASYMPTOTIC ERROR ANALYSIS FOR THE DISCRETE ITERATED
3
+ GALERKIN SOLUTION OF URYSOHN INTEGRAL EQUATIONS WITH
4
+ GREEN’S KERNELS
5
+ GOBINDA RAKSHIT
6
+ ABSTRACT. Consider a Urysohn integral equation x βˆ’ K(x) = f, where f and the integral
7
+ operator K with kernel of the type of Green’s function are given. In the computation of
8
+ approximate solutions of the given integral equation by Galerkin method, all the integrals
9
+ are needed to be evaluated by some numerical integration formula. This gives rise to the
10
+ discrete version of the Galerkin method. For r β‰₯ 1, a space of piecewise polynomials
11
+ of degree ≀ r βˆ’ 1 with respect to a uniform partition is chosen to be the approximating
12
+ space. For the appropriate choice of a numerical integration formula, an asymptotic series
13
+ expansion of the discrete iterated Galerkin solution is obtained at the above partition points.
14
+ Richardson extrapolation is used to improve the order of convergence. Using this method
15
+ we can restore the rate of convergence when the error is measured in the continuous case. A
16
+ numerical example is given to illustrate this theory.
17
+ 1 Introduction
18
+ Let X = L∞[0, 1]. Consider the problem of solving Urysohn integral equation
19
+ x(s) βˆ’
20
+ οΏ½ 1
21
+ 0
22
+ κ(s, t, x(t)) dt = f(s), s ∈ [0, 1],
23
+ (1.1)
24
+ where f ∈ X and ΞΊ ∈ C ([0, 1] Γ— [0, 1] Γ— R) are given. Let the Urysohn integral operator
25
+ K : L∞[0, 1] β†’ C[0, 1] be defined by
26
+ (1.2)
27
+ K(x)(s) =
28
+ οΏ½ 1
29
+ 0
30
+ ΞΊ(s, t, x(t)) dt,
31
+ x ∈ X , s ∈ [0, 1].
32
+ Since the kernel ΞΊ is continuous, K is compact operator on X . Denoting the equation (1.1)
33
+ by
34
+ (1.3)
35
+ x βˆ’ Kx = f.
36
+ We assume that the above equation has a solution, say Ο•. We also assume that K is twice
37
+ FrechΓ©t differentiable and 1 is not an eigenvalue of the compact linear operator Kβ€²(Ο•). This
38
+ gives us that Ο• is an isolated solution of (1.3). See [12], [14]. We are looking for Galerkin
39
+ approximations of Ο•.
40
+ For r β‰₯ 1, consider the approximating space Xn as a space of piecewise polynomials of
41
+ degree ≀ r βˆ’ 1 with respect to a uniform partition, say βˆ†(n), of [0, 1] with n subintervals
42
+ Date: January 10, 2023.
43
+ 2020 Mathematics Subject Classification. 45G10, 65B05, 65J15, 65R20.
44
+ Key words and phrases. Urysohn integral operator, Green’s kernel, Galerkin method, NystrΓΆm approxima-
45
+ tion, Richardson extrapolation.
46
+ 1
47
+
48
+ 2
49
+ G. RAKSHIT
50
+ each of length h = 1
51
+ n. Let Ο€n be the restriction to L∞[0, 1] of the orthogonal projection from
52
+ L2[0, 1] to Xn. Then the Galerkin solution Ο•G
53
+ n satisfies the following integral equation
54
+ Ο•G
55
+ n βˆ’ Ο€nK(Ο•G
56
+ n ) = Ο€nf.
57
+ Galerkin method for Urysohn integral equation has been studied extensively in research
58
+ literature. See [5], [12], [13], [14]. The iterated Galerkin solution is defined by
59
+ Ο•S
60
+ n = K(Ο•G
61
+ n ) + f.
62
+ In [5], the following orders of convergence are also obtained.
63
+ βˆ₯Ο•G
64
+ n βˆ’ Ο•βˆ₯∞ = O (h) ,
65
+ βˆ₯Ο•S
66
+ n βˆ’ Ο•βˆ₯∞ = O
67
+ οΏ½
68
+ h2οΏ½
69
+ ,
70
+ if r = 1,
71
+ and
72
+ βˆ₯Ο•G
73
+ n βˆ’ Ο•βˆ₯∞ = O (hr) ,
74
+ βˆ₯Ο•S
75
+ n βˆ’ Ο•βˆ₯∞ = O
76
+ οΏ½
77
+ hr+2οΏ½
78
+ ,
79
+ if r β‰₯ 2.
80
+ It is also shown that the order of convergence of Ο•S
81
+ n at the points of partition βˆ†(n), is h2r.
82
+ If an asymptotic expansion for the error exists, one can apply a well-known techniques
83
+ to obtain more accurate approximations. Richardson extrapolation one such method for
84
+ application. In [24], an asymptotic expansion for the iterated Galerkin solution of Urysohn
85
+ integral equation with Green’s function type of kernel, is obtained at the above mentioned
86
+ partition points. Then, by [11] and using Richardson extrapolation, an approximate solution
87
+ with order of convergence h2r+2 can be obtained.
88
+ In the computation of of above approximations, various integrals are involved. There is
89
+ an integral in the definition of the Urysohn integral operator K. In the definition of the
90
+ orthogonal projection Ο€n, the standard inner product on L2[0, 1] comes into picture. In
91
+ practice, it is necessary to replace all these integrals by a numerical quadrature formula.
92
+ This gives rise to the discrete versions of the projection methods. The discrete versions of
93
+ the Galerkin methods for Urysohn integral with Green’s kernel, are investigated in [6], [4].
94
+ Whereas, in [17], a different version of discrete projection method is discussed.
95
+ In this article, we consider the Urysohn integral equation with Green’s kernel, and discrete
96
+ Galerkin method is applied for approximations. Then, an asymptotic expansion for the
97
+ discrete iterated Galerkin solution is obtained.
98
+ We choose a fine partition of [0, 1] with m subintervals each of length ˜h = 1
99
+ m and define a
100
+ composite numerical quadrature formula. Replacing the integrals in the definition of K and
101
+ Ο€n, we define the NystrΓΆm operator Km and the discrete orthogonal projection Pn. Then the
102
+ discrete Galerkin and the discrete iterated Galerkin equations are given by
103
+ zG
104
+ n βˆ’ PnKm(zG
105
+ n ) = Pnf and zS
106
+ n βˆ’ Km(PnzS
107
+ n) = f
108
+ respectively. If Ο• ∈ Cr+2[0, 1], then from [6] and [17], we have
109
+ (1.4)
110
+ οΏ½οΏ½zG
111
+ n βˆ’ Ο•
112
+ οΏ½οΏ½
113
+ ∞ = O
114
+ οΏ½
115
+ max
116
+ οΏ½
117
+ hr, ˜h2��
118
+ (1.5)
119
+ οΏ½οΏ½zS
120
+ n βˆ’ Ο•
121
+ οΏ½οΏ½
122
+ ∞ =
123
+ ο£±
124
+ ο£²
125
+ ο£³
126
+ O
127
+ οΏ½
128
+ max
129
+ οΏ½
130
+ h2, ˜h2��
131
+ ,
132
+ r = 1,
133
+ O
134
+ οΏ½
135
+ max
136
+ οΏ½
137
+ hr+2, ˜h2��
138
+ , r β‰₯ 2.
139
+
140
+ Section 2. Preliminaries
141
+ 3
142
+ In this article, first we find an asymptotic error expansion due to the discrete orthogonal
143
+ projection. Then using this, the following asymptotic expansion is obtained:
144
+ (1.6)
145
+ zS
146
+ n(ti) = Ο•(ti) + Ξ³(ti)h2r + O
147
+ οΏ½
148
+ max
149
+ οΏ½
150
+ h2r+2, ˜h2��
151
+ ,
152
+ where the function Ξ³ is independent of h. If we choose m such that ˜h ≀ h2r+2, then using
153
+ the Richardson extrapolation, an approximation of Ο• of the order of h2r+2 could be obtained.
154
+ See [11].
155
+ This article is organized as follows. Definitions, notations and some preliminary results
156
+ are given in section 2. In Section 3, a quadrature rule is defined, and using it the discrete
157
+ orthogonal projection and the Nystrâm approximations of the integral operators are defined.
158
+ Section 4 contains the asymptotic error analysis for the approximations. Numerical example
159
+ is given in Section 5.
160
+ 2 Preliminaries
161
+ For an integer Ξ± β‰₯ 0, let CΞ±[0, 1] denotes the space of all real valued Ξ±-times continuously
162
+ differentiable functions on [0, 1] with the norm
163
+ βˆ₯xβˆ₯Ξ±,∞ = max
164
+ 0≀j≀α
165
+ οΏ½οΏ½x(j)οΏ½οΏ½
166
+ ∞ ,
167
+ where x(j) is the jth derivative of the function x, and
168
+ οΏ½οΏ½οΏ½x(j)οΏ½οΏ½οΏ½
169
+ ∞ = sup
170
+ 0≀t≀1
171
+ |x(j)(t)|. Define
172
+ βˆ₯ΞΊβˆ₯Ξ±,∞ =
173
+ max
174
+ 0≀i+j+k≀α
175
+ οΏ½οΏ½οΏ½D(i,j,k)ΞΊ(s, t, u)
176
+ οΏ½οΏ½οΏ½
177
+ ∞ ,
178
+ where
179
+ D(i,j,k)ΞΊ(s, t, u) =
180
+ βˆ‚i+j+kΞΊ
181
+ βˆ‚siβˆ‚tjβˆ‚uk (s, t, u).
182
+ 2.0 Green’s function type kernel
183
+ Let r β‰₯ 1 be an integer and assume that the kernel ΞΊ has the following properties.
184
+ (1) For i = 1, 2, 3, 4, the functions ΞΊ, βˆ‚iΞΊ
185
+ βˆ‚ui ∈ C(Ω), where C(Ω) denotes the space of all
186
+ real valued continuous function on Ω = [0, 1] Γ— [0, 1] Γ— R.
187
+ (2) Let Ω1 = {(s, t, u) : 0 ≀ t ≀ s ≀ 1, u ∈ R} and Ω2 = {(s, t, u) : 0 ≀ s ≀ t ≀
188
+ 1, u ∈ R}. There are two functions ΞΊj ∈ Cr(Ωj), j = 1, 2, such that
189
+ ΞΊ(s, t, u) =
190
+ οΏ½
191
+ ΞΊ1(s, t, u),
192
+ (s, t, u) ∈ Ω1,
193
+ ΞΊ2(s, t, u),
194
+ (s, t, u) ∈ Ω2.
195
+ (3) Denote β„“(s, t, u) = βˆ‚ΞΊ
196
+ βˆ‚u(s, t, u) and Ξ»(s, t, u) = βˆ‚2ΞΊ
197
+ βˆ‚u2(s, t, u), (s, t, u) ∈ Ω. The partial
198
+ derivatives of β„“(s, t, u) and Ξ»(s, t, u) with respect to s and t have jump discontinu-
199
+ ities on s = t.
200
+ (4) There are functions β„“j, Ξ»j ∈ Cr(Ωj), j = 1, 2, with
201
+ β„“(s, t, u) =
202
+ οΏ½
203
+ β„“1(s, t, u),
204
+ (s, t, u) ∈ Ω1,
205
+ β„“2(s, t, u),
206
+ (s, t, u) ∈ Ω2,
207
+ Ξ»(s, t, u) =
208
+ οΏ½
209
+ Ξ»1(s, t, u),
210
+ (s, t, u) ∈ Ω1,
211
+ Ξ»2(s, t, u),
212
+ (s, t, u) ∈ Ω2.
213
+
214
+ 4
215
+ G. RAKSHIT
216
+ Under the above assumptions, the operator K is four times FrΓ©chet differentiable, and its
217
+ Fréchet derivatives at x ∈ X are given by
218
+ Kβ€²(x)v1(s) =
219
+ οΏ½ 1
220
+ 0
221
+ βˆ‚ΞΊ
222
+ βˆ‚u (s, t, x(t)) v1(t) dt,
223
+ K(i)(x)(v1, . . . , vi)(s) =
224
+ οΏ½ 1
225
+ 0
226
+ βˆ‚iΞΊ
227
+ βˆ‚ui (s, t, x(t)) v1(t) Β· Β· Β·vi(t) dt,
228
+ i = 2, 3, 4,
229
+ where
230
+ βˆ‚iΞΊ
231
+ βˆ‚ui (s, t, x(t)) = βˆ‚iΞΊ
232
+ βˆ‚ui (s, t, u)|u=x(t),
233
+ i = 1, 2, 3, 4
234
+ and v1, v2, v3, v4 ∈ X . Note that Kβ€²(x) : X β†’ X is linear and K(i)(x) : X i β†’ X are multi-
235
+ linear operators, where X i is the cartesian product of i copies of X . See [25]. The norms of
236
+ these operators are defined by
237
+ οΏ½οΏ½K(i)(x)
238
+ οΏ½οΏ½ = sup
239
+ οΏ½οΏ½οΏ½K(i)(x)(v1, . . . , vi)
240
+ οΏ½οΏ½
241
+ ∞ : βˆ₯vjβˆ₯∞ ≀ 1, j = 1, . . . , i
242
+ οΏ½
243
+ for i = 1, 2, 3, 4. It follows that
244
+ οΏ½οΏ½K(i)(x)
245
+ οΏ½οΏ½
246
+ ≀
247
+ sup
248
+ 0≀s,t≀1
249
+ οΏ½οΏ½οΏ½οΏ½
250
+ βˆ‚iΞΊ
251
+ βˆ‚ui (s, t, x(t))
252
+ οΏ½οΏ½οΏ½οΏ½ ,
253
+ i = 1, 2, 3, 4.
254
+ Note that, if f ∈ CΞ±[0, 1] for any positive integer Ξ±, then Ο• ∈ CΞ±[0, 1]. See [5, Corollary
255
+ 3.2], [6, Corollary 4.2].
256
+ 3 Discretization of Integrals by numerical quadrature
257
+ Rule
258
+ In this section, first we consider a numerical integration formula. We replace the integral
259
+ in the standard inner product of L2[0, 1] (i.e. ⟨x , y⟩ =
260
+ οΏ½ 1
261
+ 0 x(t)y(t)dt) by the quadrature rule
262
+ and define a discrete inner product. Subsequently, the corresponding discrete orthogonal
263
+ projection is defined. After that, an asymptotic error expansion for the discrete orthogonal
264
+ projection is obtained. Next we define the Nystrâm approximations of the integral operator
265
+ K and its FrΓ©chet derivatives.
266
+ Consider a basic numerical integration formula by
267
+ (3.1)
268
+ οΏ½ 1
269
+ 0
270
+ x(t)dt β‰ˆ
271
+ ρ
272
+ οΏ½
273
+ q=1
274
+ wq x(Β΅q),
275
+ which is exact at least for polynomials of degree ≀ 3r. If r = 0, then it is assumed that the
276
+ quadrature rule is exact atleast for linear polynomials. It follows that �ρ
277
+ q=1 wq = 1.
278
+ Let n ∈ N and consider the following uniform partition of [0, 1] :
279
+ (3.2)
280
+ βˆ†(n) :
281
+ 0 < 1
282
+ n < Β· Β· Β· < n βˆ’ 1
283
+ n
284
+ < 1.
285
+ Define tj = j
286
+ n, βˆ†j = [tjβˆ’1, tj] and h = tj βˆ’ tjβˆ’1 = 1
287
+ n, j = 1, . . . , n. Define the subspace
288
+ CΞ±
289
+ βˆ†(n)[0, 1] = {x ∈ X : x ∈ CΞ±[tjβˆ’1, tj], j = 1, 2, 3, . . ., n} . For r β‰₯ 1, the approximating
290
+ space
291
+ Xn =
292
+ οΏ½
293
+ x ∈ X : x|βˆ†j is a polynomial of degree ≀ r βˆ’ 1
294
+ οΏ½
295
+ .
296
+
297
+ Section 3. Discretization of Integrals by numerical quadrature Rule
298
+ 5
299
+ Let p be a positive integer and m = pn. Consider the following uniform partition of [0, 1] :
300
+ (3.3)
301
+ βˆ†(m) :
302
+ 0 < 1
303
+ m < Β· Β· Β· < m βˆ’ 1
304
+ m
305
+ < 1.
306
+ Let ˜h = 1
307
+ m
308
+ and
309
+ si = i
310
+ m,
311
+ i = 0, . . . , m.
312
+ Note : As our goal to find the equation (1.6), where the higher order term is max
313
+ οΏ½
314
+ h2r+2, ˜h2�
315
+ ,
316
+ we choose the partition βˆ†(m) such that ˜h2 ≀ hr.
317
+ A composite integration rule with respect to the partition (3.3) is then defined as
318
+ οΏ½ 1
319
+ 0
320
+ x(t) dt
321
+ =
322
+ m
323
+ οΏ½
324
+ i=1
325
+ οΏ½ si
326
+ siβˆ’1
327
+ x(t) dt β‰ˆ ˜h
328
+ m
329
+ οΏ½
330
+ i=1
331
+ ρ
332
+ οΏ½
333
+ q=1
334
+ wq x(siβˆ’1 + Β΅q˜h).
335
+ Thus,
336
+ οΏ½ tj
337
+ tjβˆ’1
338
+ x(t) dt =
339
+ οΏ½ jh
340
+ (jβˆ’1)h
341
+ x(t) dt =
342
+ p
343
+ οΏ½
344
+ Ξ½=1
345
+ οΏ½ (jβˆ’1)h+ν˜h
346
+ (jβˆ’1)h+(Ξ½βˆ’1)˜h
347
+ x(t) dt.
348
+ Since h = p˜h,
349
+ οΏ½ tj
350
+ tjβˆ’1
351
+ x(t)dt
352
+ =
353
+ p
354
+ οΏ½
355
+ Ξ½=1
356
+ οΏ½
357
+ (jβˆ’1)p+Ξ½
358
+ p
359
+ h
360
+ (jβˆ’1)p+Ξ½βˆ’1
361
+ p
362
+ h
363
+ x(t) dt.
364
+ Substituting t =
365
+ (jβˆ’1)p+Ξ½βˆ’1
366
+ p
367
+ h + ˜hΟƒ =
368
+ (jβˆ’1)p+Ξ½βˆ’1+Οƒ
369
+ p
370
+ h in the above equation, we obtain
371
+ οΏ½ tj
372
+ tjβˆ’1
373
+ x(t) dt
374
+ =
375
+ h
376
+ p
377
+ p
378
+ οΏ½
379
+ Ξ½=1
380
+ οΏ½ 1
381
+ 0
382
+ x
383
+ οΏ½(j βˆ’ 1)p + Ξ½ βˆ’ 1 + Οƒ
384
+ p
385
+ h
386
+ οΏ½
387
+ dσ
388
+ =
389
+ h
390
+ p
391
+ p
392
+ οΏ½
393
+ Ξ½=1
394
+ οΏ½ 1
395
+ 0
396
+ x
397
+ οΏ½
398
+ tjβˆ’1 + Ξ½ βˆ’ 1 + Οƒ
399
+ p
400
+ h
401
+ οΏ½
402
+ dσ.
403
+ Note that Ξ½βˆ’1+Οƒ
404
+ p
405
+ ∈ [0, 1]. Now using the numerical quadrature formula (3.1), we obtain
406
+ οΏ½ tj
407
+ tjβˆ’1
408
+ x(t) dt
409
+ β‰ˆ
410
+ h
411
+ p
412
+ p
413
+ οΏ½
414
+ Ξ½=1
415
+ ρ
416
+ οΏ½
417
+ q=1
418
+ wq x
419
+ οΏ½
420
+ tjβˆ’1 + Ξ½ βˆ’ 1 + Β΅q
421
+ p
422
+ h
423
+ οΏ½
424
+ .
425
+ Let
426
+ Β΅qΞ½ = Ξ½ βˆ’ 1 + Β΅q
427
+ p
428
+ ,
429
+ q = 1, 2, . . . , ρ; ν = 1, 2, . . . , p.
430
+ Then,
431
+ οΏ½ tj
432
+ tjβˆ’1
433
+ x(t) dt
434
+ β‰ˆ
435
+ h
436
+ p
437
+ p
438
+ οΏ½
439
+ Ξ½=1
440
+ ρ
441
+ οΏ½
442
+ q=1
443
+ wq x(tjβˆ’1 + Β΅qΞ½h).
444
+ (3.4)
445
+ We prove the following lemma which will be used to find an asymptotic error expansion for
446
+ the discrete orthogonal projection.
447
+ Lemma 3.1. Let LΞ· be the Legendre polynomial of degree Ξ· ∈ {0, 1, . . . , r βˆ’ 1} defined on
448
+ [0, 1]. Then for any k = 1, 2, . . . , 2r βˆ’ 1,
449
+ 1
450
+ p
451
+ rβˆ’1
452
+ οΏ½
453
+ Ξ·=0
454
+ p
455
+ οΏ½
456
+ Ξ½=1
457
+ ρ
458
+ οΏ½
459
+ q=1
460
+ wq LΞ·(Β΅qΞ½)LΞ·(Ο„)(Β΅qΞ½ βˆ’ Ο„)k
461
+ k!
462
+ =
463
+ οΏ½ 1
464
+ 0
465
+ Ξ›r(Ο„, s)(s βˆ’ Ο„)k
466
+ k!
467
+ ds,
468
+
469
+ 6
470
+ G. RAKSHIT
471
+ where
472
+ rβˆ’1
473
+ οΏ½
474
+ Ξ·=0
475
+ LΞ·(Ο„)LΞ·(s) = Ξ›r(Ο„, s), for Ο„, s ∈ [0, 1].
476
+ Proof. Since LΞ· is a polynomial of degree 0 ≀ Ξ· ≀ r βˆ’ 1,
477
+ 1
478
+ p
479
+ p
480
+ οΏ½
481
+ Ξ½=1
482
+ ρ
483
+ οΏ½
484
+ q=1
485
+ wq LΞ·
486
+ οΏ½Ξ½ βˆ’ 1 + Β΅q
487
+ p
488
+ οΏ½
489
+ = 1
490
+ p
491
+ p
492
+ οΏ½
493
+ Ξ½=1
494
+ οΏ½ 1
495
+ 0
496
+ LΞ·
497
+ οΏ½Ξ½ βˆ’ 1 + t
498
+ p
499
+ οΏ½
500
+ dt
501
+ =
502
+ p
503
+ οΏ½
504
+ Ξ½=1
505
+ οΏ½
506
+ Ξ½
507
+ p
508
+ Ξ½βˆ’1
509
+ p
510
+ LΞ·(s) ds
511
+ =
512
+ οΏ½ 1
513
+ 0
514
+ LΞ·(s) ds.
515
+ Since the basic quadrature formula (3.1) is exact for polynomials of degree ≀ 3r,
516
+ 1
517
+ p
518
+ p
519
+ οΏ½
520
+ Ξ½=1
521
+ ρ
522
+ οΏ½
523
+ q=1
524
+ wq LΞ·
525
+ οΏ½Ξ½ βˆ’ 1 + Β΅q
526
+ p
527
+ οΏ½ οΏ½Ξ½ βˆ’ 1 + Β΅q
528
+ p
529
+ βˆ’ Ο„
530
+ οΏ½k
531
+ =
532
+ οΏ½ 1
533
+ 0
534
+ LΞ·(s) (s βˆ’ Ο„)k ds.
535
+ It follows that
536
+ 1
537
+ p
538
+ p
539
+ οΏ½
540
+ Ξ½=1
541
+ ρ
542
+ οΏ½
543
+ q=1
544
+ wq LΞ·(Β΅qΞ½) (Β΅qΞ½ βˆ’ Ο„)k
545
+ k!
546
+ =
547
+ οΏ½ 1
548
+ 0
549
+ LΞ·(s) (s βˆ’ Ο„)k
550
+ k!
551
+ ds,
552
+ where Β΅qΞ½ = Ξ½ βˆ’ 1 + Β΅q
553
+ p
554
+ . This gives
555
+ 1
556
+ p
557
+ rβˆ’1
558
+ οΏ½
559
+ Ξ·=0
560
+ p
561
+ οΏ½
562
+ Ξ½=1
563
+ ρ
564
+ οΏ½
565
+ q=1
566
+ wq LΞ·(Ο„)LΞ·(Β΅qΞ½) (Β΅qΞ½ βˆ’ Ο„)k
567
+ k!
568
+ =
569
+ rβˆ’1
570
+ οΏ½
571
+ Ξ·=0
572
+ LΞ·(Ο„)
573
+ οΏ½ 1
574
+ 0
575
+ LΞ·(s) (s βˆ’ Ο„)k
576
+ k!
577
+ ds.
578
+ Let
579
+ rβˆ’1
580
+ οΏ½
581
+ Ξ·=0
582
+ LΞ·(Ο„)LΞ·(s) = Ξ›r(Ο„, s),
583
+ Ο„, s ∈ [0, 1].
584
+ Then,
585
+ 1
586
+ p
587
+ rβˆ’1
588
+ οΏ½
589
+ Ξ·=0
590
+ p
591
+ οΏ½
592
+ Ξ½=1
593
+ ρ
594
+ οΏ½
595
+ q=1
596
+ wq LΞ·(Ο„)LΞ·(Β΅qΞ½) (Β΅qΞ½ βˆ’ Ο„)k
597
+ k!
598
+ =
599
+ οΏ½ 1
600
+ 0
601
+ Ξ›r(Ο„, s) (s βˆ’ Ο„)k
602
+ k!
603
+ ds.
604
+ Hence the required result follows.
605
+ β–‘
606
+ 3.1 Discrete Orthogonal Projection
607
+ Let j ∈ {1, 2, . . . , n} and x, y ∈ C(βˆ†j). Define a discrete inner product on βˆ†j by
608
+ (3.5)
609
+ ⟨x , yβŸ©βˆ†j,m = ˜h
610
+ p
611
+ οΏ½
612
+ Ξ½=1
613
+ ρ
614
+ οΏ½
615
+ q=1
616
+ wq x(tjβˆ’1 + Β΅qΞ½h) y(tjβˆ’1 + Β΅qΞ½h).
617
+ Note that, this is an indefinite inner product. For more details on indefinite inner product
618
+ spaces, see [8]. However, the properties which we need to define a discrete orthogonal
619
+ projection, hold true for (3.5). For Ξ· = 0, 1, . . . , rβˆ’1, let LΞ· denote the Legendre polynomial
620
+ of degree Ξ· on [0, 1]. For j = 2, . . . , n, and for Ξ· = 0, 1, . . . , r βˆ’ 1, define
621
+ Ο•j,Ξ·(t)
622
+ =
623
+ οΏ½ οΏ½
624
+ 1
625
+ hLΞ·
626
+ οΏ½
627
+ tβˆ’tjβˆ’1
628
+ h
629
+ οΏ½
630
+ ,
631
+ t ∈ (tjβˆ’1, tj],
632
+ 0,
633
+ otherwise
634
+
635
+ 3.1
636
+ Discrete Orthogonal Projection
637
+ 7
638
+ and, Ο•1,Ξ·(t) =
639
+ οΏ½
640
+ 1
641
+ hLΞ·
642
+ οΏ½ tβˆ’t0
643
+ h
644
+ οΏ½
645
+ if t ∈ [t0, t1] and 0 otherwise. Note that
646
+ (3.6)
647
+ Ο•j,Ξ· (tjβˆ’1 + Β΅qΞ½h) = hβˆ’ 1
648
+ 2LΞ·(Β΅qΞ½)
649
+ for all j = 1, 2, . . . , n.
650
+ Note that {Ο•j,Ξ· : j = 1, . . . , n, Ξ· = 0, 1, . . . , r βˆ’ 1} be a set of orthonormal basis for Xn,
651
+ where Ο•j,Ξ· is the Legendre polynomial of degree Ξ· defined on [tjβˆ’1, tj]. Since the basic
652
+ numerical integration (3.1) has degree of precision 3r, the set {Ο•j,Ξ·} is also orthonormal
653
+ with respect to the discrete inner product (3.5). Let Pr,βˆ†j be the space of polynomials of
654
+ degree ≀ r βˆ’ 1 on βˆ†j. Define the discrete orthogonal projection Pn,j : C[tjβˆ’1, tj] β†’ Pr,βˆ†j
655
+ as follows:
656
+ (3.7)
657
+ Pn,jx =
658
+ rβˆ’1
659
+ οΏ½
660
+ Ξ·=0
661
+ ⟨x , Ο•j,Ξ·βŸ©βˆ†j Ο•j,Ξ·.
662
+ See [4], [6] for more details. A discrete orthogonal projection Pn : C[0, 1] β†’ Xn is defined
663
+ by
664
+ Pnx =
665
+ n
666
+ οΏ½
667
+ j=1
668
+ Pn,jx.
669
+ (3.8)
670
+ It follows that Pnx(t) = Pn,jx(t), for all t ∈ [tjβˆ’1, tj]. We also have the following error
671
+ bound:
672
+ βˆ₯Pnβˆ₯ < ∞ and also, if x ∈ Cr[tjβˆ’1, tj], then
673
+ (3.9)
674
+ βˆ₯x βˆ’ Pn,jxβˆ₯βˆ†j,∞ ≀ C1
675
+ οΏ½οΏ½x(r)οΏ½οΏ½
676
+ βˆ†j,∞ hr,
677
+ if x ∈ Cr[0, 1], then
678
+ (3.10)
679
+ βˆ₯x βˆ’ Pnxβˆ₯βˆ†j,∞ ≀ C1
680
+ οΏ½οΏ½x(r)οΏ½οΏ½
681
+ ∞ hr,
682
+ where βˆ₯xβˆ₯βˆ†j,∞ =
683
+ sup
684
+ t∈[tjβˆ’1,tj]
685
+ |x(t)| and, C1 is a constant independent of h. For details see
686
+ [17].
687
+ In (3.10) we have a error bound for the discrete orthogonal projection. But, by the follow-
688
+ ing lemma we obtain an asymptotic error expansion for the discrete orthogonal projection,
689
+ which is more stronger result than (3.10).
690
+ Lemma 3.2. Let Pn be the discrete orthogonal projection defined by (3.7) - (3.8). Let
691
+ x ∈ C2r+2
692
+ βˆ†(n) [0, 1] and t = tjβˆ’1 + Ο„h with Ο„ ∈ [0, 1]. Then
693
+ Pnx(t) βˆ’ x(t) =
694
+ 2r+1
695
+ οΏ½
696
+ k=1
697
+ Jk(Ο„) x(k)(tjβˆ’1 + Ο„h) hk + O
698
+ οΏ½
699
+ h2r+2οΏ½
700
+ ,
701
+ where Jk(Ο„) =
702
+ οΏ½ 1
703
+ 0
704
+ Ξ›r(Ο„, s)(s βˆ’ Ο„)k
705
+ k!
706
+ ds,
707
+ k = 1, 2, . . . , 2r + 1.
708
+ Proof. Define a function vj : [tjβˆ’1, tj] β†’ R by
709
+ vj(t) = 1,
710
+ t ∈ [tjβˆ’1, tj].
711
+ For Ο„ ∈ [0, 1], let t = tjβˆ’1 + hΟ„ ∈ [tjβˆ’1, tj]. From (3.7) it is easy to see that
712
+ Pn,jvj = vj.
713
+
714
+ 8
715
+ G. RAKSHIT
716
+ It follows that
717
+ rβˆ’1
718
+ οΏ½
719
+ Ξ·=0
720
+ ⟨vj , Ο•j,Ξ·βŸ©βˆ†j Ο•j,Ξ·(t) = 1.
721
+ Since Ο•j,Ξ· is a polynomial of degree 0 ≀ Ξ· ≀ r βˆ’ 1 on [tjβˆ’1, tj],
722
+ ⟨vj , Ο•j,Ξ·βŸ©βˆ†j =
723
+ οΏ½ tj
724
+ tjβˆ’1
725
+ Ο•j,Ξ·(s) ds
726
+ =
727
+ h
728
+ p
729
+ p
730
+ οΏ½
731
+ Ξ½=1
732
+ ρ
733
+ οΏ½
734
+ q=1
735
+ wq Ο•j,Ξ·(tjβˆ’1 + Β΅qΞ½h).
736
+ Thus for any function x : [tjβˆ’1, tj] β†’ R, we have
737
+ x(t) = x(t) h
738
+ p
739
+ rβˆ’1
740
+ οΏ½
741
+ Ξ·=0
742
+ p
743
+ οΏ½
744
+ Ξ½=1
745
+ ρ
746
+ οΏ½
747
+ q=1
748
+ wq Ο•j,Ξ·(tjβˆ’1 + Β΅qΞ½h) Ο•j,Ξ·(t).
749
+ It follows that
750
+ Pn,jx(t) βˆ’ x(t)
751
+ = h
752
+ p
753
+ rβˆ’1
754
+ οΏ½
755
+ Ξ·=0
756
+ p
757
+ οΏ½
758
+ Ξ½=1
759
+ ρ
760
+ οΏ½
761
+ q=1
762
+ wq Ο•j,Ξ·(tjβˆ’1 + Β΅qΞ½h) [x(tjβˆ’1 + Β΅qΞ½h) βˆ’ x(t)] Ο•j,Ξ·(t),
763
+ where t = tjβˆ’1 + hΟ„ ∈ [tjβˆ’1, tj] and Ο„ ∈ [0, 1]. From (3.6), we have
764
+ (3.11)
765
+ Pn,jx(t) βˆ’ x(t) = 1
766
+ p
767
+ rβˆ’1
768
+ οΏ½
769
+ Ξ·=0
770
+ p
771
+ οΏ½
772
+ Ξ½=1
773
+ ρ
774
+ οΏ½
775
+ q=1
776
+ wq LΞ·(Β΅qΞ½) [x(tjβˆ’1 + Β΅qΞ½h) βˆ’ x(t)] LΞ·(Ο„).
777
+ Since x ∈ C2r+2[tjβˆ’1, tj], using Taylor series expansion we obtain
778
+ x(tjβˆ’1 + Β΅qΞ½h) βˆ’ x(tjβˆ’1 + hΟ„) =
779
+ 2r+1
780
+ οΏ½
781
+ k=1
782
+ x(k)(tjβˆ’1 + Ο„h) (Β΅qΞ½ βˆ’ Ο„)k
783
+ k!
784
+ hk + O
785
+ οΏ½
786
+ h2r+2οΏ½
787
+ .
788
+ Thus
789
+ Pnx(t) βˆ’ x(t) = Pn,jx(t) βˆ’ x(t)
790
+ =
791
+ 2r+1
792
+ οΏ½
793
+ k=1
794
+ x(k)(tjβˆ’1 + Ο„h) hk
795
+ οΏ½
796
+ 1
797
+ p
798
+ rβˆ’1
799
+ οΏ½
800
+ Ξ·=0
801
+ p
802
+ οΏ½
803
+ Ξ½=1
804
+ ρ
805
+ οΏ½
806
+ q=1
807
+ wqLΞ·(Β΅qΞ½)LΞ·(Ο„)(Β΅qΞ½ βˆ’ Ο„)k
808
+ k!
809
+ οΏ½
810
+ + O
811
+ οΏ½
812
+ h2r+2οΏ½
813
+ .
814
+ Let
815
+ Jk(Ο„) = 1
816
+ p
817
+ rβˆ’1
818
+ οΏ½
819
+ Ξ·=0
820
+ p
821
+ οΏ½
822
+ Ξ½=1
823
+ ρ
824
+ οΏ½
825
+ q=1
826
+ wq LΞ·(Β΅qΞ½) LΞ·(Ο„) (Β΅qΞ½ βˆ’ Ο„)k
827
+ k!
828
+ ,
829
+ Ο„ ∈ [0, 1].
830
+ By Lemma 3.1, we can write Jk(Ο„) =
831
+ οΏ½ 1
832
+ 0 Ξ›r(Ο„, s) (sβˆ’Ο„)k
833
+ k!
834
+ ds. Hence
835
+ Pnx(t) βˆ’ x(t) =
836
+ 2r+1
837
+ οΏ½
838
+ k=1
839
+ Jk(Ο„) x(k)(tjβˆ’1 + Ο„h) hk + O
840
+ οΏ½
841
+ h2r+2οΏ½
842
+ .
843
+ The result follows.
844
+ β–‘
845
+
846
+ 3.2
847
+ Approximation of the Integral Operator
848
+ 9
849
+ Let
850
+ L = (I βˆ’ Kβ€²(Ο•))βˆ’1 Kβ€²(Ο•).
851
+ Then L is a compact linear integral operator with kernel Λœβ„“. Note that the smoothness of Λœβ„“ is
852
+ same as the kernel β„“. See [26], [5, Lemma 5.1] for details. It follows that
853
+ LPnx(s) βˆ’ Lx(s) =
854
+ n
855
+ οΏ½
856
+ j=1
857
+ οΏ½ tj
858
+ tjβˆ’1
859
+ Λœβ„“(s, t) (Pnx(t) βˆ’ x(t)) dt
860
+ βˆ€x ∈ X .
861
+ Then using Lemma 3.2, and following the proofs of [22, Theorem 5.1] and [23, Theorem
862
+ 3.2], it can be shown that
863
+ (3.12)
864
+ L(I βˆ’ Pn)Ο•(ti) = E2r(Ο•)(ti)h2r + O
865
+ οΏ½
866
+ h2r+2οΏ½
867
+ ,
868
+ i = 0, 1, . . . , n,
869
+ where
870
+ E2r(Ο•)(ti) = Β―b2r,2r
871
+ οΏ½ 1
872
+ 0
873
+ Λœβ„“(ti, t)(t) Ο•(2r)(t) dt
874
+ +
875
+ 2rβˆ’1
876
+ οΏ½
877
+ p=1
878
+ Β―b2r,p
879
+ οΏ½ οΏ½οΏ½ βˆ‚
880
+ βˆ‚t
881
+ οΏ½2rβˆ’pβˆ’1 οΏ½
882
+ Λœβ„“(ti, t)Ο•(p)(t)
883
+ οΏ½οΏ½t=1
884
+ t=0
885
+ βˆ’
886
+ οΏ½οΏ½ βˆ‚
887
+ βˆ‚t
888
+ οΏ½2rβˆ’pβˆ’1 οΏ½
889
+ Λœβ„“(ti, t)Ο•(p)(t)
890
+ οΏ½οΏ½t=ti+
891
+ t=tiβˆ’
892
+ οΏ½
893
+ with
894
+ Β―b2r,p =
895
+ οΏ½ 1
896
+ 0
897
+ οΏ½ 1
898
+ 0
899
+ Ξ›r(Ο„, s)(Ο„ βˆ’ s)p
900
+ p!
901
+ B2rβˆ’p(s)
902
+ (2r βˆ’ p)! dΟ„ ds
903
+ and Bk is the Bernoulli polynomial of degree k β‰₯ 0.
904
+ 3.2 Approximation of the Integral Operator
905
+ Let x ∈ X . Recall that
906
+ K(x)(s) =
907
+ οΏ½ 1
908
+ 0
909
+ ΞΊ(s, t, x(t)) dt,
910
+ s ∈ [0, 1].
911
+ Replacing the above integral by the numerical quadrature rule (3.4), we define the Nystrâm
912
+ approximation of K by
913
+ Km(x)(s) = h
914
+ p
915
+ n
916
+ οΏ½
917
+ j=1
918
+ ρ
919
+ οΏ½
920
+ q=1
921
+ p
922
+ οΏ½
923
+ Ξ½=1
924
+ wq ΞΊ(s, tjβˆ’1 + Β΅qΞ½h, x(tjβˆ’1 + Β΅qΞ½h)),
925
+ s ∈ [0, 1].
926
+ Let {Β΅j
927
+ qΞ½ = tjβˆ’1 + Β΅qΞ½h : j = 1, 2, . . . , n; q = 1, 2, . . . , ρ; Ξ½ = 1, 2, . . . , p} denotes the set
928
+ of all quadrature nodes in [0, 1]. Then
929
+ Km(x)(s) = h
930
+ p
931
+ n
932
+ οΏ½
933
+ j=1
934
+ ρ
935
+ οΏ½
936
+ q=1
937
+ p
938
+ οΏ½
939
+ Ξ½=1
940
+ wq ΞΊ
941
+ οΏ½
942
+ s, Β΅j
943
+ qΞ½, x
944
+ οΏ½
945
+ Β΅j
946
+ qΞ½
947
+ οΏ½οΏ½
948
+ ,
949
+ s ∈ [0, 1].
950
+ The Nystrâm method for solving (1.1) is to find the element xm for which
951
+ xm βˆ’ Km(xm) = f.
952
+ For sufficiently large m, the above equation has a unique solution Ο•m in a neighborhood
953
+ B(Ο•, Η«) of Ο•, and
954
+ βˆ₯Ο• βˆ’ Ο•mβˆ₯∞
955
+ ≀
956
+ C2 βˆ₯K(Ο•) βˆ’ Km(Ο•)βˆ₯∞ = O
957
+ οΏ½
958
+ ˜h2�
959
+ ,
960
+ (3.13)
961
+
962
+ 10
963
+ G. RAKSHIT
964
+ where C2 is a constant independent of m. See [2, Theorem 4]. We write
965
+ (I βˆ’ Pn) Ο•m = (I βˆ’ Pn) (Ο•m βˆ’ Ο•) + (I βˆ’ Pn) Ο•.
966
+ Then from (3.10), (3.13), we have
967
+ (3.14)
968
+ (I βˆ’ Pn) Ο•m = O
969
+ οΏ½
970
+ max
971
+ οΏ½
972
+ hr, ˜h2��
973
+ .
974
+ Let v1, v2 ∈ X and x ∈ B(Ο•, Η«). Then the FrΓ©chet derivatives of Km at x are given by
975
+ Kβ€²
976
+ m(x)v1(s) = h
977
+ p
978
+ n
979
+ οΏ½
980
+ j=1
981
+ ρ
982
+ οΏ½
983
+ q=1
984
+ p
985
+ οΏ½
986
+ Ξ½=1
987
+ wq D(0,0,1)ΞΊ
988
+ οΏ½
989
+ s, Β΅j
990
+ qΞ½, x
991
+ οΏ½
992
+ Β΅j
993
+ qΞ½
994
+ οΏ½οΏ½
995
+ v1
996
+ οΏ½
997
+ Β΅j
998
+ qΞ½
999
+ οΏ½
1000
+ ,
1001
+ s ∈ [0, 1],
1002
+ Kβ€²β€²
1003
+ m(x) (v1, v2) (s) = h
1004
+ p
1005
+ n
1006
+ οΏ½
1007
+ j=1
1008
+ ρ
1009
+ οΏ½
1010
+ q=1
1011
+ p
1012
+ οΏ½
1013
+ Ξ½=1
1014
+ wq
1015
+ βˆ‚2ΞΊ
1016
+ βˆ‚u2
1017
+ οΏ½
1018
+ s, Β΅j
1019
+ qΞ½, x
1020
+ οΏ½
1021
+ Β΅j
1022
+ qΞ½
1023
+ οΏ½οΏ½
1024
+ v1
1025
+ οΏ½
1026
+ Β΅j
1027
+ qΞ½
1028
+ οΏ½
1029
+ v2
1030
+ οΏ½
1031
+ Β΅j
1032
+ qΞ½
1033
+ οΏ½
1034
+ .
1035
+ It follows that
1036
+ βˆ₯Kβ€²β€²
1037
+ m(x) (v1, v2)βˆ₯∞ ≀
1038
+ 
1039
+ 
1040
+ ο£­
1041
+ sup
1042
+ s,t∈[0,1]
1043
+ |u|≀βˆ₯Ο•βˆ₯∞+Η«
1044
+ οΏ½οΏ½οΏ½οΏ½
1045
+ βˆ‚2ΞΊ
1046
+ βˆ‚u2(s, t, u)
1047
+ οΏ½οΏ½οΏ½οΏ½
1048
+ ο£Ά
1049
+ ο£·
1050
+ ο£Έ βˆ₯v1βˆ₯∞ βˆ₯v2βˆ₯∞ .
1051
+ This implies
1052
+ βˆ₯Kβ€²β€²
1053
+ m(x)βˆ₯ < ∞
1054
+ Similarly, it can be shown that
1055
+ οΏ½οΏ½οΏ½K(3)
1056
+ m (x)
1057
+ οΏ½οΏ½οΏ½ ≀
1058
+ 
1059
+ 
1060
+ ο£­
1061
+ sup
1062
+ s,t∈[0,1]
1063
+ |u|≀βˆ₯Ο•βˆ₯∞+Η«
1064
+ οΏ½οΏ½οΏ½οΏ½
1065
+ βˆ‚3ΞΊ
1066
+ βˆ‚u3 (s, t, u)
1067
+ οΏ½οΏ½οΏ½οΏ½
1068
+ ο£Ά
1069
+ ο£·
1070
+  = C3 < ∞
1071
+ Lemma 3.3. Let x1, x2 ∈ B(Ο•, Η«). If D(0,0,3)ΞΊ ∈ C(Ω) then
1072
+ βˆ₯Kβ€²β€²
1073
+ m(x1) βˆ’ Kβ€²β€²
1074
+ m(x2)βˆ₯ ≀ C3 βˆ₯x1 βˆ’ x2βˆ₯∞ ,
1075
+ where C3 is constant independent of n.
1076
+ Proof. For v1, v2 ∈ X , we have
1077
+ [Kβ€²β€²
1078
+ m(x1) βˆ’ Kβ€²β€²
1079
+ m(x2)] (v1, v2)(s)
1080
+ = h
1081
+ p
1082
+ n
1083
+ οΏ½
1084
+ j=1
1085
+ ρ
1086
+ οΏ½
1087
+ q=1
1088
+ p
1089
+ οΏ½
1090
+ Ξ½=1
1091
+ wq
1092
+ οΏ½βˆ‚2ΞΊ
1093
+ βˆ‚u2
1094
+ οΏ½
1095
+ s, Β΅j
1096
+ qΞ½, x1
1097
+ οΏ½
1098
+ Β΅j
1099
+ qΞ½
1100
+ οΏ½οΏ½
1101
+ βˆ’ βˆ‚2ΞΊ
1102
+ βˆ‚u2
1103
+ οΏ½
1104
+ s, Β΅j
1105
+ qΞ½, x2
1106
+ οΏ½
1107
+ Β΅j
1108
+ qΞ½
1109
+ οΏ½οΏ½οΏ½
1110
+ v1
1111
+ οΏ½
1112
+ Β΅j
1113
+ qΞ½
1114
+ οΏ½
1115
+ v2
1116
+ οΏ½
1117
+ Β΅j
1118
+ qΞ½
1119
+ οΏ½
1120
+ for all s ∈ [0, 1]. Since D(0,0,3)ΞΊ ∈ C(Ω), applying mean value theorem on βˆ‚2ΞΊ
1121
+ βˆ‚u2 with respect
1122
+ to its third variable u, we obtain
1123
+ βˆ‚2ΞΊ
1124
+ βˆ‚u2
1125
+ οΏ½
1126
+ s, Β΅j
1127
+ qΞ½, x1
1128
+ οΏ½
1129
+ Β΅j
1130
+ qΞ½
1131
+ οΏ½οΏ½
1132
+ βˆ’βˆ‚2ΞΊ
1133
+ βˆ‚u2
1134
+ οΏ½
1135
+ s, Β΅j
1136
+ qΞ½, x2
1137
+ οΏ½
1138
+ Β΅j
1139
+ qΞ½
1140
+ οΏ½οΏ½
1141
+ =
1142
+ οΏ½
1143
+ x1
1144
+ οΏ½
1145
+ Β΅j
1146
+ qΞ½
1147
+ οΏ½
1148
+ βˆ’ x2
1149
+ οΏ½
1150
+ Β΅j
1151
+ qΞ½
1152
+ οΏ½οΏ½ βˆ‚3ΞΊ
1153
+ βˆ‚u3
1154
+ οΏ½
1155
+ s, Β΅j
1156
+ qΞ½, ΞΆj
1157
+ qΞ½
1158
+ οΏ½
1159
+ ,
1160
+ where ΞΆj
1161
+ qΞ½ lies in the line segment joining the points x1
1162
+ οΏ½
1163
+ Β΅j
1164
+ qΞ½
1165
+ οΏ½
1166
+ and x2
1167
+ οΏ½
1168
+ Β΅j
1169
+ qΞ½
1170
+ οΏ½
1171
+ . Then
1172
+ οΏ½οΏ½οΏ½οΏ½
1173
+ βˆ‚2ΞΊ
1174
+ βˆ‚u2
1175
+ οΏ½
1176
+ s, Β΅j
1177
+ qΞ½, x1
1178
+ οΏ½
1179
+ Β΅j
1180
+ qΞ½
1181
+ οΏ½οΏ½
1182
+ βˆ’ βˆ‚2ΞΊ
1183
+ βˆ‚u2
1184
+ οΏ½
1185
+ s, Β΅j
1186
+ qΞ½, x2
1187
+ οΏ½
1188
+ Β΅j
1189
+ qΞ½
1190
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ C3 βˆ₯x1 βˆ’ x2βˆ₯∞ .
1191
+ Hence
1192
+ βˆ₯[Kβ€²β€²
1193
+ m(x1) βˆ’ Kβ€²β€²
1194
+ m(x2)] (v1, v2)βˆ₯∞ ≀ C3 βˆ₯x1 βˆ’ x2βˆ₯∞ βˆ₯v1βˆ₯∞ βˆ₯v2βˆ₯∞ ,
1195
+ which follows the result.
1196
+ β–‘
1197
+
1198
+ Section 4. Asymptotic Error Analysis
1199
+ 11
1200
+ We will now quote some error estimates for the NystrΓΆm approximations.
1201
+ For Ξ± β‰₯ 0, if v1, v2 ∈ CΞ±
1202
+ βˆ†(m)[0, 1], then from [27] or [4, Corollary 1], we obtain the
1203
+ following errors for numerical integration.
1204
+ βˆ₯[Kβ€²
1205
+ m(Ο•) βˆ’ Kβ€²(Ο•)] v1βˆ₯∞ = O
1206
+ οΏ½
1207
+ ˜h2�
1208
+ ,
1209
+ (3.15)
1210
+ βˆ₯[Kβ€²β€²
1211
+ m(Ο•) βˆ’ Kβ€²β€²(Ο•)] (v1, v2)βˆ₯∞ = O
1212
+ οΏ½
1213
+ ˜h2�
1214
+ .
1215
+ Also from [18, Proposition 3.3], we have
1216
+ βˆ₯Kβ€²
1217
+ m(Ο•m) βˆ’ Kβ€²
1218
+ m(Ο•)βˆ₯ ≀ C4 βˆ₯Ο•m βˆ’ Ο•βˆ₯∞ = O
1219
+ οΏ½
1220
+ ˜h2�
1221
+ .
1222
+ (3.16)
1223
+ Therefore combining (3.15) and the above equation, we obtain
1224
+ βˆ₯[Kβ€²
1225
+ m(Ο•m) βˆ’ Kβ€²(Ο•)] vβˆ₯∞ = O
1226
+ οΏ½
1227
+ ˜h2�
1228
+ ,
1229
+ for all v ∈ Cν
1230
+ βˆ†m[0, 1].
1231
+ (3.17)
1232
+ Similarly,
1233
+ βˆ₯[Kβ€²β€²
1234
+ m(Ο•m) βˆ’ Kβ€²β€²(Ο•)] (v1, v2)βˆ₯∞ = O
1235
+ οΏ½
1236
+ ˜h2�
1237
+ ,
1238
+ βˆ€v1, v2 ∈ CΞ½
1239
+ βˆ†m[0, 1],
1240
+ (3.18)
1241
+ for all v1, v2, v3 ∈ Cν
1242
+ βˆ†m[0, 1] implies
1243
+ οΏ½οΏ½οΏ½
1244
+ K(3)
1245
+ m (Ο•m) βˆ’ K(3)(Ο•)
1246
+ οΏ½
1247
+ (v1, v2, v3)
1248
+ οΏ½οΏ½
1249
+ ∞ = O
1250
+ οΏ½
1251
+ ˜h2�
1252
+ .
1253
+ (3.19)
1254
+ 4 Asymptotic Error Analysis
1255
+ Replacing K by Km and Ο€n by Pn in the Galerkin equation x βˆ’ Ο€nK(x) = Ο€nf, the
1256
+ discrete Galerkin equation is defined by zG
1257
+ n βˆ’ PnKm(zG
1258
+ n ) = Pnf, where zG
1259
+ n is the discrete
1260
+ Galerkin solution. Then the discrete iterated Galerkin solution is defined by
1261
+ zS
1262
+ n = Km(zG
1263
+ n ) + f.
1264
+ Note that PnzS
1265
+ n = zG
1266
+ n . From the equations Ο•m βˆ’ Km(Ο•m) = f and zS
1267
+ n βˆ’ Km(PnzS
1268
+ n) = f,
1269
+ we obtain the following error term.
1270
+ zS
1271
+ n βˆ’ Ο•m = [I βˆ’ Kβ€²
1272
+ m(Ο•m)]βˆ’1 οΏ½
1273
+ Km(zG
1274
+ n ) βˆ’ Km(Ο•m) βˆ’ Kβ€²
1275
+ m(Ο•m)(zG
1276
+ n βˆ’ Ο•m)
1277
+ οΏ½
1278
+ βˆ’ Lm(I βˆ’ Pn)
1279
+ οΏ½
1280
+ Km(zG
1281
+ n ) βˆ’ Km(Ο•m) βˆ’ Kβ€²
1282
+ m(Ο•m)(zG
1283
+ n βˆ’ Ο•m)
1284
+ οΏ½
1285
+ βˆ’ Lm(I βˆ’ Pn)Kβ€²
1286
+ m(Ο•m)(zG
1287
+ n βˆ’ Ο•m)
1288
+ βˆ’ Lm(I βˆ’ Pn)Ο•m,
1289
+ (4.1)
1290
+ where
1291
+ Lm = [I βˆ’ Kβ€²
1292
+ m(Ο•m)]βˆ’1 Kβ€²
1293
+ m(Ο•m).
1294
+ Using the Resolvent Identity, we get
1295
+ (I βˆ’ Kβ€²
1296
+ m(Ο•m))βˆ’1 βˆ’ (I βˆ’ Kβ€²(Ο•))βˆ’1
1297
+ = (I βˆ’ Kβ€²(Ο•))βˆ’1 [Kβ€²
1298
+ m(Ο•m) βˆ’ Kβ€²(Ο•)] (I βˆ’ Kβ€²
1299
+ m(Ο•m))βˆ’1
1300
+
1301
+ 12
1302
+ G. RAKSHIT
1303
+ Therefore
1304
+ (I βˆ’ Kβ€²
1305
+ m(Ο•m))βˆ’1 = (I βˆ’ Kβ€²(Ο•))βˆ’1
1306
+ + (I βˆ’ Kβ€²(Ο•))βˆ’1 [Kβ€²
1307
+ m(Ο•m) βˆ’ Kβ€²
1308
+ m(Ο•)] (I βˆ’ Kβ€²
1309
+ m(Ο•m))βˆ’1
1310
+ + (I βˆ’ Kβ€²(Ο•))βˆ’1 [Kβ€²
1311
+ m(Ο•) βˆ’ Kβ€²(Ο•)] (I βˆ’ Kβ€²
1312
+ m(Ο•m))βˆ’1
1313
+ (4.2)
1314
+ Now, we will analyze each of the terms appearing in the RHS of the equation (4.1). Error
1315
+ estimates for each of the said terms will be obtained by the following propositions.
1316
+ Proposition 4.1. Let {ti : i = 0, 1, . . . , n} be the set of partition points of [0, 1] defined by
1317
+ (3.2), then
1318
+ Lm(I βˆ’ Pn)Ο•m(ti) = E2r(Ο•)(ti)h2r + O
1319
+ οΏ½
1320
+ max
1321
+ οΏ½
1322
+ h2r+2, ˜h2��
1323
+ ,
1324
+ where E2r is defined by (3.12).
1325
+ Proof. It can be easily verified that (using (4.2))
1326
+ Lm(I βˆ’ Pn)Ο•m =
1327
+ οΏ½
1328
+ I βˆ’ Kβ€²
1329
+ m(Ο•m)
1330
+ οΏ½βˆ’1 Kβ€²
1331
+ m(Ο•m)(I βˆ’ Pn)Ο•m
1332
+ =
1333
+ οΏ½
1334
+ I βˆ’ Kβ€²(Ο•)
1335
+ οΏ½βˆ’1 Kβ€²
1336
+ m(Ο•m)(I βˆ’ Pn)Ο•m
1337
+ +
1338
+ οΏ½
1339
+ I βˆ’ Kβ€²(Ο•)
1340
+ οΏ½βˆ’1 οΏ½
1341
+ Kβ€²
1342
+ m(Ο•m) βˆ’ Kβ€²
1343
+ m(Ο•)
1344
+ οΏ½ οΏ½
1345
+ I βˆ’ Kβ€²
1346
+ m(Ο•m)
1347
+ οΏ½βˆ’1 Kβ€²
1348
+ m(Ο•m)(I βˆ’ Pn)Ο•m
1349
+ +
1350
+ οΏ½
1351
+ I βˆ’ Kβ€²(Ο•)
1352
+ οΏ½βˆ’1 οΏ½
1353
+ Kβ€²
1354
+ m(Ο•) βˆ’ Kβ€²(Ο•)
1355
+ οΏ½ οΏ½
1356
+ I βˆ’ Kβ€²
1357
+ m(Ο•m)
1358
+ οΏ½βˆ’1 Kβ€²
1359
+ m(Ο•m)(I βˆ’ Pn)Ο•m.
1360
+ (4.3)
1361
+ Consider the first term of the above equation, we have
1362
+ (I βˆ’ Kβ€²(Ο•))βˆ’1Kβ€²
1363
+ m(Ο•m)(I βˆ’ Pn)Ο•m
1364
+ = (I βˆ’ Kβ€²(Ο•))βˆ’1 Kβ€²(Ο•)(I βˆ’ Pn)Ο•
1365
+ + (I βˆ’ Kβ€²(Ο•))βˆ’1 Kβ€²(Ο•)(I βˆ’ Pn)(Ο•m βˆ’ Ο•)
1366
+ + (I βˆ’ Kβ€²(Ο•))βˆ’1 [Kβ€²
1367
+ m(Ο•m) βˆ’ Kβ€²(Ο•)] (I βˆ’ Pn)Ο•m.
1368
+ Using (3.12), (3.13) and (3.17), we obtain
1369
+ (4.4)
1370
+ (I βˆ’ Kβ€²(Ο•))βˆ’1 Kβ€²
1371
+ m(Ο•m)(I βˆ’Pn)Ο•m(ti) = E2r(Ο•)(ti)h2r +O
1372
+ οΏ½
1373
+ max
1374
+ οΏ½
1375
+ h2r+2, ˜h2��
1376
+ .
1377
+ Note that
1378
+ βˆ₯Kβ€²
1379
+ m(Ο•m)βˆ₯ ≀
1380
+ sup
1381
+ s,t∈[0,1]
1382
+ |u|≀βˆ₯Ο•βˆ₯∞+Η«
1383
+ |ΞΊu(s, t, u)|
1384
+ and from [18, Proposition 4.2], we have
1385
+ οΏ½οΏ½(I βˆ’ Kβ€²
1386
+ m(Ο•m))βˆ’1οΏ½οΏ½ < ∞. Thus, from (3.15) and
1387
+ (3.16), we have the followings
1388
+ οΏ½οΏ½οΏ½
1389
+ οΏ½
1390
+ I βˆ’ Kβ€²(Ο•)
1391
+ οΏ½βˆ’1 οΏ½
1392
+ Kβ€²
1393
+ m(Ο•m) βˆ’ Kβ€²
1394
+ m(Ο•)
1395
+ οΏ½ οΏ½
1396
+ I βˆ’ Kβ€²
1397
+ m(Ο•m)
1398
+ οΏ½βˆ’1 Kβ€²
1399
+ m(Ο•m)(I βˆ’ Pn)Ο•m
1400
+ οΏ½οΏ½οΏ½
1401
+ ∞ = O
1402
+ οΏ½
1403
+ ˜h2�
1404
+ ,
1405
+ οΏ½οΏ½οΏ½
1406
+ οΏ½
1407
+ I βˆ’ Kβ€²(Ο•)
1408
+ οΏ½βˆ’1 οΏ½
1409
+ Kβ€²
1410
+ m(Ο•m) βˆ’ Kβ€²
1411
+ m(Ο•)
1412
+ οΏ½ οΏ½
1413
+ I βˆ’ Kβ€²
1414
+ m(Ο•m)
1415
+ οΏ½βˆ’1 Kβ€²
1416
+ m(Ο•m)(I βˆ’ Pn)Ο•m
1417
+ οΏ½οΏ½οΏ½
1418
+ ∞ = O
1419
+ οΏ½
1420
+ ˜h2�
1421
+ .
1422
+ Hence the required result follows from (4.3), (4.4) and the above two estimates.
1423
+ β–‘
1424
+ Before each of the following propositions, we prove lemmas and its corollaries which are
1425
+ used to prove next propositions.
1426
+
1427
+ Section 4. Asymptotic Error Analysis
1428
+ 13
1429
+ Lemma 4.1. Let Pn be the discrete orthogonal projection defined by (3.8). If D(0,0,3)κ ∈
1430
+ C(Ω) and v ∈ Cr+2([0, 1]), then for r β‰₯ 1
1431
+ Kβ€²β€²(Ο•)(Pnv βˆ’ v)2 = T(v)h2r + O(h2r+2),
1432
+ (4.5)
1433
+ where
1434
+ T(v) =
1435
+ οΏ½οΏ½ 1
1436
+ 0
1437
+ Jr(Ο„)2 dΟ„
1438
+ οΏ½
1439
+ Kβ€²β€²(Ο•)
1440
+ οΏ½
1441
+ v(r)οΏ½2 .
1442
+ Furthermore, when r = 1, then
1443
+ (4.6)
1444
+ K(3)(Ο•)(Pnv βˆ’ v)3 = O
1445
+ οΏ½
1446
+ h4οΏ½
1447
+ .
1448
+ Proof. The proofs of (4.5) and (4.6) follows from Lemma 3.2, [16, Lemma 2.4] and [16,
1449
+ Remark 2.4] respectively.
1450
+ β–‘
1451
+ Given that (I βˆ’ Kβ€²(Ο•))βˆ’1 is a bounded linear operator. Let
1452
+ M = (I βˆ’ Kβ€²(Ο•))βˆ’1 Kβ€²β€²(Ο•).
1453
+ Note that M is a compact bi-linear integral operator. Also the smoothness of the kernel of
1454
+ M, is same as the kernels of Kβ€²β€²(Ο•). See [4], [26].
1455
+ As a consequence of the above lemma, we get the following result.
1456
+ Corollary 4.1. For r β‰₯ 1,
1457
+ M(Pnv βˆ’ v)2 = T (v)h2r + O(h2r+2),
1458
+ (4.7)
1459
+ where
1460
+ T (v) =
1461
+ οΏ½οΏ½ 1
1462
+ 0
1463
+ Jr(Ο„)2 dΟ„
1464
+ οΏ½
1465
+ M
1466
+ οΏ½
1467
+ v(r)οΏ½2
1468
+ .
1469
+ Lemma 4.2. If Ο• ∈ Cr+2([0, 1]), then for r β‰₯ 1,
1470
+ Kβ€²β€²
1471
+ m(Ο•m)(zG
1472
+ n βˆ’ Ο•m)2 = T(Ο•)h2r + O
1473
+ οΏ½
1474
+ max
1475
+ οΏ½
1476
+ h2r+2, ˜h2��
1477
+ ,
1478
+ where T is defined in Lemma 4.1.
1479
+ Proof. Note that
1480
+ zG
1481
+ n βˆ’ Ο•m = PnzS
1482
+ n βˆ’ PnΟ•m βˆ’ Ο•m + PnΟ•m + Ο•
1483
+ = Pn
1484
+ οΏ½
1485
+ zS
1486
+ n βˆ’ Ο•m
1487
+ οΏ½
1488
+ βˆ’ (I βˆ’ Pn) Ο•m
1489
+ (4.8)
1490
+ Thus,
1491
+ Kβ€²β€²
1492
+ m(Ο•m)(zG
1493
+ n βˆ’ Ο•m)2
1494
+ = Kβ€²β€²
1495
+ m(Ο•m)
1496
+ οΏ½
1497
+ Pn
1498
+ οΏ½
1499
+ zS
1500
+ n βˆ’ Ο•m
1501
+ οΏ½οΏ½2 βˆ’ 2Kβ€²β€²
1502
+ m(Ο•m)
1503
+ οΏ½
1504
+ Pn
1505
+ οΏ½
1506
+ zS
1507
+ n βˆ’ Ο•m
1508
+ οΏ½
1509
+ , (I βˆ’ Pn) Ο•m
1510
+ οΏ½
1511
+ + Kβ€²β€²
1512
+ m(Ο•m) ((I βˆ’ Pn) Ο•m)2
1513
+ (4.9)
1514
+ Since βˆ₯Kβ€²β€²
1515
+ m(Ο•m)βˆ₯ < ∞ and βˆ₯Pnβˆ₯ < ∞, from (1.4) it is easy to see that
1516
+ (4.10)
1517
+ οΏ½οΏ½οΏ½Kβ€²β€²
1518
+ m(Ο•m)
1519
+ οΏ½
1520
+ Pn
1521
+ οΏ½
1522
+ zS
1523
+ n βˆ’ Ο•m
1524
+ οΏ½οΏ½2οΏ½οΏ½οΏ½
1525
+ ∞ = O
1526
+ οΏ½
1527
+ max
1528
+ οΏ½
1529
+ h2r+4, ˜h4��
1530
+ ,
1531
+ (4.11)
1532
+ οΏ½οΏ½Kβ€²β€²
1533
+ m(Ο•m)
1534
+ οΏ½
1535
+ Pn
1536
+ οΏ½
1537
+ zS
1538
+ n βˆ’ Ο•m
1539
+ οΏ½
1540
+ , (I βˆ’ Pn) Ο•m
1541
+ οΏ½οΏ½οΏ½
1542
+ ∞ = O
1543
+ οΏ½
1544
+ ˜h2 max
1545
+ οΏ½
1546
+ hr+2, ˜h2��
1547
+ .
1548
+
1549
+ 14
1550
+ G. RAKSHIT
1551
+ Now we write
1552
+ Kβ€²β€²
1553
+ m(Ο•m) ((I βˆ’ Pn) Ο•m)2 =
1554
+ οΏ½
1555
+ Kβ€²β€²
1556
+ m(Ο•m) βˆ’ Kβ€²β€²(Ο•)
1557
+ οΏ½
1558
+ ((I βˆ’ Pn) Ο•m)2
1559
+ + Kβ€²β€²(Ο•) ((I βˆ’ Pn) Ο•m)2
1560
+ = Kβ€²β€²(Ο•) ((I βˆ’ Pn) Ο•)2 + Kβ€²β€²(Ο•) ((I βˆ’ Pn) (Ο• βˆ’ Ο•m))2
1561
+ +
1562
+ οΏ½
1563
+ Kβ€²β€²
1564
+ m(Ο•m) βˆ’ Kβ€²β€²(Ο•)
1565
+ οΏ½
1566
+ ((I βˆ’ Pn) Ο•m)2 .
1567
+ Since βˆ₯Kβ€²β€²
1568
+ m(Ο•m)βˆ₯ < ∞ and βˆ₯Pnβˆ₯ < ∞, from (3.13), (3.18) and the above estimate, we
1569
+ obtain
1570
+ Kβ€²β€²
1571
+ m(Ο•m) ((I βˆ’ Pn) Ο•m)2 = Kβ€²β€²(Ο•) ((I βˆ’ Pn) Ο•)2 + O
1572
+ οΏ½
1573
+ ˜h2�
1574
+ .
1575
+ Therefore, from (4.5) we obtain
1576
+ Kβ€²β€²
1577
+ m(Ο•m) ((I βˆ’ Pn) Ο•m)2 = T(Ο•)h2r + O
1578
+ οΏ½
1579
+ max
1580
+ οΏ½
1581
+ h2r+2, ˜h2��
1582
+ .
1583
+ Hence, the required result follows from (4.9), (4.10), (4.11) and the above equation.
1584
+ β–‘
1585
+ From the above lemma, we obtain
1586
+ (4.12)
1587
+ (I βˆ’ Kβ€²(Ο•))βˆ’1 Kβ€²β€²
1588
+ m(Ο•m)(zG
1589
+ n βˆ’ Ο•m)2 = T (Ο•)h2r + O
1590
+ οΏ½
1591
+ max
1592
+ οΏ½
1593
+ h2r+2, ˜h2��
1594
+ ,
1595
+ where T is defined by (4.7).
1596
+ Lemma 4.3. If Ο• ∈ Cr+2([0, 1]), then
1597
+ [I βˆ’ Kβ€²
1598
+ m(Ο•m)]βˆ’1 K(3)
1599
+ m (Ο•m)(zG
1600
+ n βˆ’ Ο•m)3 =
1601
+ ο£±
1602
+ ο£²
1603
+ ο£³
1604
+ O
1605
+ οΏ½
1606
+ max
1607
+ οΏ½
1608
+ h4, ˜h2��
1609
+ ,
1610
+ r = 1,
1611
+ O
1612
+ οΏ½
1613
+ max
1614
+ οΏ½
1615
+ h3r, ˜h6��
1616
+ , r β‰₯ 2.
1617
+ Proof. First, we consider the case when r β‰₯ 2.
1618
+ Since
1619
+ οΏ½οΏ½[I βˆ’ Kβ€²
1620
+ m(Ο•m)]βˆ’1οΏ½οΏ½ < ∞ and
1621
+ οΏ½οΏ½οΏ½K(3)
1622
+ m (Ο•m)
1623
+ ��� < ∞, from (1.4) we obtain
1624
+ οΏ½οΏ½οΏ½[I βˆ’ Kβ€²
1625
+ m(Ο•m)]βˆ’1 K(3)
1626
+ m (Ο•m)(zG
1627
+ n βˆ’ Ο•m)3οΏ½οΏ½οΏ½
1628
+ ∞ = O
1629
+ οΏ½
1630
+ max
1631
+ οΏ½
1632
+ h3r, ˜h6��
1633
+ .
1634
+ Now consider the case, when r = 1. We rewrite (4.8) as
1635
+ (zG
1636
+ n βˆ’ Ο•m)3 =
1637
+ οΏ½
1638
+ Pn
1639
+ οΏ½
1640
+ zS
1641
+ n βˆ’ Ο•m
1642
+ οΏ½
1643
+ βˆ’ (I βˆ’ Pn) Ο•m
1644
+ οΏ½3 .
1645
+ Thus
1646
+ K(3)
1647
+ m (Ο•m)(zG
1648
+ n βˆ’ Ο•m)3 = K(3)
1649
+ m (Ο•m)
1650
+ οΏ½
1651
+ Pn
1652
+ οΏ½
1653
+ zS
1654
+ n βˆ’ Ο•m
1655
+ οΏ½οΏ½3 βˆ’ K(3)
1656
+ m (Ο•m) ((I βˆ’ Pn) Ο•m)3
1657
+ βˆ’ K(3)
1658
+ m (Ο•m)
1659
+ οΏ½οΏ½
1660
+ Pn
1661
+ οΏ½
1662
+ zS
1663
+ n βˆ’ Ο•m
1664
+ οΏ½οΏ½2 , (I βˆ’ Pn) Ο•m
1665
+ οΏ½
1666
+ + K(3)
1667
+ m (Ο•m)
1668
+ οΏ½
1669
+ Pn
1670
+ οΏ½
1671
+ zS
1672
+ n βˆ’ Ο•m
1673
+ οΏ½
1674
+ , ((I βˆ’ Pn) Ο•m)2οΏ½
1675
+ .
1676
+ (4.13)
1677
+ Since
1678
+ οΏ½οΏ½οΏ½K(3)
1679
+ m (Ο•m)
1680
+ οΏ½οΏ½οΏ½ < ∞ and βˆ₯Pnβˆ₯ < ∞, from (1.5) and (3.14) we obtain
1681
+ K(3)
1682
+ m (Ο•m)
1683
+ οΏ½
1684
+ Pn
1685
+ οΏ½
1686
+ zS
1687
+ n βˆ’ Ο•m
1688
+ οΏ½οΏ½3 = O
1689
+ οΏ½
1690
+ h6οΏ½
1691
+ ,
1692
+ K(3)
1693
+ m (Ο•m)
1694
+ οΏ½οΏ½
1695
+ Pn
1696
+ οΏ½
1697
+ zS
1698
+ n βˆ’ Ο•m
1699
+ οΏ½οΏ½2 , (I βˆ’ Pn) Ο•m
1700
+ οΏ½
1701
+ = O
1702
+ οΏ½
1703
+ h5οΏ½
1704
+ ,
1705
+ K(3)
1706
+ m (Ο•m)
1707
+ οΏ½
1708
+ Pn
1709
+ οΏ½
1710
+ zS
1711
+ n βˆ’ Ο•m
1712
+ οΏ½
1713
+ , ((I βˆ’ Pn) Ο•m)2οΏ½
1714
+ = O
1715
+ οΏ½
1716
+ h4οΏ½
1717
+ .
1718
+
1719
+ Section 4. Asymptotic Error Analysis
1720
+ 15
1721
+ Note that, we have used the fact ˜h ≀ h in the above three expressions. On the other hand,
1722
+ from (3.13), (3.19) we have
1723
+ K(3)
1724
+ m (Ο•m) ((I βˆ’ Pn) Ο•m)3 = K(3)(Ο•) ((I βˆ’ Pn) Ο•)3 + O
1725
+ οΏ½
1726
+ ˜h2�
1727
+ .
1728
+ From (4.6), it follows that
1729
+ (4.14)
1730
+ K(3)
1731
+ m (Ο•m) ((I βˆ’ Pn) Ο•m)3 = O
1732
+ οΏ½
1733
+ max
1734
+ οΏ½
1735
+ h4, ˜h2��
1736
+ .
1737
+ Now, combining the results (4.13) - (4.14), we obtain
1738
+ K(3)
1739
+ m (Ο•m)(zG
1740
+ n βˆ’ Ο•m)3 = O
1741
+ οΏ½
1742
+ max
1743
+ οΏ½
1744
+ h4, ˜h2��
1745
+ .
1746
+ Therefore
1747
+ [I βˆ’ Kβ€²
1748
+ m(Ο•m)]βˆ’1 K(3)
1749
+ m (Ο•m)(zG
1750
+ n βˆ’ Ο•m)3 = O
1751
+ οΏ½
1752
+ max
1753
+ οΏ½
1754
+ h4, ˜h2��
1755
+ .
1756
+ Hence follows the result.
1757
+ β–‘
1758
+ Proposition 4.2. Let Ο• ∈ Cr+2([0, 1]). Then for r β‰₯ 1,
1759
+ [I βˆ’ Kβ€²
1760
+ m(Ο•m)]βˆ’1 οΏ½
1761
+ Km(zG
1762
+ n ) βˆ’ Km(Ο•m) βˆ’ Kβ€²
1763
+ m(Ο•m)(zG
1764
+ n βˆ’ Ο•m)
1765
+ οΏ½
1766
+ (s)
1767
+ = 1
1768
+ 2T (Ο•)(s)h2r + O
1769
+ οΏ½
1770
+ max
1771
+ οΏ½
1772
+ h2r+2, ˜h2��
1773
+ ,
1774
+ for all s ∈ [0, 1].
1775
+ Proof. Applying the generalized Taylor’s series expansion of Km about Ο•m in the neigh-
1776
+ bourhood B(Ο•, Η«), we obtain
1777
+ (4.15)
1778
+ Km(zG
1779
+ n ) βˆ’ Km(Ο•m) βˆ’ Kβ€²
1780
+ m(Ο•m)(zG
1781
+ n βˆ’ Ο•m)
1782
+ = 1
1783
+ 2Kβ€²β€²
1784
+ m(Ο•m)(zG
1785
+ n βˆ’ Ο•m)2 + 1
1786
+ 6K(3)
1787
+ m (Ο•m)(zG
1788
+ n βˆ’ Ο•m)3 + R4,m
1789
+ οΏ½
1790
+ zG
1791
+ n βˆ’ Ο•m
1792
+ οΏ½
1793
+ ,
1794
+ where
1795
+ R4,m
1796
+ οΏ½
1797
+ zG
1798
+ n βˆ’ Ο•m
1799
+ οΏ½
1800
+ =
1801
+ οΏ½ 1
1802
+ 0
1803
+ (1 βˆ’ ΞΈ)3
1804
+ 3!
1805
+ K(4)
1806
+ m
1807
+ οΏ½
1808
+ Ο•m + ΞΈ(zG
1809
+ n βˆ’ Ο•m)
1810
+ οΏ½
1811
+ (zG
1812
+ n βˆ’ Ο•m)4 dΞΈ.
1813
+ Note that for any x ∈ B(Ο•, Η«), v ∈ X ,
1814
+ K(4)
1815
+ m (x)v4(s) = h
1816
+ p
1817
+ n
1818
+ οΏ½
1819
+ j=1
1820
+ ρ
1821
+ οΏ½
1822
+ q=1
1823
+ p
1824
+ οΏ½
1825
+ Ξ½=1
1826
+ wq
1827
+ βˆ‚4ΞΊ
1828
+ βˆ‚u4
1829
+ οΏ½
1830
+ s, Β΅j
1831
+ qΞ½, x
1832
+ οΏ½
1833
+ Β΅j
1834
+ qΞ½
1835
+ οΏ½οΏ½
1836
+ v4οΏ½
1837
+ Β΅j
1838
+ qΞ½
1839
+ οΏ½
1840
+ ,
1841
+ s ∈ [0, 1].
1842
+ It follows that
1843
+ οΏ½οΏ½K(4)
1844
+ m (x)v4οΏ½οΏ½
1845
+ ∞ ≀
1846
+ 
1847
+ 
1848
+ ο£­
1849
+ sup
1850
+ s,t∈[0,1]
1851
+ |u|≀βˆ₯Ο•βˆ₯∞+Η«
1852
+ οΏ½οΏ½οΏ½οΏ½
1853
+ βˆ‚4ΞΊ
1854
+ βˆ‚u4 (s, t, u)
1855
+ οΏ½οΏ½οΏ½οΏ½
1856
+ ο£Ά
1857
+ ο£·
1858
+ ο£Έ βˆ₯vβˆ₯4
1859
+ ∞ = C5 βˆ₯vβˆ₯4
1860
+ ∞ .
1861
+ Since Ο•m and zG
1862
+ n ∈ B(Ο•, Η«), Ο•m + ΞΈ(zG
1863
+ n βˆ’ Ο•m) ∈ B(Ο•, Η«) and therefore
1864
+ οΏ½οΏ½K(4)
1865
+ m
1866
+ οΏ½
1867
+ Ο•m + ΞΈ(zG
1868
+ n βˆ’ Ο•m)
1869
+ οΏ½
1870
+ (zG
1871
+ n βˆ’ Ο•m)4οΏ½οΏ½
1872
+ ∞ ≀ C5
1873
+ οΏ½οΏ½zG
1874
+ n βˆ’ Ο•m
1875
+ οΏ½οΏ½4
1876
+ ∞ = O
1877
+ οΏ½
1878
+ h4rοΏ½
1879
+ .
1880
+ It follows that
1881
+ R4,m
1882
+ οΏ½
1883
+ zG
1884
+ n βˆ’ Ο•m
1885
+ οΏ½
1886
+ = O
1887
+ οΏ½
1888
+ h4rοΏ½
1889
+ .
1890
+
1891
+ 16
1892
+ G. RAKSHIT
1893
+ Using the resolvent identity (3.17) and (4.2), we obtain
1894
+ [I βˆ’ Kβ€²
1895
+ m(Ο•m)]βˆ’1 Kβ€²β€²
1896
+ m(Ο•m)(zG
1897
+ n βˆ’Ο•m)2 = [I βˆ’ Kβ€²(Ο•)]βˆ’1 Kβ€²β€²
1898
+ m(Ο•m)(zG
1899
+ n βˆ’Ο•m)2 +O
1900
+ οΏ½
1901
+ ˜h2�
1902
+ .
1903
+ By (4.12), it follows that
1904
+ [I βˆ’ Kβ€²
1905
+ m(Ο•m)]βˆ’1 Kβ€²β€²
1906
+ m(Ο•m)(zG
1907
+ n βˆ’ Ο•m)2 = T (Ο•)h2r + O
1908
+ οΏ½
1909
+ max
1910
+ οΏ½
1911
+ h2r+2, ˜h2��
1912
+ .
1913
+ From the Lemma 4.3, we have
1914
+ (4.16)
1915
+ [I βˆ’ Kβ€²
1916
+ m(Ο•m)]βˆ’1 K(3)
1917
+ m (Ο•m)(zG
1918
+ n βˆ’ Ο•m)3 =
1919
+ ο£±
1920
+ ο£²
1921
+ ο£³
1922
+ O
1923
+ οΏ½
1924
+ max
1925
+ οΏ½
1926
+ h4, ˜h2��
1927
+ ,
1928
+ r = 1,
1929
+ O
1930
+ οΏ½
1931
+ max
1932
+ οΏ½
1933
+ h3r, ˜h6��
1934
+ , r β‰₯ 2.
1935
+ Combining the results from (4.15) to (4.16), we obtain for r β‰₯ 1,
1936
+ [I βˆ’ Kβ€²
1937
+ m(Ο•m)]βˆ’1 οΏ½
1938
+ Km(zG
1939
+ n ) βˆ’ Km(Ο•m) βˆ’ Kβ€²
1940
+ m(Ο•m)(zG
1941
+ n βˆ’ Ο•m)
1942
+ οΏ½
1943
+ = 1
1944
+ 2T (Ο•)h2r + O
1945
+ οΏ½
1946
+ max
1947
+ οΏ½
1948
+ h2r+2, ˜h2��
1949
+ ,
1950
+ which completes the proof.
1951
+ β–‘
1952
+ Lemma 4.4. If v ∈ X , then for r β‰₯ 1,
1953
+ max
1954
+ 0≀i≀n |Kβ€²
1955
+ m(Ο•m)(I βˆ’ Pn)v(ti)|
1956
+ ≀
1957
+ C7 βˆ₯(I βˆ’ Pn)vβˆ₯∞ hr,
1958
+ where C7 is a constant independent of h.
1959
+ Proof. We write
1960
+ (4.17)
1961
+ Kβ€²
1962
+ m(Ο•m)(I βˆ’ Pn)v = Kβ€²
1963
+ m(Ο•)(I βˆ’ Pn)v + [Kβ€²
1964
+ m(Ο•m) βˆ’ Kβ€²
1965
+ m(Ο•)](I βˆ’ Pn)v.
1966
+ For fixed s ∈ [0, 1], let
1967
+ β„“βˆ—,s(t) = β„“βˆ—(s, t) = β„“(s, t, Ο•(t)) = βˆ‚ΞΊ
1968
+ βˆ‚u(s, t, Ο•(t)),
1969
+ t ∈ [0, 1].
1970
+ From the definition of Kβ€²
1971
+ m(Ο•) and the discrete inner product, we have
1972
+ Kβ€²
1973
+ m(Ο•)(I βˆ’ Pn)v(s)
1974
+ =
1975
+ n
1976
+ οΏ½
1977
+ j=1
1978
+ βŸ¨β„“βˆ—,s , (I βˆ’ Pn,j)vβŸ©βˆ†j,m .
1979
+ Since Pn,j is self-adjoint on C(βˆ†j), so as I βˆ’ Pn,j. Therefore
1980
+ Kβ€²
1981
+ m(Ο•)(I βˆ’ Pn)v(s)
1982
+ =
1983
+ n
1984
+ οΏ½
1985
+ j=1
1986
+ ⟨(I βˆ’ Pn,j)β„“βˆ—,s , (I βˆ’ Pn,j)vβŸ©βˆ†j,m .
1987
+ Note that, if s = ti for some i ∈ {0, 1, . . . , n}, then β„“βˆ—,s ∈ Cr[tjβˆ’1, tj] for all j = 1, . . . , n.
1988
+ Hence from (3.9),
1989
+ βˆ₯(I βˆ’ Pn,j)β„“βˆ—,tiβˆ₯βˆ†j,∞
1990
+ ≀
1991
+ C1
1992
+ οΏ½
1993
+ sup
1994
+ t∈[tjβˆ’1,tj]
1995
+ |D(0,r)β„“βˆ—(ti, t)|
1996
+ οΏ½
1997
+ hr
1998
+ Thus,
1999
+ max
2000
+ 0≀i≀n |Kβ€²
2001
+ m(Ο•)(I βˆ’ Pn)v(ti)| ≀
2002
+ n
2003
+ οΏ½
2004
+ j=1
2005
+ βˆ₯(I βˆ’ Pn,j)β„“m,sβˆ₯βˆ†j,∞ βˆ₯(I βˆ’ Pn,j)vβˆ₯βˆ†j,∞h
2006
+ ≀ C6 βˆ₯(I βˆ’ Pn)vβˆ₯∞ hr,
2007
+
2008
+ Section 4. Asymptotic Error Analysis
2009
+ 17
2010
+ where C6 is a constant independent of h. Now, from (3.16), (4.17) and the above estimate,
2011
+ we obtain
2012
+ max
2013
+ 0≀i≀n |Kβ€²
2014
+ m(Ο•m)(I βˆ’ Pn)v(ti)| ≀ C7 βˆ₯(I βˆ’ Pn)vβˆ₯∞ max
2015
+ οΏ½
2016
+ hr, ˜h2�
2017
+ ,
2018
+ where C7 = C4 + C6. Since hr β‰₯ ˜h2, the result follows.
2019
+ β–‘
2020
+ Recall that Lm = [I βˆ’ Kβ€²
2021
+ m(Ο•m)]βˆ’1 Kβ€²
2022
+ m(Ο•m). Therefore, the proof of the following result
2023
+ is similar to that of the above lemma.
2024
+ Corollary 4.2. If v ∈ X , then for r β‰₯ 1,
2025
+ max
2026
+ 0≀i≀n|Lm(I βˆ’ Pn)v(ti)|
2027
+ ≀
2028
+ C8 βˆ₯(I βˆ’ Pn)vβˆ₯∞ hr,
2029
+ where C8 is a constant independent of h.
2030
+ Lemma 4.5. Let v ∈ X . If r = 1, that is, when the range of Pn is the space of piecewise
2031
+ polynomials of degree zero, then
2032
+ βˆ₯(I βˆ’ Pn)Kβ€²β€²
2033
+ m(Ο•)(v, v)βˆ₯∞ ≀ C9h βˆ₯vβˆ₯2
2034
+ ∞ ,
2035
+ where C9 is a constant independent of h.
2036
+ Proof. Given that Xn is the space of piecewise constant functions with respect to the parti-
2037
+ tion (3.2). Note that
2038
+ βˆ₯(I βˆ’ Pn)Kβ€²β€²
2039
+ m(Ο•)(v, v)βˆ₯∞
2040
+ =
2041
+ max
2042
+ 1≀j≀n
2043
+ sup
2044
+ s∈[tjβˆ’1,tj]
2045
+ |(I βˆ’ Pn,j)Kβ€²β€²
2046
+ m(Ο•)(v, v)(s)|.
2047
+ Let s ∈ [tjβˆ’1, tj]. Since the Legendre polynomial of zero L0(t) = 1 for all t ∈ [0, 1], we
2048
+ have from (3.11),
2049
+ (4.18)
2050
+ (I βˆ’ Pn)Kβ€²β€²
2051
+ m(Ο•)(v, v)(s)
2052
+ = 1
2053
+ p
2054
+ p
2055
+ οΏ½
2056
+ Ξ½=1
2057
+ ρ
2058
+ οΏ½
2059
+ q=1
2060
+ wq [Kβ€²β€²
2061
+ m(Ο•)(v, v)(s) βˆ’ Kβ€²β€²
2062
+ m(Ο•)(v, v)(tjβˆ’1 + Β΅qΞ½h)] .
2063
+ We also have
2064
+ Kβ€²β€²
2065
+ m(Ο•)(v, v)(s) βˆ’ Kβ€²β€²
2066
+ m(Ο•)(v, v)(tjβˆ’1 + Β΅qΞ½h)
2067
+ = h
2068
+ p
2069
+ n
2070
+ οΏ½
2071
+ k=1
2072
+ ρ
2073
+ οΏ½
2074
+ q=1
2075
+ p
2076
+ οΏ½
2077
+ Ξ½=1
2078
+ wq
2079
+ οΏ½βˆ‚2ΞΊ
2080
+ βˆ‚u2
2081
+ οΏ½
2082
+ s, Β΅k
2083
+ qΞ½, Ο•
2084
+ οΏ½
2085
+ Β΅k
2086
+ qΞ½
2087
+ οΏ½οΏ½
2088
+ βˆ’ βˆ‚2ΞΊ
2089
+ βˆ‚u2
2090
+ οΏ½
2091
+ Β΅j
2092
+ qΞ½, Β΅k
2093
+ qΞ½, Ο•
2094
+ οΏ½
2095
+ Β΅k
2096
+ qΞ½
2097
+ οΏ½οΏ½οΏ½
2098
+ v2οΏ½
2099
+ Β΅k
2100
+ qΞ½
2101
+ οΏ½
2102
+ ,
2103
+ where Β΅j
2104
+ qΞ½ = tjβˆ’1 + Β΅qΞ½h. For fixed s ∈ [0, 1], let
2105
+ Ξ»βˆ—,s(t) = Ξ»βˆ—(s, t) = Ξ»(s, t, Ο•(t)) = βˆ‚2ΞΊ
2106
+ βˆ‚u2 (s, t, Ο•(t)),
2107
+ t ∈ [0, 1].
2108
+ Then, by (3.5)
2109
+ Kβ€²β€²
2110
+ m(Ο•)(v, v)(s) βˆ’ Kβ€²β€²
2111
+ m(Ο•)(v, v)(Β΅j
2112
+ qΞ½) =
2113
+ n
2114
+ οΏ½
2115
+ k=1
2116
+ οΏ½
2117
+ Ξ»βˆ—,s βˆ’ Ξ»βˆ—,Β΅iqΞ½, v2οΏ½
2118
+ βˆ†k,m
2119
+ =
2120
+ n
2121
+ οΏ½
2122
+ k=1
2123
+ kΜΈ=j
2124
+ οΏ½
2125
+ Ξ»βˆ—,s βˆ’ Ξ»βˆ—,Β΅j
2126
+ qΞ½ , v2οΏ½
2127
+ βˆ†k,m +
2128
+ οΏ½
2129
+ Ξ»βˆ—,s βˆ’ Ξ»βˆ—,Β΅j
2130
+ qΞ½ , v2οΏ½
2131
+ βˆ†j,m .
2132
+ (4.19)
2133
+
2134
+ 18
2135
+ G. RAKSHIT
2136
+ First consider the case when k ̸= j. Applying Mean Value Theorem on the first component
2137
+ of Ξ»βˆ—(Β·, Β·) in the interval [s, Β΅j
2138
+ qΞ½], we obtain
2139
+ οΏ½
2140
+ Ξ»βˆ—,s βˆ’ Ξ»βˆ—,Β΅j
2141
+ qΞ½ , v2οΏ½
2142
+ βˆ†k,m = (s βˆ’ Β΅j
2143
+ qΞ½)
2144
+ οΏ½
2145
+ D(1,0)Ξ»βˆ—(ΞΈj
2146
+ qΞ½, Β·) , v2οΏ½
2147
+ βˆ†k,m ,
2148
+ for some ΞΈj
2149
+ qΞ½ ∈ (tjβˆ’1, tj), and the function D(1,0)Ξ»βˆ—(s, t), t ∈ [tkβˆ’1, tk] is given by
2150
+ D(1,0)Ξ»βˆ—(s, t) =
2151
+ οΏ½
2152
+ D(1,0)Ξ»1,βˆ—(s, t) =
2153
+ βˆ‚
2154
+ βˆ‚sΞ»1(s, t, Ο•(t)),
2155
+ 0 ≀ t ≀ s ≀ 1,
2156
+ D(1,0)Ξ»2,βˆ—(s, t) =
2157
+ βˆ‚
2158
+ βˆ‚sΞ»2(s, t, Ο•(t)),
2159
+ 0 ≀ s ≀ t ≀ 1.
2160
+ Therefore, for k ΜΈ= j,
2161
+ οΏ½οΏ½οΏ½οΏ½
2162
+ οΏ½
2163
+ Ξ»βˆ—,s βˆ’ Ξ»βˆ—,Β΅j
2164
+ qΞ½ , v2οΏ½
2165
+ βˆ†k,m
2166
+ οΏ½οΏ½οΏ½οΏ½
2167
+ ≀
2168
+ οΏ½οΏ½s βˆ’ Β΅j
2169
+ qΞ½
2170
+ οΏ½οΏ½
2171
+ οΏ½
2172
+ sup
2173
+ sΜΈ=t
2174
+ οΏ½οΏ½D(1,0)Ξ»βˆ—(s, t)
2175
+ οΏ½οΏ½
2176
+ οΏ½
2177
+ βˆ₯vβˆ₯2
2178
+ ∞ h
2179
+ ≀
2180
+ οΏ½
2181
+ sup
2182
+ sΜΈ=t
2183
+ οΏ½οΏ½D(1,0)Ξ»βˆ—(s, t)
2184
+ οΏ½οΏ½
2185
+ οΏ½
2186
+ βˆ₯vβˆ₯2
2187
+ ∞ h2,
2188
+ where
2189
+ sup
2190
+ sΜΈ=t
2191
+ |D(1,0)Ξ»βˆ—(s, t)|
2192
+ = max
2193
+ οΏ½
2194
+ sup
2195
+ 0≀t<s≀1
2196
+ |D(1,0)Ξ»1,βˆ—(s, t)| ,
2197
+ sup
2198
+ 0≀s<t≀1
2199
+ |D(1,0)Ξ»2,βˆ—(s, t)|
2200
+ οΏ½
2201
+ = max
2202
+ ο£±
2203
+ 
2204
+ ο£²
2205
+ 
2206
+ ο£³
2207
+ sup
2208
+ 0≀t<s≀1
2209
+ |u|≀βˆ₯Ο•βˆ₯∞
2210
+ |D(1,0,2)ΞΊ1(s, t, u)| ,
2211
+ sup
2212
+ 0≀s<t≀1
2213
+ |u|≀βˆ₯Ο•βˆ₯∞
2214
+ |D(1,0,2)ΞΊ2(s, t, u)|
2215
+ ο£Ό
2216
+ 
2217
+ ο£½
2218
+ 
2219
+ ο£Ύ
2220
+ .
2221
+ On the other hand,
2222
+ |
2223
+ οΏ½
2224
+ Ξ»βˆ—,s βˆ’ Ξ»βˆ—,Β΅j
2225
+ qΞ½ , v2οΏ½
2226
+ βˆ†j,m| ≀ 2
2227
+ οΏ½
2228
+ sup
2229
+ 0≀s,t≀1
2230
+ |Ξ»βˆ—(s, t)|
2231
+ οΏ½
2232
+ βˆ₯vβˆ₯2
2233
+ ∞ h,
2234
+ where
2235
+ sup
2236
+ 0≀s,t≀1
2237
+ |Ξ»βˆ—(s, t)| =
2238
+ sup
2239
+ 0≀s,t≀1
2240
+ |u|≀βˆ₯Ο•βˆ₯∞
2241
+ |D(0,0,2)ΞΊ(s, t, u)|. Then, from (4.19) we obtain
2242
+ |Kβ€²β€²
2243
+ m(Ο•)(v, v)(s) βˆ’ Kβ€²β€²
2244
+ m(Ο•)(v, v)(Β΅j
2245
+ qΞ½)|
2246
+ ≀
2247
+ 
2248
+ 
2249
+ ο£­
2250
+ n
2251
+ οΏ½
2252
+ k=1
2253
+ kΜΈ=j
2254
+ οΏ½
2255
+ sup
2256
+ sΜΈ=t
2257
+ |D(1,0)Ξ»βˆ—(s, t)|
2258
+ οΏ½
2259
+ βˆ₯vβˆ₯2
2260
+ ∞ h2
2261
+ ο£Ά
2262
+ ο£·
2263
+ ο£Έ + 2
2264
+ οΏ½
2265
+ sup
2266
+ 0≀s,t≀1
2267
+ |Ξ»βˆ—(s, t)|
2268
+ οΏ½
2269
+ βˆ₯vβˆ₯2
2270
+ ∞ h.
2271
+ It follows that
2272
+ |Kβ€²β€²
2273
+ m(Ο•)(v, v)(s) βˆ’ Kβ€²β€²
2274
+ m(Ο•)(v, v)(Β΅j
2275
+ qΞ½)| ≀ C9 βˆ₯vβˆ₯2 h,
2276
+ where C9 =
2277
+ οΏ½
2278
+ sup
2279
+ sΜΈ=t
2280
+ |D(1,0)Ξ»βˆ—(s, t)|
2281
+ οΏ½
2282
+ + 2
2283
+ οΏ½
2284
+ sup
2285
+ 0≀s,t≀1
2286
+ |Ξ»βˆ—(s, t)|
2287
+ οΏ½
2288
+ .
2289
+ The result now follows from (4.18) and the above estimate.
2290
+ β–‘
2291
+ Corollary 4.3. Let v ∈ X . If r = 1, that is, when the range of Pn is the space of piecewise
2292
+ polynomials of degree zero, then
2293
+ βˆ₯(I βˆ’ Pn)Kβ€²
2294
+ m(Ο•)vβˆ₯∞ ≀ C10h βˆ₯vβˆ₯∞ ,
2295
+ where C10 is a constant independent of h.
2296
+
2297
+ Section 4. Asymptotic Error Analysis
2298
+ 19
2299
+ Proof. The proof is similar to that of Lemma 4.5.
2300
+ β–‘
2301
+ Proposition 4.3. Let ti be any point of the partition βˆ†(n) defined by (3.2). Then
2302
+ Lm(I βˆ’ Pn)
2303
+ οΏ½
2304
+ Km(zG
2305
+ n ) βˆ’ Km(Ο•m) βˆ’ Kβ€²
2306
+ m(Ο•m)(zG
2307
+ n βˆ’ Ο•m)
2308
+ οΏ½
2309
+ (ti)
2310
+ =
2311
+ ο£±
2312
+ ο£²
2313
+ ο£³
2314
+ O (h4) ,
2315
+ r = 1,
2316
+ O
2317
+ οΏ½
2318
+ max
2319
+ οΏ½
2320
+ h3r, hr˜h4��
2321
+ ,
2322
+ r β‰₯ 2.
2323
+ Proof. Generalized Taylor’s series expansion gives
2324
+ (4.20)
2325
+ Lm(I βˆ’ Pn)
2326
+ οΏ½
2327
+ Km(zG
2328
+ n ) βˆ’ Km(Ο•m) βˆ’ Kβ€²
2329
+ m(Ο•m)(zG
2330
+ n βˆ’ Ο•m)
2331
+ οΏ½
2332
+ = 1
2333
+ 2Lm(I βˆ’ Pn)Kβ€²β€²
2334
+ m(Ο•m)(zG
2335
+ n βˆ’ Ο•m)2 + Lm(I βˆ’ Pn)R3,m
2336
+ οΏ½
2337
+ zG
2338
+ n βˆ’ Ο•m
2339
+ οΏ½
2340
+ ,
2341
+ where
2342
+ R3,m
2343
+ οΏ½
2344
+ zG
2345
+ n βˆ’ Ο•m
2346
+ οΏ½
2347
+ =
2348
+ οΏ½ 1
2349
+ 0
2350
+ (1 βˆ’ ΞΈ)2
2351
+ 2!
2352
+ K(3)
2353
+ m
2354
+ οΏ½
2355
+ Ο•m + ΞΈ(zG
2356
+ n βˆ’ Ο•m)
2357
+ οΏ½
2358
+ (zG
2359
+ n βˆ’ Ο•m)3 dΞΈ.
2360
+ It follows that
2361
+ οΏ½οΏ½R3,m
2362
+ οΏ½
2363
+ zG
2364
+ n βˆ’ Ο•m
2365
+ οΏ½οΏ½οΏ½
2366
+ ∞ ≀ 1
2367
+ 6
2368
+ 
2369
+ 
2370
+ ο£­
2371
+ sup
2372
+ s,t∈[0,1]
2373
+ |u|≀βˆ₯Ο•βˆ₯∞+Η«
2374
+ οΏ½οΏ½οΏ½οΏ½
2375
+ βˆ‚3ΞΊ
2376
+ βˆ‚u3 (s, t, u)
2377
+ οΏ½οΏ½οΏ½οΏ½
2378
+ ο£Ά
2379
+ ο£·
2380
+ ο£Έ
2381
+ οΏ½οΏ½zG
2382
+ n βˆ’ Ο•m
2383
+ οΏ½οΏ½3
2384
+ ∞ .
2385
+ Therefore, by (1.4)
2386
+ οΏ½οΏ½R3,m
2387
+ οΏ½
2388
+ zG
2389
+ n βˆ’ Ο•m
2390
+ οΏ½οΏ½οΏ½
2391
+ ∞ = O
2392
+ οΏ½
2393
+ max
2394
+ οΏ½
2395
+ h3r, ˜h6��
2396
+ .
2397
+ Since βˆ₯I βˆ’ Pnβˆ₯ ≀ 1 + βˆ₯Pnβˆ₯ < ∞, from Corollary 4.2, it is easy to see that
2398
+ Lm(I βˆ’ Pn)R3,m
2399
+ οΏ½
2400
+ zG
2401
+ n βˆ’ Ο•m
2402
+ οΏ½
2403
+ (ti) = O
2404
+ οΏ½
2405
+ max
2406
+ οΏ½
2407
+ h4r, hr˜h6��
2408
+ .
2409
+ (4.21)
2410
+ First consider the case r β‰₯ 2. Since βˆ₯Kβ€²β€²
2411
+ m(Ο•m)βˆ₯ < ∞ and βˆ₯I βˆ’ Pnβˆ₯∞ < ∞, by (1.4) and
2412
+ the Corollary 4.2, we have
2413
+ 1
2414
+ 2Lm(I βˆ’ Pn)Kβ€²β€²
2415
+ m(Ο•m)(zG
2416
+ n βˆ’ Ο•m)2(ti) = O
2417
+ οΏ½
2418
+ max
2419
+ οΏ½
2420
+ h3r, hr˜h4��
2421
+ ,
2422
+ r β‰₯ 2.
2423
+ When r = 1, we write
2424
+ Lm(I βˆ’ Pn)Kβ€²β€²
2425
+ m(Ο•m)(zG
2426
+ n βˆ’ Ο•m)2
2427
+ = Lm(I βˆ’ Pn)
2428
+ οΏ½
2429
+ Kβ€²β€²
2430
+ m(Ο•m) βˆ’ Kβ€²β€²
2431
+ m(Ο•)
2432
+ οΏ½
2433
+ (zG
2434
+ n βˆ’ Ο•m)2 + Lm(I βˆ’ Pn)Kβ€²β€²
2435
+ m(Ο•)(zG
2436
+ n βˆ’ Ο•m)2.
2437
+ By (1.4), (3.13) and the Lemma 3.3, we have
2438
+ Lm(I βˆ’ Pn) [Kβ€²β€²
2439
+ m(Ο•m) βˆ’ Kβ€²β€²
2440
+ m(Ο•)] (zG
2441
+ n βˆ’ Ο•m)2 = O
2442
+ οΏ½
2443
+ h4οΏ½
2444
+ .
2445
+ On the other hand
2446
+ Lm(I βˆ’ Pn)Kβ€²β€²
2447
+ m(Ο•)(zG
2448
+ n βˆ’ Ο•m)2(ti) =
2449
+ n
2450
+ οΏ½
2451
+ j=1
2452
+ οΏ½
2453
+ β„“m,ti , (I βˆ’ Pn,j)Kβ€²β€²
2454
+ m(Ο•)(zG
2455
+ n βˆ’ Ο•m)2οΏ½
2456
+ βˆ†j,m .
2457
+
2458
+ 20
2459
+ G. RAKSHIT
2460
+ Since I βˆ’ Pn,j is self-adjoint,
2461
+ Lm(I βˆ’ Pn)Kβ€²β€²
2462
+ m(Ο•)(zG
2463
+ n βˆ’ Ο•m)2(ti)
2464
+ =
2465
+ n
2466
+ οΏ½
2467
+ j=1
2468
+ οΏ½
2469
+ (I βˆ’ Pn,j)β„“m,ti , (I βˆ’ Pn,j)Kβ€²β€²
2470
+ m(Ο•)(zG
2471
+ n βˆ’ Ο•m)2οΏ½
2472
+ βˆ†j,m .
2473
+ It follows that
2474
+ max
2475
+ 0≀i≀n
2476
+ οΏ½οΏ½Lm(I βˆ’ Pn)Kβ€²β€²
2477
+ m(Ο•)(zG
2478
+ n βˆ’ Ο•m)2(ti)
2479
+ οΏ½οΏ½
2480
+ ≀
2481
+ n
2482
+ οΏ½
2483
+ j=1
2484
+ βˆ₯(I βˆ’ Pn,j)β„“m,tiβˆ₯βˆ†j,∞
2485
+ οΏ½οΏ½(I βˆ’ Pn,j)Kβ€²β€²
2486
+ m(Ο•)(zG
2487
+ n βˆ’ Ο•m)2οΏ½οΏ½
2488
+ βˆ†j,∞ h.
2489
+ By Corollary 4.2 and Lemma 4.5, we obtain
2490
+ max
2491
+ 0≀i≀n|Lm(I βˆ’ Pn)Kβ€²β€²
2492
+ m(Ο•)(zG
2493
+ n βˆ’ Ο•m)2(ti)| = O
2494
+ οΏ½
2495
+ h4οΏ½
2496
+ ,
2497
+ r = 1.
2498
+ Therefore
2499
+ 1
2500
+ 2Lm(I βˆ’ Pn)Kβ€²β€²
2501
+ m(Ο•m)(zG
2502
+ n βˆ’ Ο•m)2(ti) =
2503
+ ο£±
2504
+ ο£²
2505
+ ο£³
2506
+ O (h4) ,
2507
+ r = 1,
2508
+ O
2509
+ οΏ½
2510
+ max
2511
+ οΏ½
2512
+ h3r, hr˜h4��
2513
+ ,
2514
+ r β‰₯ 2.
2515
+ (4.22)
2516
+ Then combing (4.20), (4.21) and (4.22), we obtain
2517
+ Lm(I βˆ’ Pn)
2518
+ οΏ½
2519
+ Km(zG
2520
+ n ) βˆ’ Km(Ο•m) βˆ’ Kβ€²
2521
+ m(Ο•m)(zG
2522
+ n βˆ’ Ο•m)
2523
+ οΏ½
2524
+ (ti)
2525
+ =
2526
+ ο£±
2527
+ ο£²
2528
+ ο£³
2529
+ O (h4) ,
2530
+ r = 1,
2531
+ O
2532
+ οΏ½
2533
+ max
2534
+ οΏ½
2535
+ h3r, hr˜h4��
2536
+ ,
2537
+ r β‰₯ 2.
2538
+ This follows the result.
2539
+ β–‘
2540
+ We quote the following result from By [17, Proposition 1, Proposition 6], which will be
2541
+ used in the next proposition.
2542
+ (4.23)
2543
+ βˆ₯(I βˆ’ Pn)Kβ€²
2544
+ m(Ο•)(I βˆ’ Pn)Ο•βˆ₯∞ =
2545
+ οΏ½
2546
+ O (h3) ,
2547
+ r = 1,
2548
+ O (hr+2) ,
2549
+ r β‰₯ 2.
2550
+ Proposition 4.4. If Ο•m and zG
2551
+ n are respectively the NystrΓΆm and the discrete Galerkin ap-
2552
+ proximation of Ο•, then
2553
+ Lm(I βˆ’ Pn)Kβ€²
2554
+ m(Ο•m)(zG
2555
+ n βˆ’ Ο•m)(ti) = O
2556
+ οΏ½
2557
+ max
2558
+ οΏ½
2559
+ h2r+2, ˜h2��
2560
+ .
2561
+ Proof. Adding and subtracting Kβ€²
2562
+ m(Ο•), we have
2563
+ Lm(I βˆ’ Pn)Kβ€²
2564
+ m(Ο•m)(zG
2565
+ n βˆ’ Ο•m)
2566
+ = Lm(I βˆ’ Pn) [Kβ€²
2567
+ m(Ο•m) βˆ’ Kβ€²
2568
+ m(Ο•)] (zG
2569
+ n βˆ’ Ο•m) + Lm(I βˆ’ Pn)Kβ€²
2570
+ m(Ο•)(zG
2571
+ n βˆ’ Ο•m).
2572
+ Then, using (1.4), (3.16) and the Corollary 4.2, we obtain for r β‰₯ 1,
2573
+ (4.24)
2574
+ οΏ½οΏ½Lm(I βˆ’ Pn) [Kβ€²
2575
+ m(Ο•m) βˆ’ Kβ€²
2576
+ m(Ο•)] (zG
2577
+ n βˆ’ Ο•m)(ti)
2578
+ οΏ½οΏ½
2579
+ ≀ C4C8 (1 + βˆ₯Pnβˆ₯) hr˜h2 οΏ½
2580
+ max
2581
+ οΏ½
2582
+ hr, ˜h2��
2583
+
2584
+ Section 4. Asymptotic Error Analysis
2585
+ 21
2586
+ Note that
2587
+ zG
2588
+ n βˆ’ Ο•m = PnzS
2589
+ n βˆ’ Ο•m = PnzS
2590
+ n βˆ’ PnΟ• + PnΟ• βˆ’ Ο• + Ο• βˆ’ Ο•m
2591
+ = Pn
2592
+ οΏ½
2593
+ zS
2594
+ n βˆ’ Ο•
2595
+ οΏ½
2596
+ βˆ’ (I βˆ’ Pn) Ο• + (Ο• βˆ’ Ο•m) .
2597
+ Then
2598
+ Lm(I βˆ’ Pn)Kβ€²
2599
+ m(Ο•)(zG
2600
+ n βˆ’ Ο•m) = Lm(I βˆ’ Pn)Kβ€²
2601
+ m(Ο•)Pn
2602
+ οΏ½
2603
+ zS
2604
+ n βˆ’ Ο•
2605
+ οΏ½
2606
+ βˆ’ Lm(I βˆ’ Pn)Kβ€²
2607
+ m(Ο•) (I βˆ’ Pn) Ο•
2608
+ + Lm(I βˆ’ Pn)Kβ€²
2609
+ m(Ο•) (Ο• βˆ’ Ο•m) .
2610
+ By the Corollary 4.2
2611
+ οΏ½οΏ½Lm(I βˆ’ Pn)Kβ€²
2612
+ m(Ο•)Pn
2613
+ οΏ½
2614
+ zS
2615
+ n βˆ’ Ο•
2616
+ οΏ½
2617
+ (ti)
2618
+ οΏ½οΏ½ ≀ C8
2619
+ οΏ½οΏ½(I βˆ’ Pn) Kβ€²
2620
+ m(Ο•)Pn
2621
+ οΏ½
2622
+ zS
2623
+ n βˆ’ Ο•
2624
+ οΏ½οΏ½οΏ½ hr,
2625
+ then by (1.5) and Corollary 4.3, we obtain
2626
+ οΏ½οΏ½Lm(I βˆ’ Pn)Kβ€²
2627
+ m(Ο•)Pn
2628
+ οΏ½
2629
+ zS
2630
+ n βˆ’ Ο•
2631
+ οΏ½
2632
+ (ti)
2633
+ οΏ½οΏ½ = O
2634
+ οΏ½
2635
+ max
2636
+ οΏ½
2637
+ h2r+2, ˜h2��
2638
+ ,
2639
+ for r β‰₯ 1.
2640
+ Also, the Corollary 4.2 and (4.23) implies
2641
+ Lm(I βˆ’ Pn)Kβ€²
2642
+ m(Ο•) (I βˆ’ Pn) Ο•(ti) = O
2643
+ οΏ½
2644
+ h2r+2οΏ½
2645
+ ,
2646
+ for r β‰₯ 1.
2647
+ It is easy to see (from (3.13) and Corollary 4.2) that
2648
+ Lm(I βˆ’ Pn)Kβ€²
2649
+ m(Ο•) (Ο• βˆ’ Ο•m) = O
2650
+ οΏ½
2651
+ hr˜h2�
2652
+ ,
2653
+ for r β‰₯ 1.
2654
+ Therefore, for r β‰₯ 1,
2655
+ Lm(I βˆ’ Pn)Kβ€²
2656
+ m(Ο•)(zG
2657
+ n βˆ’ Ο•m)(ti) = O
2658
+ οΏ½
2659
+ max
2660
+ οΏ½
2661
+ h2r+2, ˜h2��
2662
+ .
2663
+ Hence, the result follows from (4.24) and the above equation.
2664
+ β–‘
2665
+ We prove the main theorem as follows.
2666
+ Theorem 4.1. Let K be the Urysohn integral operator with Green’s function type kernel ΞΊ,
2667
+ defined by (1.2). Let Ο• be the unique solution of the equation (1.1). Assume that 1 is not
2668
+ an eigenvalue of Kβ€²(Ο•). Let Xn be the space of piecewise polynomials of degree ≀ r βˆ’ 1
2669
+ with respect to the partition βˆ†(n) := 0 = t0 < t1 < Β· Β· Β· < tn = 1 defined by (3.2). Let
2670
+ Pn : L∞[0, 1] β†’ Xn be the discrete orthogonal projection defined by (3.8) and ˜zS
2671
+ n be the
2672
+ discrete iterated Galerkin approximation of Ο•. Then
2673
+ οΏ½
2674
+ zS
2675
+ n βˆ’ Ο•
2676
+ οΏ½
2677
+ (ti) =
2678
+ οΏ½
2679
+ E2r(Ο•)(ti) + 1
2680
+ 2T (Ο•)(ti)
2681
+ οΏ½
2682
+ h2r + O
2683
+ οΏ½
2684
+ max
2685
+ οΏ½
2686
+ h2r+2, ˜h2��
2687
+ ,
2688
+ where the operators E2r and T are respectively defined by (3.12) and (4.5).
2689
+ Proof. We have from (4.1)
2690
+ zS
2691
+ n βˆ’ Ο• = [I βˆ’ Kβ€²
2692
+ m(Ο•m)]βˆ’1 οΏ½
2693
+ Km(zG
2694
+ n ) βˆ’ Km(Ο•m) βˆ’ Kβ€²
2695
+ m(Ο•m)(zG
2696
+ n βˆ’ Ο•m)
2697
+ οΏ½
2698
+ βˆ’ Lm(I βˆ’ Pn)
2699
+ οΏ½
2700
+ Km(zG
2701
+ n ) βˆ’ Km(Ο•m) βˆ’ Kβ€²
2702
+ m(Ο•m)(zG
2703
+ n βˆ’ Ο•m)
2704
+ οΏ½
2705
+ βˆ’ Lm(I βˆ’ Pn)Kβ€²
2706
+ m(Ο•m)(zG
2707
+ n βˆ’ Ο•m)
2708
+ βˆ’ Lm(I βˆ’ Pn)Ο•m
2709
+ + Ο•m βˆ’ Ο•.
2710
+
2711
+ 22
2712
+ G. RAKSHIT
2713
+ The result now follows from (3.13), Proposition 4.1, Proposition 4.2, Proposition 4.3 and
2714
+ Proposition 4.4.
2715
+ β–‘
2716
+ We now apply Richardson extrapolation to obtain an approximation of Ο• with higher
2717
+ order of convergence. Define
2718
+ zEX
2719
+ n
2720
+ = 24rzS
2721
+ 2n βˆ’ zS
2722
+ n
2723
+ 24r βˆ’ 1
2724
+ .
2725
+ We choose the partitions βˆ†(m) and βˆ†(n) such that m2 β‰₯ n2r+2. Then, it is easy to see from
2726
+ the Theorem 4.1, that
2727
+ (4.25)
2728
+ οΏ½
2729
+ zEX
2730
+ n
2731
+ βˆ’ Ο•
2732
+ οΏ½
2733
+ (ti) = O
2734
+ οΏ½
2735
+ h2r+2οΏ½
2736
+ ,
2737
+ for all i = 1, 2, . . . , n.
2738
+ 5 Numerical results
2739
+ For the numerical results, we consider the following example from [19]. Consider
2740
+ (5.1)
2741
+ Ο•(s) βˆ’
2742
+ οΏ½ 1
2743
+ 0
2744
+ ΞΊ(s, t) [ψ (t, Ο•(t))] dt = f(s),
2745
+ 0 ≀ s ≀ 1,
2746
+ where
2747
+ ΞΊ(s, t) =
2748
+ 1
2749
+ Ξ³ sinh Ξ³
2750
+ οΏ½
2751
+ sinh Ξ³s sinh Ξ³(1 βˆ’ t),
2752
+ 0 ≀ t ≀ s ≀ 1,
2753
+ Ξ³(1 βˆ’ s) sinh Ξ³t,
2754
+ 0 ≀ s ≀ t ≀ 1,
2755
+ with Ξ³ =
2756
+ √
2757
+ 12, and
2758
+ ψ(t, Ο•(t)) = Ξ³2Ο•(t) βˆ’ 2 (Ο•(t))3 ,
2759
+ t ∈ [0, 1].
2760
+ We have f(s) =
2761
+ 1
2762
+ sinh Ξ³
2763
+ οΏ½
2764
+ 2 sinh Ξ³(1 βˆ’ s) + 2
2765
+ 3 sinh Ξ³s
2766
+ οΏ½
2767
+ . The exact solution of (5.1) is given
2768
+ by
2769
+ Ο•(s) =
2770
+ 2
2771
+ 2s + 1,
2772
+ s ∈ [0, 1].
2773
+ Let Xn be the space of piecewise constant functions with respect to the uniform partition
2774
+ βˆ†(n) of the interval [0, 1]. Let Pn : L∞[0, 1] β†’ Xn be the discrete orthogonal projection
2775
+ defined by (3.8).
2776
+ Let ti = iβˆ’1
2777
+ 20 , i = 1, 2, . . . , 21 be the partition points with step size h =
2778
+ 1
2779
+ 20. The numerical
2780
+ quadrature is chosen to be the composite 2 point Gaussian quadrature rule with respect to
2781
+ partition βˆ†(m) with m = n2 subintervals. Then ˜h = h2. Therefore, it is expected from the
2782
+ Theorem 4.1 and equation (4.25), that
2783
+ Η«S
2784
+ n(ti) = |Ο•(ti) βˆ’ zS
2785
+ n(ti)| = O
2786
+ οΏ½
2787
+ h2οΏ½
2788
+ and Η«EX
2789
+ n
2790
+ (ti) = |Ο•(ti) βˆ’ zEX
2791
+ n
2792
+ (ti)| = O
2793
+ οΏ½
2794
+ h4οΏ½
2795
+ ,
2796
+ where
2797
+ zEX
2798
+ n
2799
+ (ti) = 4zS
2800
+ 2n(ti) βˆ’ zS
2801
+ n(ti)
2802
+ 3
2803
+ .
2804
+ Let Ξ΄S and Ξ΄EX be respectively the orders of convergence of zS
2805
+ n and zEX
2806
+ n
2807
+ at the partition
2808
+ points. We expect Ξ΄S = 2 and Ξ΄EX = 4.
2809
+
2810
+ Section 5. Numerical results
2811
+ 23
2812
+ Table 1
2813
+ ti
2814
+ Η«S
2815
+ n(ti) : n = 20
2816
+ Ξ΄S
2817
+ Η«EX
2818
+ n
2819
+ (ti) : n = 20
2820
+ Ξ΄EX
2821
+ 0.05
2822
+ 8.6 Γ— 10βˆ’3
2823
+ 2.00
2824
+ 2.98 Γ— 10βˆ’6
2825
+ 3.99
2826
+ 0.1
2827
+ 7.56 Γ— 10βˆ’3
2828
+ 2.00
2829
+ 2.23 Γ— 10βˆ’6
2830
+ 3.99
2831
+ 0.15
2832
+ 6.79 Γ— 10βˆ’3
2833
+ 2.00
2834
+ 1.59 Γ— 10βˆ’6
2835
+ 3.99
2836
+ 0.2
2837
+ 6.22 Γ— 10βˆ’3
2838
+ 2.00
2839
+ 1.09 Γ— 10βˆ’6
2840
+ 3.97
2841
+ 0.25
2842
+ 5.78 Γ— 10βˆ’3
2843
+ 2.00
2844
+ 7.13 Γ— 10βˆ’7
2845
+ 3.96
2846
+ 0.3
2847
+ 5.45 Γ— 10βˆ’3
2848
+ 2.00
2849
+ 4.46 Γ— 10βˆ’7
2850
+ 3.94
2851
+ 0.35
2852
+ 5.19 Γ— 10βˆ’3
2853
+ 2.00
2854
+ 2.7 Γ— 10βˆ’7
2855
+ 3.91
2856
+ 0.4
2857
+ 4.98 Γ— 10βˆ’3
2858
+ 2.00
2859
+ 1.69 Γ— 10βˆ’7
2860
+ 3.86
2861
+ 0.45
2862
+ 4.82 Γ— 10βˆ’3
2863
+ 2.00
2864
+ 1.3 Γ— 10βˆ’7
2865
+ 3.83
2866
+ 0.5
2867
+ 4.68 Γ— 10βˆ’3
2868
+ 2.00
2869
+ 1.41 Γ— 10βˆ’7
2870
+ 3.85
2871
+ 0.55
2872
+ 4.55 Γ— 10βˆ’3
2873
+ 2.00
2874
+ 1.91 Γ— 10βˆ’7
2875
+ 3.89
2876
+ 0.6
2877
+ 4.44 Γ— 10βˆ’3
2878
+ 2.00
2879
+ 2.72 Γ— 10βˆ’7
2880
+ 3.93
2881
+ 0.65
2882
+ 4.33 Γ— 10βˆ’3
2883
+ 2.00
2884
+ 3.75 Γ— 10βˆ’7
2885
+ 3.95
2886
+ 0.7
2887
+ 4.22 Γ— 10βˆ’3
2888
+ 2.00
2889
+ 4.95 Γ— 10βˆ’7
2890
+ 3.97
2891
+ 0.75
2892
+ 4.10 Γ— 10βˆ’3
2893
+ 2.00
2894
+ 6.26 Γ— 10βˆ’7
2895
+ 3.98
2896
+ 0.8
2897
+ 3.98 Γ— 10βˆ’3
2898
+ 2.00
2899
+ 7.6 Γ— 10βˆ’7
2900
+ 3.99
2901
+ 0.85
2902
+ 3.84 Γ— 10βˆ’3
2903
+ 2.00
2904
+ 8.94 Γ— 10βˆ’7
2905
+ 3.99
2906
+ 0.9
2907
+ 3.69 Γ— 10βˆ’3
2908
+ 2.00
2909
+ 1.02 Γ— 10βˆ’6
2910
+ 3.99
2911
+ 0.95
2912
+ 3.52 Γ— 10βˆ’3
2913
+ 2.00
2914
+ 1.14 Γ— 10βˆ’6
2915
+ 4
2916
+ From the above table, it is clear that the obtained orders of convergence match well with
2917
+ the theoretical orders of convergence. Also the order of convergence of the extrapolated
2918
+ solution improves upon the discrete iterated Galerkin solution.
2919
+ 5 References
2920
+ [1] K. E. Atkinson. The numerical solutions of integral equations of the second kind, Cambridge University
2921
+ Press, Cambridge, (1997).
2922
+ [2] K. E. Atkinson. The numerical evaluation of fixed points for completely continuous operators. SIAM Jour-
2923
+ nal on Numerical Analysis, 10(5) , 799–807 (1973).
2924
+ [3] K. E. Atkinson and A. Bogomolny. The discrete Galerkin method for integral equations. Mathematics of
2925
+ Computation, 48(178), 595–616 (1987).
2926
+ [4] K. E. Atkinson and F. A. Potra. On the discrete Galerkin method for Fredholm integral equations of the
2927
+ second kind. IMA Journal of Numerical Analysis, 9(3), 385-403 (1989).
2928
+ [5] K. E. Atkinson and F. A. Potra. Projection and iterated projection methods for nonlinear integral equations.
2929
+ SIAM Journal on Numerical Analysis, 24(6), 1352-1373 (1987).
2930
+ [6] K. E. Atkinson and F. A. Potra. The discrete Galerkin method for nonlinear integral equations. Journal of
2931
+ Integral Equations and Applications, 1(1), 17-54 (1988).
2932
+ [7] C. T. Baker. The numerical treatment of integral equations. Oxford University Press, (1977).
2933
+ [8] J. BognÑr. Indefinite inner product spaces (Vol. 78). Springer Science & Business Media, (2012).
2934
+ [9] H. Brunner, Y. Lin and S. Zhang. Higher accuracy methods for second-kind Volterra integral equations
2935
+ based on asymptotic expansions of iterated Galerkin methods. Journal of Integral Equations and Applica-
2936
+ tions, 10(4), 375-396 (1998).
2937
+ [10] F. Chatelin and R. Lebbar. Superconvergence results for the iterated projection method applied to a Fred-
2938
+ holm integral equation of the second kind and the corresponding eigenvalue problem. Journal of Integral
2939
+ Equations, 6(1), 71-91 (1984).
2940
+
2941
+ 24
2942
+ G. RAKSHIT
2943
+ [11] W. F. Ford, J. A. Pennline, Y. Xu and Y. Zhao. Asymptotic error analysis of a quadrature method for
2944
+ integral equations with Green’s function kernels. Journal of Integral Equations and Applications, 12(4), 349-
2945
+ 384 (2000).
2946
+ [12] M. A. Krasnoselskii. Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon
2947
+ Press, London, (1964).
2948
+ [13] M. A. Krasnoselskii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii and V. Ya.Stetsenko. Approximate
2949
+ Solution of Operator Equations, P. Noordhoff, Groningen, (1972).
2950
+ [14] M. A. Krasnoselskii and P. P. Zabreiko. Geometrical Methods of Nonlinear Analysis, Springer-Verlag,
2951
+ Berlin, (1984).
2952
+ [15] R. P. Kulkarni and L. Grammont. Extrapolation using a modified projection method. Numerical Func-
2953
+ tional Analysis and Optimization, 30(11-12), 1339-1359 (2009).
2954
+ [16] R. P. Kulkarni and T. J. Nidhin, Asymptotic error analysis of projection and modified projection methods
2955
+ for nonlinear integral equations. Journal of Integral Equations and Applications, 27(1), 67-101 (2015).
2956
+ [17] R. P. Kulkarni and G. Rakshit. Discrete modified projection methods for Urysohn integral equations with
2957
+ Green’s function type kernels. Mathematical Modelling and Analysis, 25(3), 421 - 440 (2020).
2958
+ [18] R. P. Kulkarni and G. Rakshit. Discrete modified projection method for Urysohn integral equations with
2959
+ smooth kernels. Applied Numerical Mathematics, 126, 180-198 (2018).
2960
+ [19] R. P. Kulkarni and A. S. Rane. Asymptotic expansions for approximate solutions of Hammerstein integral
2961
+ equations with Green’s function Type Kernels. Mathematical Modelling and Analysis, 19(1), 127-143 (2014).
2962
+ [20] Q. Lin, I. H. Sloan and R. Xie. Extrapolation of the Iterated–Collocation Method for Integral Equations
2963
+ of the Second Kind. SIAM Journal on Numerical Analysis, 27(6), 1535-1541 (1990).
2964
+ [21] P. Linz. Theoretical Numerical Analysis: Introduction to Advanced Techniques, Courier Dover Publica-
2965
+ tions, (2019).
2966
+ [22] W. McLean. Asymptotic error expansions for numerical solutions of integral equations. IMA Journal of
2967
+ Numerical Analysis, 9(3), 373-384 (1989).
2968
+ [23] G. Rakshit and A. S. Rane. Asymptotic expansion of iterated Galerkin solution of Fredholm integral
2969
+ equations of the second kind with Green’s kernel. Journal of Integral Equations and Applications, 32(4),
2970
+ 495-507 (2020).
2971
+ [24] G. Rakshit, A. S. Rane and K. Patil. Richardson extrapolation for the iterated Galerkin solution of
2972
+ Urysohn integral equations with Green’s kernels. International Journal of Computer Mathematics, 19(8),
2973
+ 1538 - 1556 (2022).
2974
+ [25] L. B. Rall, Computational solution of nonlinear operator equations, Wiley New York (1969).
2975
+ [26] F. Riesz, and B. S. Nagy. Functional Analysis. Courier Corporation, (2012).
2976
+ [27] L. Schumaker. Spline functions: Basic Theory. Cambridge University Press, (2007).
2977
+ [28] I. H. Sloan. Improvement by iteration for compact operator equations. Mathematics of Computation,
2978
+ 30(136), 758-764 (1976).
2979
+ DEPARTMENT OF MATHEMATICAL SCIENCES, RAJIV GANDHI INSTITUTE OF PETROLEUM TECHNOL-
2980
+ OGY, JAIS CAMPUS, UTTAR PRADESH 229304, INDIA., ORCID ID : 0000-0002-5813-4656
2981
+ Email address: [email protected]
2982
+
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1
+ Test Reuse Based on Adaptive Semantic Matching across Android Mobile Applications
2
+ Shuqi Liu1, Yu Zhou1,βˆ—, Tingting Han2, and Taolue Chen2
3
+ 1College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China
4
+ 2Department of Computer Science and Data Science, Birkbeck, University of London, UK
5
+ {liushuqi, zhouyu}@nuaa.edu.cn, {t.han, t.chen}@bbk.ac.uk
6
+ *corresponding author
7
+ Abstractβ€”Automatic test generation can help verify and de-
8
+ velop the behavior of mobile applications. Test reuse based on
9
+ semantic similarities between applications of the same category
10
+ has been utilized to reduce the manual effort of Graphical User
11
+ Interface (GUI) testing. However, most of the existing studies fail
12
+ to solve the semantic problem of event matching, which leads to
13
+ the failure of test reuse. To overcome this challenge, we propose
14
+ TRASM (Test Reuse based on Adaptive Semantic Matching), a
15
+ test reuse approach based on adaptive strategies to find a better
16
+ event matching across android mobile applications. TRASM
17
+ first performs GUI events deduplication on the initial test set
18
+ obtained from test generation, and then employs an adaptive
19
+ strategy to find better event matching, which enables reusing
20
+ the existing test. Preliminary experiments with comparison to
21
+ baseline methods on 15 applications demonstrate that TRASM
22
+ can improve the precision of GUI event matching while reducing
23
+ the failure of test reuse and the running time required for test
24
+ reuse.
25
+ Keywordsβ€”adaptive semantic matching; android mobile appli-
26
+ cations; GUI event; test reuse; oracle generation
27
+ I. INTRODUCTION
28
+ Graphical User Interface (GUI) testing is commonly em-
29
+ ployed to verify and develop the behaviors of applications by
30
+ designing and executing test cases of GUI applications [1].
31
+ However, with the ever increasing functionalities in mobile
32
+ applications, it takes more effort for developers to manually
33
+ design GUI test cases (GUI test in short) [2–4], which in turn
34
+ decreases the efficiency of testing processes.
35
+ Considering the necessity of reducing time consumption,
36
+ many researchers have conducted a series of investigation on
37
+ automatic test generation [5–15] in GUI testing. Recently,
38
+ some researchers observed that test reuse [16–24] could
39
+ be achieved by exploiting the semantic similarity of GUIs
40
+ between similar applications to generate tests automatically.
41
+ Figure 1 shows a simple example in which the existing test (a)
42
+ of application To-Do List is successfully reused to application
43
+ Minimal, and the reused test (b) is obtained. As Figure 1
44
+ shows, events em
45
+ 1 , em
46
+ 2 , em
47
+ 3 , and em
48
+ 4 in test (b) are similar to
49
+ et
50
+ 1, et
51
+ 2, et
52
+ 3, and et
53
+ 4 in test (a), respectively.
54
+ Existing research mainly focuses on how to accurately select
55
+ specific characteristics of widgets in GUIs such as β€˜text’
56
+ and β€˜resource-id’. Combining the selected characteristics, they
57
+ design the semantic similarity calculation method between
58
+ widgets to generate meaningful tests. They attempt to select
59
+ widgets with high similarity in a similar application for match-
60
+ ing each event of the existing test. However, little attention
61
+ has been paid to optimizing the matching process. Taking
62
+ Figure 1 as an example, the widget wt
63
+ 3 of To-Do List and
64
+ the correctly similar widget wm
65
+ 3
66
+ of the application Minimal
67
+ are laid out differently in the GUI. When reusing test (b) to
68
+ the application To-Do List, adopting the existing approach may
69
+ always incorrectly match the widget wm
70
+ 3 with other widgets.
71
+ This may cause subsequent events to match incorrectly or even
72
+ result in failed test reuse. In cases where the existing methods
73
+ do not work well, it is necessary to adopt other corresponding
74
+ measures. The lower the similarity of the generated event,
75
+ the more likely the match is inappropriate. Hence, mining
76
+ such events and exploring other widgets with more similar
77
+ semantics to form events for substitution is considered.
78
+ In addition, in Figure 1, the application To-Do List needs
79
+ to skip the boot page that the application Minimal does not
80
+ before entering the home page. And it is assumed that the
81
+ event step is et
82
+ 0 = click(wt
83
+ 0). Under this assumption, the
84
+ widget wt
85
+ 0 will match the widget with the highest similarity
86
+ on the home page of the application Minimal. Obviously, the
87
+ event produced by this step is redundant in the generated test.
88
+ This simple example explains that we need to solve the event
89
+ redundancy issue in the process of test reuse caused by some
90
+ particular functionality in the existing test.
91
+ (a) The existing test for To-Do List
92
+ (b) The reused test for Minimal
93
+ Figure 1. A simple example of test reuse. The test (b) is obtained by reusing
94
+ the existing test (a).
95
+ Inspired by the above observation, in this paper, we propose
96
+ a novel approach TRASM (Test Reuse based on Adaptive
97
+ Semantic Matching) to reuse the existing tests across android
98
+ arXiv:2301.00530v1 [cs.SE] 2 Jan 2023
99
+
100
+ θ‡ͺ7:36
101
+ 47:36
102
+ 7:37
103
+ 7:37
104
+ δΈ‰
105
+ To-Do List
106
+ Q
107
+ οΌ‹
108
+ To-Do List
109
+ δΈ‰
110
+ To-Do List
111
+ Q+"
112
+ :
113
+ Q
114
+ CLICK+FOR NEW LIST
115
+ CLICK+FORNEWLIST
116
+ No Deadline
117
+ New To-Do task
118
+ e’ = exist(Test')(w')
119
+ e, = fill(w)
120
+ Welcome!
121
+ Test
122
+ 8
123
+ scription
124
+ 菌 Deadline
125
+ @Reminder
126
+ Notasks available
127
+ Progress:
128
+ 0%
129
+ Thisappdoesnotuseanypermissions
130
+ Priority:
131
+ Medium
132
+ List:
133
+ Click to select!
134
+ CANCEL
135
+ OKAY
136
+ e, = click(w')
137
+ e. = click(wΒ²)
138
+ +
139
+ +
140
+ ADDNEWTASK>
141
+ Γ—
142
+ ADDNEWTASK>
143
+ sKIP
144
+ NEXTζ—₯7:35
145
+ 47:35
146
+ ζ—₯7:36
147
+ Minimal
148
+ X
149
+ Minimal
150
+ itl
151
+ Test
152
+ e" = exist(Test')(w")
153
+ e’ = fill(wz)
154
+ You don't have any todos
155
+ " = click(w"mobile applications. In addition, we carry out comparative
156
+ experiments with the-state-of-the-art baseline approaches to
157
+ evaluate our work. Overall, our main contributions are as
158
+ follows:
159
+ 1) We propose a novel approach TRASM, which utilizes an
160
+ adaptive strategy to reuse more existing tests. TRASM
161
+ can get more semantic matches in the generated test.
162
+ 2) TRASM includes a GUI events deduplication method,
163
+ which could eliminate duplicated events caused by
164
+ reusing particular functionality contained in the existing
165
+ test to improve the quality of the generated test.
166
+ 3) We carry out extensive experiments which confirm that
167
+ TRASM improves the accuracy of GUI event matching
168
+ while reducing test reuse failures and reduces the run-
169
+ ning time required for test reuse.
170
+ The rest of this paper is organized as follows. Section
171
+ II introduces related work. Section III describes the main
172
+ idea and the proposed approach in detail. Section IV carries
173
+ out experimental evaluation. Finally, Section V concludes the
174
+ paper and outlines future research.
175
+ II. RELATED WORK
176
+ A. Test Generation
177
+ In order to improve the efficiency of developers, based on
178
+ different exploration strategies, several studies on automatic
179
+ test generation have been proposed, which has laid a solid
180
+ foundation.
181
+ Sapienz [5] combined random fuzzing, systematic and
182
+ search-based exploration, exploiting seeding and multi-level
183
+ instrumentation to explore and optimize test sequences auto-
184
+ matically. Gu et al. [7] dynamically abstracted the model by
185
+ leveraging decision tree-based representation and updated the
186
+ model by utilizing the evolution mechanism, which balances
187
+ the accuracy and size of the model. ConmboDroid [15] ob-
188
+ tained the use cases for verifying the unique functions of the
189
+ application and then systematically enumerates and combines
190
+ them to generate higher quality input. The advantage of their
191
+ work is that they can mine more hidden bugs or achieve as high
192
+ coverage as possible. Nevertheless, the test generated by their
193
+ method is seldom standardized for verifying the application’s
194
+ functionality.
195
+ Different from their purpose and inspired by their explo-
196
+ ration method, we focus on generating more meaningful tests
197
+ based on semantic information.
198
+ B. Test Reuse
199
+ Test reuse, as an alternative method to automatically gener-
200
+ ate GUI test, makes full use of existing resources to provide
201
+ convenience for verifying the application’s behavior.
202
+ Lin et al. proposed CRAFTDROID [19], an approach of
203
+ test transfer across applications, which utilizes the GUI model
204
+ extracted by static analysis to match event sequences similar to
205
+ the semantics of the existing test in order. They realized the
206
+ successful transfer of GUI and oracle events, which guides
207
+ for improving test transfer. To more accurately express the
208
+ similarity of widgets in test events, Mao et al. [20] raised a
209
+ semantic-based event fuzzy mapping strategy when matching
210
+ candidate widgets to generate target events. They always
211
+ greedily preferentially explore and match the widgets with the
212
+ highest similarity. Unfortunately, when their similarity calcu-
213
+ lation method does not work well, the correctness of event
214
+ matching will be threatened. Considering that the success
215
+ of test reuse heavily depends on the semantic matching of
216
+ test events, there is still space for improvement by adopting
217
+ appropriate strategies to increase the quality of reused tests
218
+ from the perspective of application functionality.
219
+ Leonardo et al. [21] conducted extensive research and
220
+ pointed out that some attributes representing widgets play
221
+ a negative role and how designing the semantic matching
222
+ process is the most influential component to matched results.
223
+ Their key findings point to an entry point for better reuse of
224
+ test. Up to now, there is still no effective method to solve
225
+ semantic problems [25, 26]. Trying to optimize the generated
226
+ test sequence to ensure the quality of reused tests should be
227
+ an optional strategy.
228
+ III. OUR APPROACH
229
+ Figure 2 shows an overview of the proposed test reuse
230
+ approach TRASM. Based on semantic matching of events,
231
+ TRASM considers the test (source test) of the existing appli-
232
+ cation (source app), and the new application (target app) as
233
+ inputs and outputs target test. TRASM employs two significant
234
+ phases to implement test reuse: preliminary preparation and
235
+ source test reuse. For the former, the existing data is processed
236
+ through test augmentation and model extraction to facilitate
237
+ the implementation of source test reuse. For the latter, the
238
+ processed data obtained by the former is used together to reuse
239
+ the source test on the target app.
240
+ Test augmentation and model extraction are preliminary
241
+ preparation steps that follow existing work [19], and we will
242
+ only briefly introduce them. In detail, we focus on introducing
243
+ our main contributions.
244
+ A. Test augmentation
245
+ The main task of test augmentation is to extract semantic
246
+ information of widgets during the execution of collected
247
+ source tests. The semantically represented widgets, together
248
+ with actions, compose augmented tests, which are used to
249
+ match widgets in the GUI of the target app.
250
+ After the source app executes each event, the adb tool1 is
251
+ used to extract the semantic information of the corresponding
252
+ widget in the executed event according to the reached GUI
253
+ state. Multiple attributes (including class, resource-id, text,
254
+ content-desc, clickable, password, parent text, sibling text,
255
+ activity, and package) uniquely represent a widget in the GUI.
256
+ These non-empty attributes and their values constitute the
257
+ widget’s semantic information. For example, for widget wt
258
+ 1
259
+ of test (a) in Figure 1, the collected semantic information is
260
+ shown in Table 1.
261
+ 1https://developer.android.com/studio/command-line/adb.html
262
+
263
+ Figure 2.
264
+ The overview of TRASM.
265
+ Table 1. The semantic information of widget wt
266
+ 1.
267
+ Attribute
268
+ Value
269
+ class
270
+ android.widget.ImageButton
271
+ resource-id
272
+ fab new task
273
+ clickable
274
+ true
275
+ password
276
+ false
277
+ activity
278
+ .view.MainActivity
279
+ package
280
+ org.secuso.privacyfriendlytodolist
281
+ B. Model extraction
282
+ The model extraction aims to statically analyze the source
283
+ code obtained by the target app to obtain the window transition
284
+ graph (WTG). Following with existing work, we employ tool2
285
+ to obtain the WTG. And the steps of constructing WTG can
286
+ refer to literature [19].
287
+ The WTG can visually represent the interaction between
288
+ application activities and is composed of node sets and edges.
289
+ Among WTG, the node represents the activity of the applica-
290
+ tion, and the directed edge represents the activity transition
291
+ that the event can trigger. For example, for the test (b)
292
+ of application Minimal in Figure 1, the window transition
293
+ triggered by the execution of the event is shown in Figure 3,
294
+ where the nodes Main and AddToDo respectively represent
295
+ the two activities of the application Minimal. By triggering
296
+ the event on edge, state transition occurs between activities.
297
+ The obtained WTG can provide matching candidate widgets
298
+ for widgets in the source test. However, the WTG obtained
299
+ may be incomplete. More fully, we adopt updating the WTG
300
+ based on the feedback running information when executing
301
+ the application.
302
+ C. Test generation
303
+ The purpose of test generation is to generate the initial test
304
+ on the target app according to the semantic information of the
305
+ augmented test and the WTG of the target app.
306
+ Every event in the augmented test iteratively matches the
307
+ corresponding events in the target app. The candidate widgets
308
+ are first obtained by similarity calculation to match the event.
309
+ 2https://drive.google.com/file/d/1HEFS96c5nNKnzBPkWlRdwBiunOHgOs-/
310
+ view?usp=sharing
311
+ Figure 3.
312
+ Window transition triggered by test (b).
313
+ Similarity calculation. For each widget, the semantic in-
314
+ formation is captured from the current GUI page of the
315
+ target app to build a word list of attributes. For example,
316
+ the attribute β€˜resource-id’ of widget wt
317
+ 2 and widget wm
318
+ 2
319
+ are β€˜et new task name’ and β€˜userToDoEditText’ respec-
320
+ tively. After preprocessing [10, 19], we get two word lists
321
+ a=[β€˜edit’, β€˜text’, β€˜new’, β€˜task’, β€˜name’] and aβ€²=[β€˜user’, β€˜todo’,
322
+ β€˜edit’, β€˜text’]. For any word w ∈ a and wβ€²
323
+ ∈ aβ€², the
324
+ highest similarity max sim(w, wβ€²) between words w and wβ€²
325
+ is synthesized as the similarity of attributes:
326
+ sim(a, aβ€²) =
327
+ οΏ½
328
+ w∈a maxwβ€²βˆˆaβ€² sim(w, wβ€²)
329
+ |a|
330
+ (1)
331
+ where a and aβ€² represent the attributes corresponding to
332
+ widget S in source test and widget T in target app respectively,
333
+ sim(w, wβ€²) expresses the cosine distance of the word vectors
334
+ βˆ’β†’
335
+ Vw and βˆ’β†’
336
+ Vwβ€², obtained by the Word2Vec model[27]:
337
+ sim(w, wβ€²) =
338
+ βˆ’β†’
339
+ Vw Β· βˆ’β†’
340
+ Vwβ€²
341
+ |βˆ’β†’
342
+ Vw||βˆ’β†’
343
+ Vwβ€²|
344
+ (2)
345
+ Based on Equation (1), we get the similarity between widget
346
+ S and widget T:
347
+ sim(S, T) =
348
+ οΏ½
349
+ a∈S sim(a, aβ€²) βˆ— wg(a)
350
+ |S|
351
+ (3)
352
+ where, wg(a) represents the weight of attribute a among
353
+ all attributes. Based above calculation, we build candidate
354
+
355
+ Source Test
356
+ Preliminary preparation
357
+ β‘‘ Source test reuse
358
+ Candidate
359
+ Widgets
360
+ 口
361
+ A. Test
362
+ E. Test
363
+ adaptation
364
+ augmentation
365
+ 口
366
+ APP
367
+ Augmented
368
+ Processed Test
369
+ Source Test
370
+ Target Test
371
+ Similarity
372
+ Source App
373
+ Window
374
+ calculation
375
+ Transition Graph
376
+ B. Model
377
+ APP
378
+ C. Test
379
+ D. GUI events
380
+ extraction
381
+ generation
382
+ deduplication
383
+ Target App
384
+ -
385
+ Initial Test= click(w") em = fill(wΒ²)
386
+ em
387
+ m
388
+ = click(wm) em = exist(Test')(wm)
389
+ m
390
+ m
391
+ Main
392
+ AddToDo
393
+ m
394
+ mwidgets by selecting several widgets in the target app with
395
+ high similarity.
396
+ We identify a reachable widget based on the obtained
397
+ candidate widgets and assign an action to form an event.
398
+ All the paths from the current activity to the activity of
399
+ each candidate widget can be queried from the WTG. These
400
+ paths are executed to identify the reachable widget and return
401
+ leading events. In addition, to avoid repeated path exploration,
402
+ we adopt a strategy to preserve the path that has been explored
403
+ and the corresponding leading events. For example, when
404
+ matching event et
405
+ 2 in test (a), the application Minimal reaches
406
+ the AddToDo activity after executing event em
407
+ 1 as shown in
408
+ Figure 3, and then candidate widgets on the current page are
409
+ collected. From the stored explored paths, it is found that there
410
+ is a reachable path between activity AddToDo and activity
411
+ AddToDo. Widget wm
412
+ 2 is located in the reachable path, which
413
+ is identified as a reachable widget. Finally, according to the
414
+ source event, the action is allocated to the widget wm
415
+ 2 .
416
+ D. GUI events deduplication
417
+ Invalid repeated events will increase the complexity of test
418
+ execution. Although repeated events in the test will not affect
419
+ the triggering of the behavior of an application, GUI events
420
+ deduplication intends to reduce the time consumption occupied
421
+ by such events.
422
+ Since the GUIs of the two applications are different, the
423
+ target app may not have the special functionality contained
424
+ in the source test. As explained in Section
425
+ I, the reuse of
426
+ special functionality in test (a), that is, the matching of event
427
+ et
428
+ 0 on application Minimal, is meaningless. To remove such
429
+ GUI events in the test sequence, deduplication is performed.
430
+ However, it is a challenge to identify the meaningless events
431
+ in the test. We take the operation of detecting and deleting
432
+ duplicate events unrelated to the generated initial test. Con-
433
+ sidering the variety of possible duplicate event patterns, we
434
+ set two rules to distinguish them. First, if only a single event
435
+ is repeated in the initial test, we delete the repeated events
436
+ at the beginning of the test sequence. Second, if the test
437
+ sequence starts with ⟨en0, en1⟩ and also contains ⟨en1, en0⟩
438
+ such events, we delete the pair of events. After this operation,
439
+ to maintain the correctness, we check whether the test after
440
+ deduplication, that is, the processed test, can maintain the
441
+ functionality as the initial test. If not, we will give up the
442
+ GUI events deduplication.
443
+ E. Test adaptation
444
+ The goal of test adaptation is to explore whether there is a
445
+ better test sequence than the processed test using the designed
446
+ adaptation strategy.
447
+ Test generation always prioritizes the widget with the high-
448
+ est similarity for matching. When the method of calculating
449
+ similarity does not work well, it may not be possible to
450
+ distinguish the best widgets to match, which will affect the
451
+ accuracy of the result. The design idea of test adaptation
452
+ is to find indexes that may have more semantically similar
453
+ events in the processed test and then rematch them. However,
454
+ determining such indexes in the sequences of the processed
455
+ test is a challenging task. In this paper, we first record the
456
+ indexes for which widgets in the processed test have higher
457
+ similarity to another widget in the augmented test, except
458
+ for the current matching event. Then, we choose the indexes
459
+ with the lowest similarity of matching events in the processed
460
+ test, which tries to mine the event with the incorrect match.
461
+ After these two processing stages, we obtain the index sets
462
+ of events that can perform rematching. Based on the above,
463
+ we successively rematch the events of each index set from
464
+ the candidate widgets obtained by Section III-C. We set the
465
+ early termination condition to obtain a new test sequence that
466
+ is more semantically similar than the original ones.
467
+ We explain how test adaptation solves the problem of
468
+ reusing test (b) to application To-Do List in Figure
469
+ 1. As
470
+ mentioned in Section
471
+ I, different GUI designs make the
472
+ similarity between the correct widget and the source widget
473
+ low, resulting in the incorrect match of event em
474
+ 3 . Through the
475
+ strategies mentioned above, we get the index of event em
476
+ 3 to be
477
+ rematched. Then, combined with the WTG obtained from the
478
+ model extraction, the correct reachable widget wt
479
+ 3 is searched
480
+ again from the obtained candidate widgets on this index to
481
+ form event et
482
+ 3. Finally, the process ends after the oracle event
483
+ et
484
+ 4 matching.
485
+ IV. EXPERIMENTAL EVALUATION
486
+ We implement our approach TRASM as a tool. Moreover,
487
+ we compared TRASM with the baseline approach CRAFT-
488
+ DROID [19], a test transfer method across mobile applications
489
+ through semantic mapping, to verify the effectiveness and
490
+ efficiency of TRASM. In this section, we introduce the exper-
491
+ imental setup and experimental results to evaluate TRASM.
492
+ A. Experimental setup
493
+ For consistency, we reused the dataset3 of [19] to evaluate
494
+ the proposed TRASM. Following the steps of the baseline, we
495
+ conducted reuse tests on 15 applications in three categories,
496
+ including browser, Tip Calculator, and To-Do List. These
497
+ applications come from Google Play and F-Droid, which are
498
+ often used in the GUI testing field to explore the functionalities
499
+ of application [5, 24, 25]. Concretely, Table 2 details the
500
+ category, name (version), and source of each application.
501
+ Specifically, for each application category, two typical func-
502
+ tionalities are selected, and the corresponding tests of each
503
+ application are collected according to the functionalities. To
504
+ achieve the goal of verifying the implemented functionality,
505
+ the last event of each test case is set as an oracle. In general,
506
+ there are six functionalities in three categories of applications,
507
+ as shown in Table 3. Table 3 lists the number of test cases for
508
+ each functionality and the average number of GUI and oracle
509
+ events.
510
+ Our experiment was implemented on a Nexus 5X Emulator
511
+ running Android 6.0 (API 23) installed on a Ubuntu desktop
512
+ with a 3.4 GHz Intel Core i7 CPU and 32 GB RAM.
513
+ 3https://sites.google.com/view/craftdroid/
514
+
515
+ Table 2. The specific information of applications.
516
+ Category
517
+ Application (version)
518
+ Source
519
+ a1-Browser
520
+ a11-Lightning (4.5.1)
521
+ F-Droid
522
+ a12-Browser for Android (6.0)
523
+ Google Play
524
+ a13-Privacy Browser (2.10)
525
+ F-Droid
526
+ a14-FOSS Browser (5.8)
527
+ F-Droid
528
+ a15-Firefox Focus (6.0)
529
+ Google Play
530
+ a2-Tip Calculator
531
+ a21-Tip Calculator (1.1)
532
+ Google Play
533
+ a22-Tip Calc (1.11)
534
+ Google Play
535
+ a23-Simple Tip Calculator (1.2)
536
+ Google Play
537
+ a24-Tip Calculator Plus (2.0)
538
+ Google Play
539
+ a25-Free Tip Calculator (1.0.0.9)
540
+ Google Play
541
+ a3-To Do List
542
+ a31-Minimal (1.2)
543
+ F-Droid
544
+ a32-Clear List (1.5.6)
545
+ F-Droid
546
+ a33-To-Do List (2.1)
547
+ F-Droid
548
+ a34-Simply Do (0.9.1)
549
+ F-Droid
550
+ a35-Shopping List (0.10.1)
551
+ F-Droid
552
+ Table 3. Tests for the typical functionalities.
553
+ Functionality
554
+ Test
555
+ Avg
556
+ Avg
557
+ Cases
558
+ GUIs
559
+ Oracles
560
+ b11-Access website by URL
561
+ 5
562
+ 3.4
563
+ 1
564
+ b12-Back button
565
+ 5
566
+ 7.4
567
+ 3
568
+ b21-Calculate total bill with tip
569
+ 5
570
+ 3.8
571
+ 1
572
+ b22-Split bill
573
+ 5
574
+ 4.8
575
+ 1
576
+ b31-Add task
577
+ 5
578
+ 4
579
+ 1
580
+ b32-Remove task
581
+ 5
582
+ 6.8
583
+ 2
584
+ Total
585
+ 30
586
+ 5.1
587
+ 1.5
588
+ B. Experimental results
589
+ This
590
+ subsection
591
+ presents
592
+ the
593
+ experimental
594
+ results
595
+ of
596
+ TRASM and the baseline approach CRAFTDROID under
597
+ the same evaluation metrics. For each functionality of each
598
+ category, we reuse the test of one application on the remaining
599
+ four applications respectively, and the total number of test
600
+ reuse is 5(test cases) Γ— 4(target applications) = 20. This paper
601
+ shows the average result of 20 different test reuses. In order
602
+ to avoid randomness, for each test reuse, we take the average
603
+ of the multiple results recorded.
604
+ Effectiveness. By comparison, the tests reused by TRASM
605
+ perform higher usability than CRAFTDROID. The following
606
+ two aspects, including the evaluation of successful reuse and
607
+ the evaluation of matching events, can support the usability of
608
+ the TRASM approach reuse test.
609
+ Regarding reuse success rate, TRASM has significantly
610
+ improved test reuse in 3 of the six functionalities, as shown
611
+ in the last column of Table 4. For functionalities b21 and
612
+ b22, successfully reused tests achieved a 10% increase. For
613
+ functionality b32, successful reuse also increased from 20%
614
+ to 25%. In addition, 2 of the six functionalities, namely b11
615
+ and b12, have shown the highest successful reuse, i.e., 100%,
616
+ no matter whether it is approach CRAFTDROID or TRASM.
617
+ For the evaluation of matching events, the third and fourth
618
+ columns of Table 4 list the precision and recall of GUI
619
+ events and oracle events, respectively. As shown in the table,
620
+ compared with CRAFTDROID, TRASM improves the preci-
621
+ sion of the GUI by 5% to 15% in different functionalities.
622
+ Unfortunately, while improving the precision, the recall rate
623
+ of GUI events for functionalities b22 and b32 has decreased
624
+ slightly by 2% and 3%, respectively. The success of the reused
625
+ test depends on whether the match of the last oracle event
626
+ in the test sequence is correct. Therefore, the improvement
627
+ of successful reuse also represents the increase in the recall
628
+ rate of oracle events, as listed in Table 4. Among them, the
629
+ most significant is that for functionalities b21 and b22, the
630
+ recall rate is improved by 10%. In general, the improvement
631
+ in the precision and the recall of oracle events shows that
632
+ the proposed TRASM indeed increases the availability of the
633
+ reused test.
634
+ Efficiency. Figure
635
+ 4 lists the average test reuse time on
636
+ each functionality. It is obvious that the average time spent on
637
+ reuse testing of TRASM is less than that of CRAFTDROID
638
+ for each functionality. Even the most significant effect is that
639
+ for functionality b21, the average reuse test of CRAFTDROID
640
+ takes 2581 seconds (43 minutes), while our TRASM only takes
641
+ 890 seconds (15 minutes), which is close to 35% of the time
642
+ of CRAFTDROID. In summary, the results are attributed to
643
+ two factors. One is that the storage of explored paths avoids
644
+ repeated time consumption, and the other is that the adaptive
645
+ strategy improves the efficiency of widget matching. The
646
+ above results prove that we can break through the limitation
647
+ on efficiency in CRAFTDROID.
648
+ Figure 4. The average test reuse time on each functionality.
649
+ While evaluating the efficiency of TRASM, an important
650
+ finding is that implementing an adaptive strategy can improve
651
+ the success of some reused tests. Inevitably, the potential
652
+ drawback is that more suitable event matching can not be
653
+ found will bring additional time consumption. We need to
654
+ address this crucial point further to balance efficiency and
655
+ effectiveness.
656
+ V. CONCLUSION
657
+ Test reuse as an alternative method of test generation
658
+ can help developers verify the behavior of applications. In
659
+ this paper, a novel test reuse approach has been proposed
660
+ to alleviate the challenge of semantic problems in event
661
+ matching. From the initial test set, we have extended GUI
662
+ events deduplication and test adaptation to build up target tests.
663
+ The experimental results indicate that our proposed approach
664
+ achieves better performance than the baseline approaches with
665
+ increased usability of the reused tests.
666
+
667
+ 12000
668
+ CRAFTDROID
669
+ TRASM
670
+ 10000
671
+ (sec)
672
+ 8000
673
+ time
674
+ reuse
675
+ 6000
676
+ Average
677
+ 4000
678
+ 2000
679
+ 0
680
+ b11
681
+ b12
682
+ b21
683
+ b22
684
+ b31
685
+ b32
686
+ FunctionalityTable 4. Effectiveness and Efficiency Evaluation
687
+ Functionality
688
+ Approach
689
+ GUI Event
690
+ Oracle Event
691
+ Successful Reuse
692
+ Precision
693
+ Recall
694
+ Precision
695
+ Recall
696
+ b11
697
+ CRAFTDROID
698
+ 79%
699
+ 100%
700
+ 100%
701
+ 100%
702
+ 20/20(100%)
703
+ TRASM
704
+ 100%
705
+ 100%
706
+ 100%
707
+ 100%
708
+ 20/20(100%)
709
+ b12
710
+ CRAFTDROID
711
+ 85%
712
+ 100%
713
+ 100%
714
+ 100%
715
+ 20/20(100%)
716
+ TRASM
717
+ 100%
718
+ 100%
719
+ 100%
720
+ 100%
721
+ 20/20(100%)
722
+ b21
723
+ CRAFTDROID
724
+ 82%
725
+ 100%
726
+ 100%
727
+ 80%
728
+ 16/20(80%)
729
+ TRASM
730
+ 93%
731
+ 100%
732
+ 100%
733
+ 90%
734
+ 18/20(90%)
735
+ b22
736
+ CRAFTDROID
737
+ 80%
738
+ 100%
739
+ 100%
740
+ 65%
741
+ 13/20(65%)
742
+ TRASM
743
+ 85%
744
+ 98%
745
+ 100%
746
+ 75%
747
+ 15/20(75%)
748
+ b31
749
+ CRAFTDROID
750
+ 78%
751
+ 100%
752
+ 85%
753
+ 100%
754
+ 17/20(85%)
755
+ TRASM
756
+ 87%
757
+ 100%
758
+ 85%
759
+ 100%
760
+ 17/20(85%)
761
+ b32
762
+ CRAFTDROID
763
+ 69%
764
+ 100%
765
+ 85%
766
+ 80%
767
+ 11/20(55%)
768
+ TRASM
769
+ 81%
770
+ 97%
771
+ 93%
772
+ 81%
773
+ 12/20(60%)
774
+ We believe that matching events are a promising direction,
775
+ and we plan to study how to improve the matching strategy
776
+ further in the future. In addition, we plan to verify the
777
+ generalization of the method and further explore the effect
778
+ of test reuse on more applications.
779
+ ACKNOWLEDGMENTS
780
+ The work is partially supported by the National Natural
781
+ Science Foundation of China (No. 61972197), the Natural Sci-
782
+ ence Foundation of Jiangsu Province (No. BK20201292), and
783
+ the Collaborative Innovation Center of Novel Software Tech-
784
+ nology and Industrialization. T. Chen is partially supported
785
+ by an oversea grant from the State Key Laboratory of Novel
786
+ Software Technology, Nanjing University (KFKT2022A03),
787
+ Birkbeck BEI School Project (EFFECT), National Natural
788
+ Science Foundation of China (NSFC) under Grants (No.
789
+ 62072309, 62272397).
790
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791
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1
+ Performance of the r2SCAN functional in transition metal oxides
2
+ S. Swathilakshmi1, Reshma Devi1, and Gopalakrishnan Sai Gautam1,*
3
+ 1Department of Materials Engineering, Indian Institute of Science, Bengaluru, 560012,
4
+ India
5
+ *Email: [email protected]
6
+ Abstract
7
+ We assess the accuracy and computational efficiency of the recently developed meta-generalized gra-
8
+ dient approximation (metaGGA) functional, the restored regularized strongly constrained and appropri-
9
+ ately normed (r2SCAN), in transition metal oxide (TMO) systems and compare its performance against
10
+ SCAN. Specifically, we benchmark the r2SCAN-calculated oxidation enthalpies, lattice parameters, on-
11
+ site magnetic moments, and band gaps of binary 3d TMOs against the SCAN-calculated and experimental
12
+ values. Additionally, we evaluate the optimal Hubbard U correction required for each transition metal
13
+ (TM) to improve the accuracy of the r2SCAN functional, based on experimental oxidation enthalpies,
14
+ and verify the transferability of the U values by comparing against experimental properties on other
15
+ TM-containing oxides. Notably, including the U -correction to r2SCAN increases the lattice parameters,
16
+ on-site magnetic moments and band gaps of TMOs, apart from an improved description of the ground
17
+ state electronic state in narrow band gap TMOs. The r2SCAN and r2SCAN+U calculated oxidation en-
18
+ thalpies follow the qualitative trends of SCAN and SCAN+U, with r2SCAN and r2SCAN+U predicting
19
+ marginally larger lattice parameters, smaller magnetic moments, and lower band gaps compared to SCAN
20
+ and SCAN+U, respectively. We observe that the overall computational time (i.e., for all ionic+electronic
21
+ steps) required for r2SCAN(+U ) to be lower than SCAN(+U ). Thus, the r2SCAN(+U ) framework can
22
+ offer a reasonably accurate description of the ground state properties of TMOs with better computational
23
+ efficiency than SCAN(+U ).
24
+ 1
25
+ Introduction
26
+ Density functional theory (DFT [1]) calculations are the bedrock of modern computational materials
27
+ science in terms of predicting thermodynamic and kinetic properties, with such property predictions being
28
+ put to use in subsequent materials discovery [2–7] and understanding underlying physical phenomena. [8–12]
29
+ In recent years, machine learning has been used to augment DFT in property predictions, thereby reducing
30
+ computational cost and accelerating materials discovery. [13–17] Note that a key approximation within
31
+ DFT is the exchange-correlation (XC) functional, the exact form of which is unknown. However, several
32
+ approximations for the XC functional have been proposed over the years, which can be categorized into
33
+ different classes depending on the degree of sophistication and accuracy, and visually represented as rungs
34
+ on the Jacob’s ladder. [1, 2, 18, 19] As with most computational tools, the higher the accuracy (higher up
35
+ Jacob’s ladder) higher is the computational cost.
36
+ 1
37
+ arXiv:2301.00535v1 [cond-mat.mtrl-sci] 2 Jan 2023
38
+
39
+ Most DFT calculations for β€œlarge” solid systems (10s to 100s of atoms) are performed using the Perdew-
40
+ Burke-Ernzerhof (PBE) parameterization of the generalized gradient approximation (GGA) XC functional,
41
+ [20] as it offers fair accuracy at reasonable computational cost for a wide variety of materials. [21–23]
42
+ Specifically, GGAs include the local electron density as well as the gradient of the electron density in
43
+ describing the XC. As a semilocal functional of electron density, PBE captures short range interactions but
44
+ fails to capture medium and long-range dispersions and also exhibits large electronic self-interaction errors
45
+ (SIEs), especially in highly correlated systems. [24, 25] Also, PBE typically underestimates the formation
46
+ energies [26,27] and semiconductor band gaps of crystalline solids, [26,28] while overestimating their lattice
47
+ volumes. [26,29]
48
+ As we move higher in the Jacob’s ladder, [19] we obtain metaGGA functionals, which may account for
49
+ medium range dispersions and exhibit lower SIEs. Some metaGGAs consider orbital kinetic energy density
50
+ in addition to the local electron density and its gradient, such as the recently developed strongly constrained
51
+ and appropriately normed (SCAN [30]) functional, which offers better numerical accuracy than PBE and
52
+ satisfies all 17 known constraints for a XC functional (namely, 6 for exchange, 6 for correlation, and 5
53
+ for both). The iso-orbital indicator (Ξ±), which includes the kinetic energy density in SCAN, distinguishes
54
+ various bonding environments in a given material and consequently improves the accuracy of SCAN over
55
+ GGA. However, SCAN suffers from numerical instability during self-consistent-field (SCF) calculations [31]
56
+ wherein denser k-grids (than PBE) are required for accurate and consistent predictions. [31–33] Thus it is
57
+ computationally expensive (per SCF step) compared to PBE. [21]
58
+ To overcome the numerical instability and reduce the computational cost of SCAN, Bartok and Yates [34]
59
+ developed regularized SCAN (rSCAN), which satisfies 13 out of the 17 known constraints. The authors
60
+ replaced the non-analytical switching Ξ± interpolation function in SCAN with a simple polynomial function,
61
+ which improves computational speed. [35] However, subsequent investigations showed a significant drop in
62
+ numerical accuracy with rSCAN (compared to SCAN), which is attributed to the failure of the polynomial
63
+ Ξ± function to fully recover the uniform gas limit. [31, 32] Subsequently, Furness et al. [32] introduced the
64
+ restored regularized SCAN (or r2SCAN), wherein the constraints broken by rSCAN were restored except
65
+ the fourth order gradient expansion constraint for exchange (or GE4X). Furness et al. claimed that the new
66
+ r2SCAN functional combines the numerical accuracy of SCAN and computational speed of rSCAN as the
67
+ smooth polynomial α function of rSCAN is modified to satisfy the uniform gas limit in r2SCAN. [32] Recently,
68
+ Kingsbury et al. [36] demonstrated that r2SCAN functional indeed delivers robust numerical accuracy (i.e.,
69
+ similar to SCAN) and better computational performance (faster and numerically stable) by comparing
70
+ r2SCAN and SCAN for solids using a high-throughput computational workflow. Specifically, the authors [36]
71
+ reported that while r2SCAN predicts a smaller band gap (for most of the strongly-bound materials) and
72
+ larger lattice volumes than SCAN, the mean atomization error with r2SCAN is ∼15-20% lower for most
73
+ solids. However, the performance of r2SCAN in correlated electron systems, i.e., transition metal oxides
74
+ (TMOs) containing open-shell d electrons, remains to be seen and forms the main focus of this work.
75
+ Despite the accuracy of SCAN, it still has shortcomings in TMOs, which can be mitigated by adding an
76
+ on-site Hubbard U correction term for the transition metal (TM) under consideration. [37,38] This approach
77
+ is similar to the one followed to mitigate the SIEs of PBE in TMOs. [39,40] However, the magnitude of the
78
+ U correction required is not known a priori, and there are both theory-based approaches such as density
79
+ functional perturbation theory, [41] linear response theory, [42–44] embedded Hartree-Fock method, [45,46]
80
+ and machine learning based Bayesian optimisation, [47] and experimental-data-based approaches to identify
81
+ the appropriate U values. For example, Gautam et al. [37, 38] used the experimental oxidation enthalpies
82
+ 2
83
+
84
+ among binary TMOs to identify optimal U values across various oxidation states of 3d TMs. A similar
85
+ experimental-data-based Hubbard U correction scheme can be developed in conjunction with r2SCAN as
86
+ well, resulting in a r2SCAN+U framework, in case r2SCAN exhibits similar SIEs as SCAN in TMOs. We
87
+ explore the usefulness of such a r2SCAN+U framework also in this work.
88
+ Here, we verify the numerical accuracy and computational efficiency of the r2SCAN and r2SCAN+U
89
+ frameworks in comparison to SCAN and SCAN+U, respectively, in describing material properties such as
90
+ lattice parameters, on-site magnetic moments, and band gaps of binary 3d TMOs, including Ti, V, Cr,
91
+ Mn, Fe, Co, Ni, and Cu. As necessary, we evaluate the optimal Hubbard U correction with r2SCAN for
92
+ each TM by using the experimental-data-based approach employed in previous works. [37,38] We find that
93
+ r2SCAN predicts marginally larger lattice constants and smaller on-site magnetic moments than SCAN for
94
+ most of the TMOs considered. On addition of the U -correction to both SCAN and r2SCAN, we observe an
95
+ increase in the calculated lattice constants, on-site magnetic moments and band gaps. In the case of narrow
96
+ band gap TMOs, SCAN+U and r2SCAN+U generally estimate a non-zero band gap, with r2SCAN+U ’s
97
+ band gap in better agreement with experiments. Also, we perform transferability checks for the optimal U
98
+ values derived in this work for each TM, by benchmarking various properties in oxides that were not used
99
+ in obtaining the U values. Finally, we compare the computational performance of r2SCAN/r2SCAN+U
100
+ relative to SCAN/SCAN+U to explore the accuracy-cost trade-off. We report that r2SCAN/r2SCAN+U
101
+ is computationally less expensive than SCAN and SCAN+U, when all required ionic and electronic steps
102
+ are taken into account for convergence during structure relaxations. We hope that our work will provide a
103
+ foundational basis for further studies on understanding material behavior and computationally discovering
104
+ new materials in the near future.
105
+ 2
106
+ Methods
107
+ 2.1
108
+ Computational Methods
109
+ We used the Vienna ab-initio simulation package (VASP 6.2.1) [48–50] for all the spin-polarized DFT
110
+ calculations, where the frozen-core PBE-based projector augmented wave (PAW) [51] potentials employed
111
+ were identical to previous work. [37, 38] The plane waves for each system were expanded up to a kinetic
112
+ energy of 520 eV, with each structure converged until the total energy differences and atomic forces became
113
+ <0.01 meV and <|0.01| eV/˚A, respectively.
114
+ We adopted a Ξ“-centered Monkhorst-Pack [52] grid with a
115
+ density of 48 k-points per ˚A for all systems.
116
+ The conjugate gradient algorithm was used to relax the
117
+ structures (i.e., cell shapes, volumes, and ionic positions), without preserving any underlying symmetry. An
118
+ β€˜accurate’ level of precision was maintained while projecting the wavefunctions in the reciprocal space. The
119
+ Fermi surface of each system was integrated with a Gaussian smearing of partial occupancies, with a width
120
+ of 0.05 eV. In terms of DFT+U calculations, we used the Dudarev framework [53] for adding a effective
121
+ U correction on the d orbitals of TM atoms. All U values used in SCAN+U calculations were taken from
122
+ previous work (see Table S1 of the Supporting Information –SI). [37,38] Since we used different computing
123
+ systems to perform our structure relaxations for different systems, we normalized the computational time
124
+ with the number of cores used in each calculation to compare the computational efficiency of the different
125
+ XC functionals considered.
126
+ For calculating band gaps, GGA functionals typically use the Kohn Sham potential as a multiplicative
127
+ term, which typically underestimates the band gap of solids even at the SCAN level. [54, 55] Here, we
128
+ 3
129
+
130
+ use the generalized Kohn Sham technique to determine the band gaps by calculating the density of states
131
+ (DOS) for all systems considered. For each DOS calculation, we used the optimized structure and the initial
132
+ charge density from a previous structure relaxation. Subsequently, we introduced a set of zero-weighted k-
133
+ points, corresponding to a density of 96 k-points per ˚A, where the k-points that were used for the structure
134
+ relaxation retained their original weights (as determined by VASP). Finally, we performed a single-SCF
135
+ calculation where the DOS was sampled between energies of -20 to 20 eV in steps of 0.005 eV.
136
+ 2.2
137
+ Structures and magnetic configurations
138
+ We considered the binary oxides of each TM, i.e., Ti, V, Cr, Mn, Fe, Co, Ni, and Cu with different
139
+ oxidation states, similar to previous studies. [37, 38] The main criteria in selection of these metal oxides
140
+ are the availability of reliable thermodynamic data (i.e., formation energies [56–58]) and the experimentally-
141
+ determined ground-state structures that are compiled in the inorganic crystal structure database (ICSD) [59]
142
+ Note that the structures from the ICSD were the initial structures in all our DFT structure relaxations,
143
+ including the systems used as transferability checks. In the case of Ni oxides, we chose NiO and LiNiO2
144
+ (similar to previous work, [38]), as reliable thermodynamic data is not available for higher-oxidation-state
145
+ binary Ni oxides (e.g., Ni2O3 and NiO2). The TM in all oxides, except select Co and Ni compounds, was
146
+ initialized in its high-spin configuration (e.g., high-spin configuration of Fe3+ consists of five unpaired d
147
+ electrons). A detailed description of all structures utilised in this work is provided in the SI, under the
148
+ ’Crystal Structures’ section, with the magnetic configurations depicted in Figure S1.
149
+ The magnetic configuration of each TMO considered (see Figure S1) was initialized to its appropriate
150
+ (in several cases, experimentally-known) ground state configuration during the structural relaxation. For
151
+ example, we considered the ferromagnetic (FM) ground state configuration for CrO2 and VO2, given that
152
+ CrO2 is metallic [60] and VO2 undergoes a metal-to-insulator transition (MIT) below 341 K. [61] The
153
+ rocksalt (RS) TMOs, namely, VO, MnO, FeO, CoO, and NiO were initialized with their experimentally-
154
+ known type-II antiferromagnetic (AFM) configuration. [62–67] Each Ni’s spin in NiO was initialized with
155
+ two unpaired d electrons (i.e., its high-spin configuration). In CuO, we arranged the magnetic moments of
156
+ Cu2+ antiferromagnetically along the Cu-O-Cu chains in the [10Β―1] direction. [68,69]
157
+ We initialized α-Mn2O3 (bixbyite structure) in a FM configuration as this configuration was found to
158
+ be the most stable in previous work. [37] AFM configurations were utilized for rutile-MnO2, [70], and the
159
+ other TM2O3 oxides, namely, V2O3, Fe2O3, Ti2O3, and Cr2O3.
160
+ Note that V2O3 becomes AFM below
161
+ its MIT temperature, [71–73] while Fe2O3 displays an AFM configuration with the magnetic moment of
162
+ Fe alternating every two consecutive layers along the c-axis. [74] Cr2O3 and Ti2O3 exhibit ↑↓↑↓ and ↑↓↓↑
163
+ magnetic configurations, respectively, on the TM centers along the a-axis. [75,76]
164
+ In case of spinels, we used different ferrimagnetic (FIM) configurations, as per experimental observations.
165
+ For example, spinel-Fe3O4 contains both Fe3+ and Fe2+, with up-spin Fe3+ occupying tetrahedral sites and
166
+ down-spin Fe3+ occupying half the octahedral sites. The remaining octahedral sites in Fe3O4 are occupied
167
+ by up-spin Fe2+. [77, 78] In Co3O4, no-spin Co3+ occupies octahedral sites, while high-spin Co2+ (three
168
+ unpaired d electrons) occupies tetrahedral sites in an AFM configuration. [79–81] For Mn3O4, we adopted
169
+ the ”FIM6” configuration, as this was found to be the ground state in previous work. [37] TiO2, CrO3,
170
+ and V2O5 are diamagnetic, since they contain TMs with empty 3d orbitals. Similarly, Cu2O is diamagnetic
171
+ owing to the completely-filled 3d orbitals of Cu.
172
+ 4
173
+
174
+ 2.3
175
+ Determining U
176
+ We determined the required U value, with r2SCAN, for each binary TMO oxidation reaction (e.g., Ti3+ β†’
177
+ Ti4+ in 2Ti2O3 + O2 β†’ 4TiO2) by comparing the experimental enthalpy (per mole of O2) with the calculated
178
+ (r2SCAN+U ) value that minimizes the error against the experimental value. Note that U = 0 eV in our
179
+ data simply reflects a r2SCAN calculation. In order to obtain the experimental oxidation enthalpy, standard
180
+ enthalpy of formation for all the considered TMOs were taken from the Wagman and/or Kubaschewski
181
+ tables, [56,57] thus ignoring the p βˆ’ V and entropic contributions, similar to previous works. [37,38,82] The
182
+ overall optimal U value for each TM was obtained by taking the average of the required U for each of the
183
+ available oxidation reactions. In the case of Ni oxides, oxidation of NiO to LiNiO2 by 2Li2O + 4NiO + O2
184
+ β†’ 4LiNiO2 was considered as a proxy for the Ni2+ β†’ Ni3+ oxidation reaction. [38]
185
+ 3
186
+ Results
187
+ 3.1
188
+ Oxidation energetics
189
+ Figure 1 displays the variation of the enthalpy of different oxidation reactions among binary TMOs, as a
190
+ function of applied U in the r2SCAN+U framework, for all TMs considered except Cr and Cu. Solid lines
191
+ in each panel of Figure 1 represent DFT-calculated oxidation enthalpies, with each color corresponding to
192
+ different oxidation reactions for the TM. For instance in V oxides (Figure 1b), the solid black line corresponds
193
+ to the oxidation reaction, VO β†’ V2O3, while the solid red and green lines indicate V2O3 β†’ VO2 and VO2 β†’
194
+ V2O5, respectively. Similarly, the experimental enthalpy of each oxidation reaction is represented by dashed
195
+ horizontal line of the same color. For example, the black dashed line in Figure 1b indicates the experimental
196
+ oxidation enthalpy (-7.36 eV) of VO β†’ V2O3. Also, dotted vertical line of a given color highlights the
197
+ required U value to minimize the error between DFT-calculated and experimental value for the oxidation
198
+ reaction enthalpy indicated by the same color. The dotted blue line in each panel signifies the overall optimal
199
+ U for the TM that is averaged across all available oxidation reactions.
200
+ We report an optimal U value of 2.3, 1.0, 1.8, 3.1, 1.8, and 2.1 eV, respectively, for Ti, V, Mn, Fe,
201
+ Co, and Ni oxides, within the r2SCAN+U framework (Figure 1). Notably, the optimal U obtained with
202
+ r2SCAN is less than that reported previously for SCAN functional (Table S1) for all 3d TMs considered
203
+ (except V and Fe), which can be attributed to better accuracy of r2SCAN compared to SCAN, as observed
204
+ in non-TMOs. [36] For V oxides, the required U value for VO2 β†’ V2O5, V2O3 β†’ VO2, VO β†’ V2O3 is
205
+ 0.0, 0.7, and 2.2 eV, respectively. Thus, the optimal U value for V is 1.0 eV (average of the three required
206
+ U values), which is identical to the U correction required with SCAN. [38] The decreasing required U with
207
+ increasing oxidation state of V in V oxides is expected due to the decrease in the strength of exchange
208
+ interactions among the d electrons as oxidation state increases. In the case of Fe, FeO β†’ Fe2O3 and FeO
209
+ β†’ Fe3O4 reactions require a U of 2.9 and 3.3 eV, respectively, resulting in an optimal U of 3.1 eV, which
210
+ is also identical to the optimal U with SCAN. [37] Moreover, we obtain the highest optimal U of 3.1 eV
211
+ for Fe, among all TMs considered in this work, which is consistent with the fact that Fe3+ has the highest
212
+ number of unpaired d electrons resulting in the strongest exchange interactions.
213
+ For Ti and Ni, we observe a marginal improvement in the U -value for r2SCAN when compared to SCAN.
214
+ Specifically, we obtain an optimal U of 2.3 eV and 2.1 eV for Ti and Ni, respectively, versus 2.5 eV for
215
+ both elements with SCAN. We find an optimal U value of 1.8 eV for both Mn (2.7 eV with SCAN) and
216
+ Co (3.0 eV with SCAN). In Mn-oxides, the required U for the oxidation of Mn2O3 β†’ MnO2, and MnO β†’
217
+ 5
218
+
219
+ Mn2O3 are 1.5 and 2.1 eV, respectively. The optimal U for Mn is transferable to other Mn oxides as well,
220
+ indicated by the robust agreement between r2SCAN+U -calculated and experimental oxidation enthalpy for
221
+ MnO β†’ Mn3O4 (green lines in Figure 1c).
222
+ For Cr and Cu oxides, we obtain reasonable agreement with experimental data without a U correction
223
+ (Figure S2), similar to our observation with SCAN. [38] In fact, for Cu, introducing U -correction worsens the
224
+ error in the calculated oxidation enthalpy for Cu2O β†’ CuO versus experiment, similar to our observation
225
+ with SCAN(+U ) as well, which can be attributed to PAW potentials derived at the PBE-level. [38] However,
226
+ the magnitude of error (versus experiment) is smaller with r2SCAN (β‰ˆ13.1%) than with SCAN (β‰ˆ25.7%).
227
+ In case of Cr, the oxidation reaction of CrO2 β†’ CrO3 requires U ∼ 0.9 eV, but introducing a U correction
228
+ worsens any agreement with experiment for Cr2O3 β†’ CrO2 (where required U = 0 eV). Thus, the optimal
229
+ U for Cr oxides is 0.45 eV (<0.5 eV), which only provides a marginal improvement in describing oxidation
230
+ enthalpies. Hence, we recommend using only r2SCAN for calculating any Cr oxide framework.
231
+ 0
232
+ 1
233
+ 2
234
+ 3
235
+ 4
236
+ U(eV)
237
+ -9
238
+ -8
239
+ -7
240
+ -6
241
+ -5
242
+ -4
243
+ Reaction Enthalpy (eV per O2)
244
+ FeO/Fe2O3
245
+ FeO/Fe3O4
246
+ Experimental
247
+ 2.9 eV
248
+ 3.3 eV
249
+ 3.1 eV
250
+ (d)
251
+ (e)
252
+ 0
253
+ 0.5
254
+ 1
255
+ 1.5
256
+ 2
257
+ 2.5
258
+ 3
259
+ U(eV)
260
+ -7
261
+ -6
262
+ -5
263
+ -4
264
+ -3
265
+ -2
266
+ -1
267
+ Reaction Enthalpy (eV per O2)
268
+ CoO/Co3O4
269
+ Experimental
270
+ 1.8 eV
271
+ 0
272
+ 0.5
273
+ 1
274
+ 1.5
275
+ 2
276
+ 2.5
277
+ U(eV)
278
+ -4
279
+ -3.5
280
+ -3
281
+ -2.5
282
+ -2
283
+ -1.5
284
+ Reaction Enthalpy (eV per O2)
285
+ NiO/LiNiO2
286
+ Experimental
287
+ 2.1 eV
288
+ (f)
289
+ (a)
290
+ 0
291
+ 0.5
292
+ 1
293
+ 1.5
294
+ 2
295
+ 2.5
296
+ U (eV)
297
+ -8.4
298
+ -8.2
299
+ -8
300
+ -7.8
301
+ -7.6
302
+ -7.4
303
+ Reaction Enthalpy (eV per O2)
304
+ Ti2O3/TiO2
305
+ Experimental
306
+ 2.3 eV
307
+ (b)
308
+ 0
309
+ 0.5
310
+ 1
311
+ 1.5
312
+ 2
313
+ 2.5
314
+ 3
315
+ U (eV)
316
+ -8
317
+ -6
318
+ -4
319
+ -2
320
+ 0
321
+ 2
322
+ Reaction Enthalpy (eV per O2)
323
+ VO/V2O3
324
+ V2O3/VO2
325
+ VO2/V2O5
326
+ Experimental
327
+ 0.0 eV
328
+ 0.7 eV
329
+ 1.0 eV
330
+ 2.2 eV
331
+ (c)
332
+ 0
333
+ 0.5
334
+ 1
335
+ 1.5
336
+ 2
337
+ 2.5
338
+ U(eV)
339
+ -7
340
+ -6
341
+ -5
342
+ -4
343
+ -3
344
+ -2
345
+ -1
346
+ 0
347
+ 1
348
+ Reaction Enthalpy (eV per O2)
349
+ MnO/Mn2O3
350
+ Mn2O3/MnO2
351
+ Experimental
352
+ MnO/Mn3O4
353
+ 1.5 eV
354
+ 1.8 eV
355
+ 2.1 eV
356
+ Figure 1: Calculated oxidation enthalpy versus the magnitude of U correction within r2SCAN+U framework
357
+ for (a) Ti, (b) V, (c) Mn, (d) Fe, (e) Co, and (f) Ni oxides. Solid, dashed, and dotted lines of a given color
358
+ indicate calculated, experimental, and required U values for a given oxidation reaction. Optimal U for each
359
+ TM is indicated by the dotted blue line in each panel.
360
+ 3.2
361
+ Lattice parameters
362
+ All r2SCAN(+U ) and SCAN(+U ) calculated lattice parameters, on-site magnetic moments, and band
363
+ gaps for each TMO are tabulated in Table S2. Additionally, the calculated lattice volumes by the four XC
364
+ functionals are plotted against experimental data in Figure 2a for all oxides. Generally, both SCAN (green
365
+ squares in Figure 2a) and r2SCAN (blue symbols) offer < 2.8% deviation from the experimental lattice
366
+ parameters for all the TMOs considered, except VO, FeO, CuO, and LiNiO2, indicating robust agreement
367
+ with experiments for both functionals. In VO, SCAN and r2SCAN overestimate (by ∼8%) the experimental
368
+ lattice constants, while the deviation in FeO and CuO is ∼3-4% and ∼8-10%, respectively.
369
+ In LiNiO2,
370
+ 6
371
+
372
+ SCAN’s Ξ² angle evaluation is ∼4.1% different from experiment.
373
+ Notably, SCAN and r2SCAN do show qualitative differences in their calculated lattice parameters (when
374
+ compared against experiments) across TMOs. For instance, both functionals overestimate the experimental
375
+ lattice constants in TiO2, Ti2O3, and VO, while they underestimate in CrO2, CrO3, MnO2, and Fe3O4.
376
+ There are also examples (MnO and Mn2O3) where SCAN underestimates the experimental lattice constants
377
+ while r2SCAN overestimates. Overall, there are cases where SCAN’s errors in lattice parameter estimations
378
+ are lower versus experiments (e.g., Cr2O3, CoO), r2SCAN’s errors are lower (e.g., CrO2, CrO3, MnO2,
379
+ Fe3O4), and both functionals exhibit identical errors (e.g., TiO2, Co3O4, NiO, Cu2O), signifying that both
380
+ functionals offer similar performance in terms of geometrical properties.
381
+ Comparing r2SCAN and SCAN, we find that r2SCAN’s lattice constants are generally larger than SCAN
382
+ across TMOs (e.g., Ti2O3, Cr2O3, CrO3, VO2, etc.). As a range, r2SCAN estimates lattice constants that
383
+ are a maximum of ∼1.5% larger than SCAN (in CrO3) and a minimum of ∼0.1% larger than SCAN (in
384
+ Mn2O3). Having said that, there are instances where r2SCAN’s lattice constant evaluations are lower than
385
+ SCAN (VO, CoO, CuO, and LiNiO2) and cases where both functionals are identical (TiO2, Co3O4, NiO,
386
+ and Cu2O). In specific TMOs, SCAN and r2SCAN calculate an identical (individual) lattice constant, while
387
+ the other lattice constants with r2SCAN are larger than SCAN. For example, a and c lattice constants with
388
+ r2SCAN are higher than SCAN in V2O5 while both functionals estimate b = 3.55 ˚A.
389
+ On introducing the optimal U correction, an increase in the value of calculated lattice constants is ob-
390
+ tained for both SCAN and r2SCAN functionals for all TMOs. The lattice constants computed by r2SCAN+U
391
+ (yellow symbols in Figure 2a) is up to 1.3% higher than r2SCAN, except FeO (∼4.2% higher). Similar to the
392
+ comparison of r2SCAN vs. SCAN, there are systems where r2SCAN+U predicts larger, smaller, and identical
393
+ lattice constants compared to SCAN+U (red triangles). For example, r2SCAN+U calculates larger lattice
394
+ constants than SCAN+U in VO2, V2O5, MnO, Mn2O3 and Fe3O4 (maximum of ∼0.5% higher in V2O5),
395
+ while for Ti2O3, CoO and NiO, r2SCAN+U ’s estimations are smaller than SCAN+U (maximum deviation
396
+ of ∼2.1% in Ti2O3). Both SCAN+U and r2SCAN+U functionals evaluate identical lattice parameters for
397
+ TiO2, Co3O4 and LiNiO2.
398
+ Overall, lattice constants calculated by SCAN+U and r2SCAN+U deviate <∼3.3% from experiments
399
+ for all TMOs, except VO and VO2 where deviations of ∼8.5% and ∼4.6% are observed, respectively. Adding
400
+ U improves the agreement with experiment for both SCAN and r2SCAN in Co3O4, while r2SCAN+U gives
401
+ the best estimate of the lattice parameters in FeO (< 1% deviation vs. experiments) compared to SCAN,
402
+ SCAN+U and r2SCAN. Notably, all functionals break the rocksalt symmetry of VO, MnO, and FeO, while
403
+ the cubic symmetry of Fe3O4 is retained only by SCAN. In Ti2O3, the hexagonal symmetry is broken by
404
+ SCAN but the symmetry is preserved by the other frameworks. In summary, we find that the differences in
405
+ lattice parameter estimations to be minimal across the four functionals on average, with notable exceptions
406
+ of a few systems.
407
+ 3.3
408
+ On-site magnetic moments
409
+ On-site magnetic moments of the TMOs (Figure 2c and Table S2) computed by SCAN and r2SCAN
410
+ generally underestimate experimental values, with the exception of MnO2, Mn2O3, CrO2, and VO2. Note
411
+ that larger magnetic moments typically indicate stronger localization of d electrons. Comparing r2SCAN
412
+ and SCAN calculations, we find that r2SCAN typically estimates smaller magnetic moments than SCAN
413
+ but with several exceptions, such as MnO, MnO2, Mn2O3, Cr2O3, and VO2. Thus, on average, SCAN’s
414
+ magnetic moment predictions are in better agreement with experiments. However, in terms of magnitude,
415
+ 7
416
+
417
+ Ti2O3
418
+ TiO2
419
+ VO
420
+ V2O3
421
+ VO2
422
+ V2O5
423
+ Cr2O3
424
+ CrO2
425
+ CrO3
426
+ MnO
427
+ Mn3O4
428
+ Mn2O3
429
+ MnO2
430
+ Fe2O3
431
+ Fe3O4
432
+ FeO
433
+ CoO
434
+ Co3O4
435
+ NiO
436
+ LiNiO2
437
+ Cu2O
438
+ CuO
439
+ SCAN
440
+ SCAN+U
441
+ r2SCAN
442
+ r2SCAN+U
443
+ 0.8
444
+ 0.6
445
+ 0.4
446
+ 0.2
447
+ 0.0
448
+ 0.2
449
+ 0.4
450
+ 0.6
451
+ 0.8
452
+ On-site magnetic moment
453
+ difference (
454
+ B)
455
+ Ti2O3
456
+ TiO2
457
+ VO
458
+ V2O3
459
+ VO2
460
+ V2O5
461
+ Cr2O3
462
+ CrO2
463
+ CrO3
464
+ MnO
465
+ Mn3O4
466
+ Mn2O3
467
+ MnO2
468
+ Fe2O3
469
+ Fe3O4
470
+ FeO
471
+ CoO
472
+ Co3O4
473
+ NiO
474
+ LiNiO2
475
+ Cu2O
476
+ CuO
477
+ SCAN
478
+ SCAN+U
479
+ r2SCAN
480
+ r2SCAN+U
481
+ 0.8
482
+ 0.6
483
+ 0.4
484
+ 0.2
485
+ 0.0
486
+ 0.2
487
+ 0.4
488
+ 0.6
489
+ 0.8
490
+ On-site magnetic moment
491
+ difference (
492
+ B)
493
+ SCAN
494
+ SCAN+U
495
+ r2SCAN
496
+ r2SCAN+U
497
+ 2.0
498
+ 1.5
499
+ 1.0
500
+ 0.5
501
+ 0.0
502
+ 0.5
503
+ 1.0
504
+ 1.5
505
+ Band gap difference (eV)
506
+ (a)
507
+ (c)
508
+ (b)
509
+ 0
510
+ 100
511
+ 200
512
+ 300
513
+ 400
514
+ 500
515
+ 600
516
+ 700
517
+ 800
518
+ 900
519
+ Experimental lattice volume (Γ…3)
520
+ 0
521
+ 100
522
+ 200
523
+ 300
524
+ 400
525
+ 500
526
+ 600
527
+ 700
528
+ 800
529
+ 900
530
+ Predicted lattice volume (Γ…3)
531
+ SCAN
532
+ SCAN+U
533
+ r2SCAN
534
+ r2SCAN+U
535
+ Figure 2: (a) Comparison of calculated and experimental lattice volume (in ˚A3) of all TMOs considered.
536
+ (b) Violin plot capturing the difference between the experimental and computed band gap (in eV) across
537
+ TMO systems using the four XC frameworks. The empty circle and horizontal line in the inner box plot
538
+ corresponds to the mean and median of the calculated band gaps, respectively. (c) Heat map representation
539
+ of the differences between the experimental and calculated on-site magnetic moments (in Β΅B) using the
540
+ four XC functionals and across all TMOs. A value of zero indicates perfect consistency, while red (blue)
541
+ colors indicate overestimation (underestimation) of magnetic moments. Hatched boxes either correspond to
542
+ experimentally undetermined magnetic moments (VO) or calculations not executed with U frameworks (Cr
543
+ and Cu oxides).
544
+ moments predicted by r2SCAN deviate by < 3% from SCAN’s estimates, except CuO (∼ 6.8% deviation),
545
+ CrO2 (∼ 3.5%), and MnO2 (∼ 3.5%), highlighting that the differences in the predictions are marginal.
546
+ Adding optimal U to both SCAN and r2SCAN increases the magnitude of the calculated on-site magnetic
547
+ moments for all TMOs (except VO2, which is predicted to be metallic by all functionals), consistent with
548
+ the expectation that the U correction facilitates d electron localization. r2SCAN+U -calculated data are
549
+ similar to the corresponding SCAN+U values (< 2.3% variation), except LiNiO2 (∼6.3% variation), and
550
+ Ti2O3 (∼3.8%). Similar to r2SCAN versus SCAN, r2SCAN+U estimates smaller magnetic moments than
551
+ SCAN+U, with notable exceptions being VO2, Mn2O3, MnO2 and FeO. Overall, we observe the accuracy
552
+ in calculated on-site magnetic moments versus experiments to follow the order SCAN+U > r2SCAN+U >
553
+ SCAN > r2SCAN for several TMOs. However, there are specific cases where specific XC frameworks offer
554
+ better accuracy in calculating magnetic moments, such as SCAN in CrO2, Mn2O3, MnO2, Fe3O4 and CuO,
555
+ 8
556
+
557
+ r2SCAN in Mn3O4 and Cr2O3, and r2SCAN+U in V2O3. Given the numerically marginal deviations in
558
+ calculated magnetic moments across the XC frameworks (∼10% deviation), we expect an increase/decrease
559
+ in accuracy to be marginal amongst the XC frameworks considered.
560
+ 3.4
561
+ Band gaps
562
+ The differences between calculated and experimental band gaps of all TMOs considered are visualized
563
+ as violin plots for SCAN (green violin), SCAN+U (red), r2SCAN (blue), and r2SCAN+U in Figure 2b.
564
+ The top and bottom ends of the individual violins mark the highest and lowest differences in the respective
565
+ calculated data. Note that the mean values (white empty circles) are similar for SCAN and r2SCAN, and
566
+ in turn are lower than their U -corrected versions. In other words, addition of the U-correction reduces
567
+ the error of calculated band gaps compared to experimental values, which is expected given that semi-local
568
+ DFT typically underestimates band gaps.
569
+ Also, we find that SCAN+U displays the lowest mean band
570
+ gap difference among the XC functionals considered, indicating that on-average SCAN+U provides better
571
+ computed band gaps.
572
+ We present calculated electronic DOS of select TMOs, namely CoO (panels a and b), V2O3 (c and d),
573
+ and Mn2O3 (e and f), in Figure 3, to illustrate qualitative trends in computed band gaps. The DOS for
574
+ the remaining TMOs, calculated by the four XC frameworks, are compiled in Figures S3-S19 of the SI. In
575
+ each DOS panel, solid orange and solid green lines correspond to the 2p-states of O and the 3d-states of the
576
+ TM, respectively. Dashed black lines represent Fermi levels in metallic compounds. Dotted vertical lines
577
+ represent valence and conduction band edges in semiconducting/insulating compounds, with the band gaps
578
+ indicated by the text annotation near the conduction band minimum (CBM). The zero of the energy scale
579
+ is set to the valence band maximum (VBM) for TMOs with a band gap and to the Fermi level in metallic
580
+ TMOs.
581
+ We observe that r2SCAN generally calculates a smaller band gap than SCAN for most TMOs (maximum
582
+ of ∼66% lower in MnO2, see Table S2), as illustrated by the case of CoO in panels a and b of Figure 3.
583
+ Notable exceptions do exist to this observation, such as V2O5 (∼1.7% larger), CrO3 (∼3.2%), MnO (∼4.3%),
584
+ and Fe2O3 (∼1.7%), where r2SCAN calculated band gaps are marginally larger than SCAN. Both SCAN
585
+ and r2SCAN incorrectly describe the ground state electronic configuration of narrow band gap TMOs (i.e.,
586
+ experimental band gaps < 1 eV), including Ti2O3 (Figure S4), V2O3(Figure 3c and S3c), VO2 (Figure S7)
587
+ and Fe3O4 (Figure S15) to be metallic, with the exception of MnO2 where both SCAN and r2SCAN estimate
588
+ a narrow gap (Figures S12a and S12c). Additionally, both functionals also calculate the wrong electronic
589
+ structure in the case of a non-narrow-gap semiconductor, Mn2O3 (Figure S3), which exhibits an experimental
590
+ gap of 1.2-1.3 eV. [83,84] However, SCAN and r2SCAN qualitatively describe the right electronic structure in
591
+ the case of wide band gap TMOs such as FeO (Figure S13), Fe2O3 (Figure S14), and NiO (Figure S17), with
592
+ a significant quantitative underestimation of the experimental gaps. In any case, the differences in electronic
593
+ structure predictions between SCAN and r2SCAN in TMOs are minimal, with SCAN being marginally better
594
+ in accuracy.
595
+ Introducing a U correction to SCAN and r2SCAN widens or opens the band gap, especially in narrow
596
+ band gap TMOs, as illustrated by the case of V2O3 (panels c and d in Figure 3). The opening of band
597
+ gap with U correction is expected since localization of d electrons, which form the VBM and/or CBM in
598
+ 3d-TMOs, is faciliated with U addition, in turn resulting in a larger gap. However, in the case of VO2
599
+ (Figure S7), adding U does not capture the MIT that occurs at low temperatures (< 341 K [61]) with either
600
+ SCAN or r2SCAN, causing the erroneous prediction of metallic behavior. Generally, SCAN+U calculates
601
+ 9
602
+
603
+ 0.36 eV
604
+ (a)
605
+ 0.91 eV
606
+ (b)
607
+ 0.61 eV
608
+ (c)
609
+ (d)
610
+ 0.24 eV
611
+ (e)
612
+ (f)
613
+ Figure 3: DOS for CoO calculated using (a) SCAN and (b) r2SCAN, DOS for V2O3 computed using (c)
614
+ r2SCAN and (d) r2SCAN+U, and DOS for Mn2O3 estimated using (e) SCAN+U and (f) r2SCAN+U.
615
+ a larger band gap than r2SCAN+U (Table S2), as highlighted by the case of Mn2O3 (panels e and f in
616
+ Figure 3). In fact, SCAN+U is the only framework (among those considered) to estimate a band gap in
617
+ 10
618
+
619
+ 6
620
+ /eV
621
+ (states)
622
+ 4
623
+ 2
624
+ Density of states (
625
+ -6
626
+ -4-3γ€€-2γ€€-1
627
+ 2
628
+ 3
629
+ 4
630
+ 5
631
+ Energy (eV)6
632
+ /eV
633
+ (states)
634
+ 4
635
+ Density of states (
636
+ -6
637
+ .5
638
+ -4γ€€-3-2
639
+ 0
640
+ 2
641
+ 3
642
+ 4
643
+ Energy (eV)/eV)
644
+ 60
645
+ Mnd
646
+ (states)
647
+ 40
648
+ 20
649
+ MWN
650
+ Density of states
651
+ -20
652
+ 40
653
+ -60
654
+ -6
655
+ -5
656
+ 4
657
+ Β₯-3
658
+ Β₯-2-101
659
+ 2
660
+ 3
661
+ 4
662
+ Energy (eV)60
663
+ /eV)
664
+ Mnd
665
+ (states)
666
+ 40
667
+ 20
668
+ Density of states
669
+ -20
670
+ 40
671
+ -60
672
+ -6
673
+ -5
674
+ 4
675
+ 2
676
+ 3
677
+ 4
678
+ Energy (eV)/ev
679
+ 10
680
+ (states)
681
+ 5
682
+ Density of states
683
+ 0
684
+ -10
685
+ -6
686
+ -5
687
+ -2
688
+ 0
689
+ 2
690
+ 3
691
+ 4
692
+ Energy (eV)/eV
693
+ 10
694
+ (states)
695
+ Density of states
696
+ Wi
697
+ -10
698
+ -6
699
+ -5
700
+ Β₯1
701
+ 2
702
+ 3
703
+ 4
704
+ Energy (eV)Mn2O3, which is consistent with experiment. Moreover, SCAN+U ’s evaluations of larger band gaps results
705
+ in better (poorer) quantitative agreement with experiments in wide (narrow) gap materials, such as MnO
706
+ and FeO (V2O3 and MnO2).
707
+ Note that SCAN+U and r2SCAN+U do underestimate the experimental band gaps, similar to SCAN and
708
+ r2SCAN, in wide gap TMOs. The only exception to this observation is CoO, where SCAN+U overestimates
709
+ the band gap versus experiment (Figure S3a and Table S2), as also observed in our previous work. [38] In select
710
+ TMOs, including Fe2O3 and V2O5, r2SCAN+U ’s band gap is larger than SCAN+U, but the magnitude of
711
+ difference (≀ 0.2 eV) is meagre. Thus, for electronic structure predictions, we expect SCAN+U to provide the
712
+ best qualitative and quantitative band gaps across TMOs, among the functionals considered here, especially
713
+ for wide gap semiconductors/insulators. However, the qualitative trends provided by r2SCAN+U are quite
714
+ robust as well and in small gap semiconductors (< 1 eV gap), r2SCAN+U ’s quantitative accuracy is often
715
+ better than SCAN+U.
716
+ 3.5
717
+ Transferability checks
718
+ To examine the transferability of the optimal U values determined in this work (with r2SCAN), to oxide
719
+ systems not used for obtaining the values, we perform checks on systems with different oxidation state and/or
720
+ coordination environment for each TM. We compare calculated values against available experimental data,
721
+ such as structural, electronic, magnetic, and/or electrochemical properties. Specifically, we choose Ba2TiO4
722
+ as a check for Ti, BiVO4 for V, K3MnO4, K2MnO4, and Mn2O7 for Mn, SrFeO3 for Fe, LiCoO2-CoO2 for
723
+ Co, and LiNiO2-NiO2 for Ni. Data related to transferability checks are compiled in Figure 4, Table 1, and
724
+ Table S3.
725
+ In the case of Ba2TiO4, we compare the calculated lattice parameters with experimental values (see
726
+ Table S3 and lattice voliume differences plotted in Figure 4). Ba2TiO4 crystallizes in a monoclinic structure
727
+ (space group P21/n) at low temperatures, where the unit cell is composed of four formula units. [85, 86]
728
+ Ti atoms are present in distorted tetrahedra composed of neighbouring oxygen atoms (TiO4) within the
729
+ Ba2TiO4 lattice, which is different from the octahedral environments sampled in TiO2 and Ti2O3. Upon
730
+ structure relaxation, we observe that both r22SCAN and r22SCAN+U functionals marginally overestimate
731
+ (by ∼2%) experimental lattice parameters (Figure 4 and Table S3). Similar to trends observed in Table S2,
732
+ adding U to r2SCAN increases the calculated lattice parameters in Ba2TiO4 (by ∼0.03 ˚A), thereby marginally
733
+ reducing the agreement with experiment.
734
+ We benchmark both structural and electronic properties of BiVO4 as a transferability check for V-based
735
+ systems. Note that BiVO4 transforms from tetragonal (I 41/a) to a monoclinic (I 2/b) β€˜scheelite’ phase below
736
+ ∼ 528 K, [87, 88] which is a reversible second order ferroelastic transition driven by soft optical phonon
737
+ modes. The BiVO4 unit cell possesses four formula units, with tetrahedrally coordinated V ions, which is
738
+ different from the coordination environments of V in VO, V2O3, VO2, and V2O5. Importantly, monoclinic-
739
+ BiVO4 spontaneously transforms to the tetragonal structure upon structure relaxation with r22SCAN and
740
+ r22SCAN+U, similar to the observation by Liu et al [87] with GGA and hybrid functionals. Thus, neither
741
+ r22SCAN nor r22SCAN+U predict the correct ground state structure. Additionally, BiVO4 possess a band
742
+ gap of 2.4–2.48 eV [89] and is a candidate photocatalyst. [87] Both r22SCAN and r2SCAN+U provide similar
743
+ band gap predictions (2.01-1.98 eV), which is in good qualitative agreement with experiment. Surprisingly,
744
+ r2SCAN+U evaluates a marginally lower band gap than r2SCAN (see panels a and b in Figure 4). However,
745
+ both r22SCAN and r2SCAN+U predict similar states occupying the valence band (Op) and conduction band
746
+ (Vd) edges.
747
+ 11
748
+
749
+ Ba2TiO4
750
+ BiVO4
751
+ K3MnO4
752
+ K2MnO4
753
+ Mn2O7
754
+ SrFeO3
755
+ r2SCAN
756
+ r2SCAN+U
757
+ 20
758
+ 10
759
+ 0
760
+ 10
761
+ 20
762
+ Lattice volume
763
+ difference (Γ…3)
764
+ (a)
765
+ 1.98 eV
766
+ (b)
767
+ (c)
768
+ 2.013 eV
769
+ Figure 4: DOS for BiVO4 calculated using (a) r2SCAN and (b) r2SCAN+U. (c) Difference between experi-
770
+ mental and calculated lattice volumes (using r2SCAN and r2SCAN+U ), plotted as a heatmap, for various
771
+ systems. Red (blue) squares indicate overestimated (underestimated) calculated lattice volumes versus ex-
772
+ periment.
773
+ The rationale behind the choice of K3MnO4, K2MnO4, and Mn2O7 as checks for Mn-based systems is to
774
+ explore the higher, unsampled oxidation states of Mn, namely +5, +6, and +7 in K3MnO4, K2MnO4, and
775
+ Mn2O7, respectively. Also, Mn resides in tetrahedral coordination in these compounds, which is different from
776
+ the octahedral coordination observed in MnO, Mn2O3, and MnO2. Although Mn2+ resides in tetrahedral
777
+ sites in spinel-Mn3O4, we had not used in the spinel structure to obtain our optimal U. We benchmark the
778
+ calculated lattice parameters versus experiments for all Mn-based transferability checks.
779
+ Mn2O7 is a volatile liquid at 298 K and solidifies to a monoclinic crystal structure (P21/c) below ∼ 279 K,
780
+ with the unit cell consisting of 8 formula units of corner sharing tetrahedral MnO4 pairs. [90, 91] Upon
781
+ structural relaxation, both r2SCAN and r2SCAN+U underestimate the lattice constants of monoclinic-
782
+ Mn2O7 by ∼1-3% (Figure 4 and Table S3). In the case of K3MnO4, the tetragonal symmetry (I 42m) [92]
783
+ is broken with r2SCAN functional resulting in an orthorhombic structure, while the symmetry is preserved
784
+ by r2SCAN+U (see Figure
785
+ 4 and Table S3). Nonetheless, both r2SCAN and r2SCAN+U significantly
786
+ underestimate the c parameter (by ∼ 13.5%) and overestimate the a or b parameter (∼ 10.2%). K2MnO4 is
787
+ an orthorhombic crystal (Pnma) with four formula units per unit cell. [93] Here, r2SCAN and r2SCAN+U
788
+ predict identical lattice parameters, which marginally underestimate experimental values (by ∼ 0.4-1%, see
789
+ Figure 4 and Table S3).
790
+ The choice of SrFeO3, a cubic perovskite, as a check for Fe is largely motivated by the 4+ oxidation
791
+ state exhibited by Fe in the structure, which is not sampled in FeO, Fe2O3, or Fe3O4. Both r2SCAN and
792
+ r2SCAN+U preserve the cubic symmetry during structure relaxation, with r2SCAN+U ’s lattice parameters
793
+ 12
794
+
795
+ (states/eV
796
+ Bi,
797
+ Bi
798
+ Density of states (
799
+ -6
800
+ -5
801
+ -4γ€€-3
802
+ Β₯-2
803
+ 0
804
+ 2
805
+ 3
806
+ 4
807
+ Energy (eV)(states/eV
808
+ Bip
809
+ Bi
810
+ Density of states (
811
+ -6
812
+ -5
813
+ -4 -3
814
+ Β₯-2
815
+ 0
816
+ 2
817
+ 3
818
+ 4
819
+ Energy (eV)(states/eV
820
+ Bi,
821
+ Bi
822
+ Density of states (
823
+ -6
824
+ -5
825
+ -4γ€€-3
826
+ Β₯-2
827
+ 0
828
+ 2
829
+ 3
830
+ 4
831
+ Energy (eV)(states/eV
832
+ Bi,
833
+ Bi
834
+ Density of states (
835
+ -6
836
+ -5
837
+ -4γ€€-3
838
+ Β₯-2
839
+ 0
840
+ 2
841
+ 3
842
+ 4
843
+ Energy (eV)(states/eV
844
+ Bi,
845
+ Bi
846
+ Density of states (
847
+ -6
848
+ -5
849
+ -4γ€€-3
850
+ Β₯-2
851
+ 0
852
+ 2
853
+ 3
854
+ 4
855
+ Energy (eV)(states/eV
856
+ Bi,
857
+ Bi
858
+ Density of states (
859
+ -6
860
+ -5
861
+ -4γ€€-3
862
+ Β₯-2
863
+ 0
864
+ 2
865
+ 3
866
+ 4
867
+ Energy (eV)identical to experiments and r2SCAN’s parameters being a slight underestimation (∼ 0.5%, see Figure 4
868
+ and Table S3). In terms of magnetic configuration of Fe in SrFeO3, Takeda et al. [94] reported a helical spin
869
+ structure via their neutron diffraction experiments, with competing FM and AFM interactions. However,
870
+ Shein et al. [95] found a FM metallic state to be the ground state of SrFeO3, over a wide range of pressures,
871
+ based on their first principles calculations, which they attributed to stronger FM than AFM interactions. We
872
+ considered a FM configuration of Fe atoms in the SrFeO3 unit cell, and the on-site magnetic moments on Fe
873
+ calculated by both r2SCAN (3.375 Β΅B, Table 1) and r2SCAN+U (3.819 Β΅B) overestimate the experimental
874
+ value (2.7±0.4 ¡B [94]). However, our calculated magnetic moments do indicate a localization of ∼4 electrons
875
+ on the d orbitals of Fe, consistent with its +4 oxidation state.
876
+ We choose CoO2 (R3m or β€˜O3β€˜ polymorph [96]), and NiO2 (P1m1 or β€˜O1’ [97]), both layered structures,
877
+ as transferability checks for Co and Ni, respectively, owing to the unsampled 4+ oxidation states of each
878
+ TM. In terms of experimental property to benchmark, we choose the average Li intercalation voltage in these
879
+ structures, i.e., LiCoO2-CoO2, and LiNiO2-NiO2 pairs, since they have been measured with high precision.
880
+ The reader is referred to previous works on calculating and benchmarking average β€˜topotactic’ intercalation
881
+ voltages. [98,99] r2SCAN underestimates the experimental average voltage [96,99–103] in LiNiO2-NiO2 (by
882
+ ∼ 8%), while it overestimates the average voltage in LiCoO2-CoO2 (by ∼ 1.7%), similar to trends observed
883
+ with SCAN. [99] The addition of U to r2SCAN leads to an improvement in agreement with the experimental
884
+ voltage in the Ni-system (deviation of ∼ 1.8%), while it worsens the agreement in the Co-system (deviation
885
+ of ∼ 4.4%). Nevertheless, r2SCAN+U does overestimate the average voltage in both Co and Ni systems,
886
+ similar to the behavior of SCAN+U. [99]
887
+ Table 1: Voltage and magnetic moments calculated by r2SCAN, and r2SCAN+U compared against experi-
888
+ mental values (denoted by β€˜Expt.’). The U values used with r2SCAN+U are the corresponding optimal U
889
+ values obtained for each TM (from Figure 1).
890
+ Composition
891
+ Source
892
+ Voltage
893
+ Magnetic moment
894
+ (space group)
895
+ (V)
896
+ (Β΅B)
897
+ LiCoO2-CoO2
898
+ Expt.
899
+ 4.05
900
+ -
901
+ (RΒ―3m)
902
+ r2SCAN
903
+ 4.12
904
+ -
905
+ r2SCAN+U
906
+ 4.23
907
+ -
908
+ LiNiO2-NiO2
909
+ Expt.
910
+ 3.85
911
+ -
912
+ (P1m1)
913
+ r2SCAN
914
+ 3.54
915
+ -
916
+ r2SCAN+U
917
+ 3.92
918
+ -
919
+ SrFeO3
920
+ Expt.
921
+ -
922
+ 2.7Β±0.4
923
+ (PmΒ―3m)
924
+ r2SCAN
925
+ -
926
+ 3.375
927
+ r2SCAN+U
928
+ -
929
+ 3.819
930
+ 4
931
+ Discussion
932
+ In this work, we evaluated the performance of the r2SCAN functional among binary TMOs consisting
933
+ of 3d-TMs by calculating the oxidation enthalpies, lattice parameters, on-site magnetic moments, and band
934
+ gaps. Additionally, for each TM-O2 system considered, we calculated the optimal Hubbard-U corrections
935
+ to be used in a r2SCAN+U framework, based on experimental oxidation enthalpies. Although theoretical
936
+ approaches exist to derive U values, [41–47] using oxidation enthalpies nominally gives an β€œaverage” correc-
937
+ tion that is suitable across several oxidation states of a given TM. Specifically, our optimal U values are 2.3,
938
+ 13
939
+
940
+ 1.0, 1.8, 3.1, 1.8, and 2.1 eV for Ti, V, Mn, Fe, Co, and Ni, respectively, while we don’t deem a U correction
941
+ necessary for Cr and Cu oxides. Interestingly, the optimal U corrections needed with r2SCAN are lower in
942
+ magnitude compared to SCAN for Ti, Mn, Co, and Ni oxides (while the corrections are identical for V and
943
+ Fe oxides), indicating that r2SCAN exhibits lower errors with oxidation enthalpies and possibly lower SIEs
944
+ than SCAN. However, this is not reflected in other physical properties. On an average, we find the accuracy,
945
+ versus experimental values, to be similar for r2SCAN compared to SCAN, and for r2SCAN+U compared to
946
+ SCAN+U, respectively, in lattice parameter, on-site magnetic moment, and band gap evaluations as seen in
947
+ Figure 2.
948
+ The general trends in lattice parameter, magnetic moment, and band gap predictions, across the XC
949
+ frameworks considered, can be summarized as follows. We observe that r2SCAN generates larger lattice
950
+ constants than SCAN and on addition of the U correction to both functionals, the lattice constants further
951
+ increase. Thus, in systems where SCAN underestimates experimental lattice constants (e.g., CrO2, CrO3,
952
+ MnO2), shifting to r2SCAN improves agreement (e.g., error in r2SCAN in CrO3 is 0.8% versus 2.3% with
953
+ SCAN). Also, there are instances where the ground state symmetry of the TMO is not preserved by some
954
+ or all of the XC frameworks considered (i.e., in VO, MnO, FeO, Fe3O4, and Ti2O3), highlighting systematic
955
+ issues in the XC treatment across the four frameworks considered. The calculated on-site magnetic moments
956
+ by r2SCAN (and r2SCAN+U ) are marginally lower than SCAN (SCAN+U ), with the U correction nom-
957
+ inally increasing the calculated moments calculated by r2SCAN and SCAN. However, calculated magnetic
958
+ moments across the four XC frameworks differ by < 10% (except LiNiO2), signifying marginal differences
959
+ in accuracy. Both SCAN and r2SCAN underestimate band gaps across all TMOs (except MnO2), with
960
+ band gaps calculated by r2SCAN typically being lower than SCAN, and adding the U opens/widens the
961
+ gap. Thus, SCAN+U offers the best quantitative accuracy versus experimental band gaps, especially for
962
+ wide gap semiconductors. Note that the qualitative trends from r2SCAN+U are consistent with the trends
963
+ exhibited by SCAN+U and should be reliable in electronic structure predictions in other TM-based oxide
964
+ systems.
965
+ r2SCAN adopts the smooth polynomial interpolation function of rSCAN to maintain numerical stability
966
+ during SCF calculations. Additionally, the reformed gradient expansion for correlation introduced in r2SCAN
967
+ (partially) negates the error introduced to the slowly varying density by the non-vanishing interpolation
968
+ function, [32] which largely accounts for the observed variation in accuracy of r2SCAN versus SCAN. Based
969
+ on our data, we observe that r2SCAN is not systematically more accurate than SCAN across all TMOs
970
+ and for all property predictions. For example, we have lower optimal U values indicating lower SIEs with
971
+ r2SCAN versus SCAN, but also lower on-site magnetic moments (except Mn and Cr oxides) signifying poorer
972
+ d-electron localization with r2SCAN. Further, the smaller band gaps with r2SCAN (versus SCAN) may be
973
+ caused by the residual SIEs, resulting in an underestimation of the CBM across TMOs. Hence, usage of
974
+ r2SCAN(+U ) in TM-based systems must be done with care and efforts should be made to benchmark as
975
+ many available experimental properties as possible before performing β€œtrue” computational predictions.
976
+ We considered the transferability of the U values estimated in this work, with r2SCAN, by examining
977
+ systems for each TM with oxidation states and/or coordination environments not sampled while calculating
978
+ the optimal U. In general, we find that r2SCAN or its Hubbard U corrected version estimate similar lattice
979
+ parameters and hence yield similar accuracies on structural properties. Analogously, the calculated on-site
980
+ magnetic moments in SrFeO3 and the band gaps in BiVO4 are similar between r2SCAN and r2SCAN+U. In
981
+ case of electrochemical properties, we do find tangible variations in the calculated average voltages of r2SCAN
982
+ and r2SCAN+U, with r2SCAN+U exhibiting overall lower errors across the Co and Ni systems. Thus, we
983
+ 14
984
+
985
+ SCAN+U
986
+ r2SCAN
987
+ r2SCAN+U
988
+ Ti
989
+ V
990
+ Cr
991
+ Mn
992
+ Fe
993
+ Co
994
+ Ni
995
+ Cu
996
+ 0
997
+ 1
998
+ 2
999
+ 3
1000
+ 4
1001
+ 5
1002
+ 6
1003
+ 7
1004
+ 8
1005
+ 9
1006
+ Ti
1007
+ V
1008
+ Cr
1009
+ Mn
1010
+ Fe
1011
+ Co
1012
+ Ni
1013
+ Cu
1014
+ 0.6
1015
+ 0.8
1016
+ 1
1017
+ 1.2
1018
+ 1.4
1019
+ 1.6
1020
+ 1.8
1021
+ (a)
1022
+ Ti
1023
+ V
1024
+ Cr
1025
+ Mn
1026
+ Fe
1027
+ Co
1028
+ Ni
1029
+ Cu
1030
+ 0.6
1031
+ 0.8
1032
+ 1
1033
+ 1.2
1034
+ (c)
1035
+ (b)
1036
+ RelaοΏ½ve overall
1037
+ computaοΏ½on οΏ½me
1038
+ RelaοΏ½ve computaοΏ½on οΏ½me
1039
+ per ionic step
1040
+ RelaοΏ½ve computaοΏ½on οΏ½me
1041
+ per electronic step
1042
+ Figure 5: (a) Overall computational time (electronic+ionic steps) (b) computational time per ionic step and
1043
+ (c) computational time per electronic loop taken for each TM-O2 binary system with SCAN+U, r2SCAN,
1044
+ and r2SCAN+U frameworks relative to SCAN. Values greater (smaller) than 1 in each panel indicates that
1045
+ a given calculation is slower (faster) than SCAN.
1046
+ 15
1047
+
1048
+ find the optimal U values obtained in this work to be transferable across oxide frameworks not sampled a
1049
+ priori. Nevertheless, more benchmarking studies to compare the performance of r2SCAN+U with r2SCAN
1050
+ (and experiments) will help in quantifying the reliability and errors associated with using r2SCAN+U.
1051
+ Given that r2SCAN(+U ) is not systematically more or less accurate than SCAN(+U ), the computational
1052
+ performance and numerical stability of r2SCAN(+U ) is critical in determining its utility in property pre-
1053
+ dictions across materials. Thus, we have quantified the computational time of r2SCAN(+U ) and SCAN+U
1054
+ relative to SCAN for each TM-O2 system considered in Figure S1. Specifically, panels a, b, and c of Fig-
1055
+ ure 5 plot the overall (electronic+ionic steps), per ionic step, and per electronic step computational time,
1056
+ respectively, taken by the SCAN+U (blue bars), r2SCAN (red), and r2SCAN+U (yellow) frameworks, rel-
1057
+ ative to the computational time taken by the SCAN functional (dotted black lines), for each TM-based
1058
+ set of oxides. Details on calculating the computational times used by the functionals is described in the
1059
+ β€˜Computational time’ section of the SI. Note that our objective is not to provide a rigorous quantification of
1060
+ computational resources required for each XC framework, but to provide a qualitative understanding of the
1061
+ relative computational costs across the frameworks considered.
1062
+ For each electronic step, r2SCAN(+U ) is typically faster than SCAN (Figure 5), signifying better numer-
1063
+ ical stability than SCAN, with Mn, Ni, and Cu oxides being marginal exceptions. In contrast, on a per-ionic
1064
+ step basis, r2SCAN and r2SCAN+U is slower than SCAN, by ∼1.05-1.78Γ— and ∼1.1-1.31Γ—, respectively,
1065
+ highlighting that r2SCAN(+U ) takes more electronic steps to converge per ionic step. Importantly, the over-
1066
+ all computational time (ionic+electronic steps, Figure 5) required for structural relaxation of TMOs using
1067
+ r2SCAN and r2SCAN+U is lower than SCAN, by ∼12.1-61.2% and ∼1.9-34.5%, respectively, except in Fe
1068
+ oxides, indicating that r2SCAN(+U ) takes lower number of ionic steps to converge, which possibly indicates
1069
+ a better description of atomic forces. The higher overall computation time in Fe oxides with r2SCAN(+U )
1070
+ than SCAN is primarily due to the difficulty in converging Fe3O4 with r2SCAN(+U ). Comparing r2SCAN
1071
+ and r2SCAN+U, we find that r2SCAN+U takes a higher overall computational time to converge, except
1072
+ in Fe and Ni oxides.
1073
+ Thus, we expect r2SCAN(+U ) to provide good utility in property predictions in
1074
+ TM-containing systems given its better computational performance and reasonable accuracy compared to
1075
+ SCAN(+U ).
1076
+ 5
1077
+ Conclusion
1078
+ 3d-TMs and their compound phases find applications in several fields such as energy storage, solar
1079
+ cells, catalysts, thermochemical water splitting, etc., and it is imperative to predict their properties such
1080
+ as lattice constants, magnetic moments, reaction enthalpies, and band gaps accurately using DFT-based
1081
+ techniques for designing better materials. Recently, the r2SCAN metaGGA XC functional was proposed
1082
+ to exhibit the accuracy of its predecessor, SCAN, and the computational performance of rSCAN in main-
1083
+ group compounds, but the accuracy of r2SCAN was not rigorously tested on TM-based systems.
1084
+ Here,
1085
+ we assessed the numerical accuracy and computational performance of r2SCAN in binary 3d-TMOs, in
1086
+ calculating the lattice parameters, on-site magnetic moments, binary oxidation enthalpies, and band gaps
1087
+ against experimental data. Notably, we observed that r2SCAN exhibited similar qualitative trends as that
1088
+ of SCAN, with marginally larger estimations of lattice parameters than SCAN, while the on-site magnetic
1089
+ moments and band gap calculations are marginally smaller than SCAN. While both r2SCAN and SCAN
1090
+ underestimated the band gaps in wide gap TMOs, with SCAN offering slightly better accuracy, they failed
1091
+ to predict the correct ground state electronic configurations of narrow band gap TMOs (e.g., Mn2O3).
1092
+ 16
1093
+
1094
+ On analysing the addition of Hubbard U -correction to improve the accuracy of the r2SCAN functional,
1095
+ we observed that a lower optimal U value, based on experimental oxidation enthalpies, was required in
1096
+ a r2SCAN+U framework for Ti, Mn, Co and Ni oxides, when compared to a SCAN+U framework. The
1097
+ optimal U values were identical in both r2SCAN+U and SCAN+U frameworks for V and Fe oxides, while we
1098
+ did not observe the need for a U correction in Cr and Cu oxides with r2SCAN, similar to SCAN. Moreover,
1099
+ introducing the U -correction to SCAN and r2SCAN increased the calculated lattice parameters, on-site
1100
+ magnetic moments and the band gaps of the TMOs.
1101
+ r2SCAN+U and SCAN+U successfully opened a band gap for narrow gap TMOs (except VO2 and
1102
+ Mn2O3 with r2SCAN+U ). Upon testing the optimal U values with r2SCAN+U on oxides with different
1103
+ oxidation states and/or coordination environments, we found that the U values derived in this work are in
1104
+ general transferable to other TM-containing oxides as well. Furthermore, we observed that r2SCAN(+U )
1105
+ took less overall computational time (ionic+electronic steps) to converge when compared to SCAN, which
1106
+ indicated that r2SCAN(+U ) was computationally more efficient than SCAN(+U ). Since r2SCAN+U offers
1107
+ a reasonably accurate prediction of material properties at a lower computational expense than SCAN+U, we
1108
+ observe that r2SCAN+U can be used in high-throughput materials discovery, after adequate benchmarking
1109
+ tests are done in each new chemical space explored.
1110
+ Acknowledgments
1111
+ G.S.G. acknowledges the Indian Institute of Science (IISc) Seed Grant, SG/MHRD/20/0020 and SR/MHRD/20/0013
1112
+ and the Science and Engineering Research Board (SERB) of the Department of Science and Technology, Gov-
1113
+ ernment of India, under sanction numbers SRG/2021/000201 and IPA/2021/000007 for financial support.
1114
+ R.D. thanks the Ministry of Human Resource Development, Government of India, for financial assistance.
1115
+ S.S. acknowledges financial support from SERB under IPA/2021/000007. All the authors acknowledge the
1116
+ computational resources provided by the Supercomputer Education and Research Centre, IISc, for enabling
1117
+ some of the density functional theory calculations showcased in this work.
1118
+ Author Contributions
1119
+ G.S.G. envisioned and designed the work. S.S. and R.D. performed the calculations. All authors con-
1120
+ tributed in data analysis and writing the paper.
1121
+ Conflicts of Interest
1122
+ The authors declare no competing financial or non-financial interests.
1123
+ Availability of data
1124
+ The data that support the findings of this study are openly available at https://github.com/sai-mat-
1125
+ group/r2SCAN-U-benchmarking.
1126
+ Supplementary Materials
1127
+ Electronic Supporting Information is available online at , with details on the crystal structures used for
1128
+ calculations, oxidation energetics of Cr and Cu oxides, densities of states of all systems not showcased in the
1129
+ 17
1130
+
1131
+ main text, and details on computational time calculations.
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+ Chemie, 412(3):271–280, 1975.
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+ elastic properties of SrFeO3 and LaFeO3 perovskites. Phys. Solid State, 47(11):2082–2088, November
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+
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1
+ Astronomy & Astrophysics manuscript no. main
2
+ Β©ESO 2022
3
+ 2022-12-31
4
+ Stellar Karaoke: Deep Blind Separation of Terrestrial Atmospheric
5
+ Effects out of Stellar Spectra by Velocity Whitening
6
+ Nima Sedaghat1, J. Bryce Kalmbach1, Brianna M. Smart1, and Erin L. Howard1
7
+ DIRAC Institute and the Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195, USA
8
+ e-mail: [email protected]
9
+ 2022-12-31
10
+ ABSTRACT
11
+ We exploit the statistical independence of stellar features and atmospheric adversarial effects in stellar spectra, to remove the latter
12
+ from observed signals using a fully unsupervised data-driven approach. Concretely, we first increase the inter-observation entropy of
13
+ telluric absorption lines by imposing a random, virtual radial velocity to the observed spectrum. This novel β€œtrick” results in a non-
14
+ standard form of β€œwhitening” in the atmospheric components of the spectrum, decorelating them across multiple observations. Then
15
+ we use deep convolutional auto-encoders, to learn a feature-space in which the two β€œsources” of information, stellar and atmospheric,
16
+ are easily separable, leading to removal of the latter. We apply the process on spectra from two different data collections: ~250,000
17
+ HARPS spectra and ~660,000 from SDSS.
18
+ We compare and analyze the results across datasets, as well as with existing tools, and discuss directions for utilizing the introduced
19
+ method as a fast and more reliable tool in the future.
20
+ Key words. stellar spectra – telluric lines – deep learning – unsupervised – whitening – decorrelation – source separation
21
+ 1. Introduction
22
+ Throughout this paper, we introduce and elaborate a method
23
+ based on signal processing tricks combined with a fully unsuper-
24
+ vised deep neural network to remove the non-linear, time-variant
25
+ effect of telluric lines out of observed spectra without any need
26
+ for manual modeling and tuning. Concretely, we show how look-
27
+ ing at an enormous dataset of spectra reveals new insights into
28
+ their features, allowing for novel solutions to existing, difficult
29
+ problems.
30
+ In ground-based observations of stellar spectra, it is not un-
31
+ usual for the source’s spectrum to reach our detectors in an
32
+ altered state. The spectrum may be affected by a number of
33
+ events in space (redshift/blueshift, emission, absorption, etc.).
34
+ When finally reaching the Earth, the photons in the spectrum
35
+ will undergo additional transformations as they travel through
36
+ the Earth’s atmosphere and are collected by telescopes. Tele-
37
+ scope effects are often well characterized and constrained (CCD
38
+ noise, fringing, mirror defects, etc.). However, other effects are
39
+ usually more difficult to model and correct.
40
+ Particularly, passing through the terrestrial molecules present
41
+ in the Earth’s atmosphere results in absorption lines in the sensed
42
+ spectrum. These absorption lines, known as telluric lines, are the
43
+ results of interaction of photons with molecules due to a num-
44
+ ber of electron, rotational, and vibrational transitions, and are a
45
+ persistent source of contamination and loss of information in the
46
+ observed spectra. The majority of the molecules responsible for
47
+ absorption are O2, H2O, CO2, CH4, O3, N2O, and the Chappuis
48
+ ozone absorption bands. The many of these absorption features
49
+ lie in the near-infrared and ultraviolet part of the electromagnetic
50
+ spectrum, with weaker absorption features from ozone, oxygen,
51
+ and water present in optical wavelengths. Additional emission
52
+ features are also added to the spectrum as photons are emitted
53
+ from the molecules. The emission features are relatively straight-
54
+ 4000
55
+ 4500
56
+ 5000
57
+ 5500
58
+ 6000
59
+ 6500
60
+ Wavelength (Γ…)
61
+ 0.4
62
+ 0.2
63
+ 0.0
64
+ 0.2
65
+ 0.4
66
+ 0.6
67
+ 0.8
68
+ 1.0
69
+ 1.2
70
+ Normalized Flux (-)
71
+ Input Spectrum
72
+ Reconstructed Spectrum
73
+ Fig. 1. We exploit the statistical properties of stellar spectra in large
74
+ datasets, pass them through a convolutional auto-encoder, and get tel-
75
+ luric lines rejected with close to zero effort.
76
+ forward to handle by observing source-free portions of the sky.
77
+ Observed emission lines in "empty" regions provide an easily ac-
78
+ cessible template for emission removal. The absorption features
79
+ are more complicated, scaling non-linearly, and are affected by
80
+ atmospheric conditions at the time of observation.
81
+ Most of the traditional methods used for the removal of tel-
82
+ luric lines consider each spectrum as a single independent en-
83
+ tity, discarding the set of observed spectra as a whole – e.g. see
84
+ HrudkovΓ‘ & Harmanec (2005). One group of such methods use
85
+ standard stars to assist with atmospheric line removal. They do
86
+ Article number, page 1 of 10
87
+ arXiv:2301.00313v1 [astro-ph.SR] 1 Jan 2023
88
+
89
+ A&A proofs: manuscript no. main
90
+ this by observing standard stars with relatively featureless spec-
91
+ tra and using them as atmospheric templates. This is often done
92
+ by observing A0V or G-type stars, though this is predominantly
93
+ used at the near-infrared and IR wavelengths (Vacca et al. 2003;
94
+ Artigau et al. 2014). The reference star should be ideally lo-
95
+ cated near the target star and observed close in time to measure
96
+ the atmospheric conditions as accurately as possible. The target
97
+ source’s spectrum is then divided by the telluric template. How-
98
+ ever, this method is limited if there are no such standard stars
99
+ available during an observing run or if observing conditions re-
100
+ sult in a poor spectrum. Atmospheric absorption can also vary
101
+ rapidly, on the order of minutes, requiring significant time and
102
+ resources dedicated to observing the template stars. The division
103
+ is an iterative process requiring wavelength and intensity scaling
104
+ adjustments to match the atmospheric effects.
105
+ Another common method is to model the Earth’s atmosphere
106
+ and create a synthetic spectrum which precisely models the at-
107
+ mospheric absorption. These models solve the radiative trans-
108
+ fer equation of the Earth’s atmosphere using numerical models
109
+ – e.g. Allart et al. (2022). These methods are dependent on at-
110
+ mospheric conditions measurements from the night’s observa-
111
+ tions. This is commonly done with programs such as molecfit
112
+ (Smette et al. 2015) and telfit (Gullikson et al. 2014). The
113
+ radiative transfer code retrieves the temperature, pressure, and
114
+ humidity from the time of observation and uses a database of
115
+ molecular parameters to create a fit for the telluric absorption.
116
+ While this technique is relatively successful, it may perform
117
+ poorly if there are a large number of intrinsic features, little or
118
+ no continuum, low signal-to-noise ratio (S/N), or large airmass
119
+ observations with high water vapor content. Moreover, current
120
+ implementations of this technique suffer from rather slow perfor-
121
+ mance to the extent that fitting a model to a single spectrum may
122
+ take up to several minutes on today’s computers. See Ulmer-
123
+ Moll et al. (2019) for a comparison of the above methods.
124
+ More recent methods, however, have incorporated implicit
125
+ modeling of the spectra by looking at a set of examples, poten-
126
+ tially eliminating the need for manual, explicit modeling. Such
127
+ methods, loosely dubbed data-driven approaches, have gained
128
+ good momentum in the past couple of decades. Particularly in
129
+ astronomy, a data-driven look at spectra has been mainly incor-
130
+ porated in the broad context of dimensionality reduction.
131
+ Arguably the most popular dimensionality reduction method
132
+ used in astronomy has been the Principal Component Analy-
133
+ sis (PCA – see Jolliffe & Cadima (2016) for a review), which
134
+ has been used as a dimensionality reduction tool on astronom-
135
+ ical spectra for many years now. Connolly et al. (1995) used
136
+ it to classify galaxy spectra with only the first two principal
137
+ components while Bailer-Jones et al. (1998) showed that PCA
138
+ can compress stellar spectra by a factor of over 30 and classify
139
+ anomalous and non-stellar spectra in the data set. Furthermore,
140
+ Bailer-Jones et al. (1998) found that the compression removed
141
+ noise as well as bogus features in the spectra such as dust ap-
142
+ pearing on the plate during plate scanning. Some works such as
143
+ Artigau et al. (2014), even attempt to remove telluric lines of a
144
+ few specific objects in the HARPS dataset using PCA decom-
145
+ position and demonstrate improved radial velocity measurement
146
+ accuracy. However, their method does not operate solely on the
147
+ stellar spectra and still requires observations of telluric standard
148
+ stars. Our approach solely requires a set of input stellar spectra
149
+ without specific telluric standard stars.
150
+ Moreover, the very important, but often overlooked point
151
+ about applicability of PCA for such applications, is its linear na-
152
+ ture, making it unable to account for modeling non-linear phe-
153
+ nomena by definition. E.g. in the case of telluric lines, a method
154
+ simply based on PCA, may not be able to tell the difference
155
+ between a telluric and stellar line, in crowded regions of the
156
+ wavelength. This inability has been well observed as early as
157
+ in Bailer-Jones et al. (1998). In this work, we show how trans-
158
+ formation to a more sophisticated feature space is necessary for
159
+ such a task.
160
+ There is a closely related family of neural networks known
161
+ as encoder-decoder networks (fig. 2). The contraction part com-
162
+ presses the input to an often low-dimensional representation,
163
+ also known as "latent representation" or simply "code". The ex-
164
+ pansion part is then used to decode this low-dimensional repre-
165
+ sentation up to the desired output. The transformation used for
166
+ coding (and decoding) is learned during training, according to
167
+ the task at hand. A special case of such networks, auto-encoders,
168
+ is an unsupervised network trained to reconstruct the input with-
169
+ out the need for labeled input data. As described in Hinton &
170
+ Salakhutdinov (2006) these networks can be used to reduce the
171
+ dimensionality of input data in a non-linear generalization of
172
+ PCA.
173
+ In Yang & Li (2015) the authors use a classical (non-
174
+ convolutional) auto-encoder to transform 3000-dimensional
175
+ spectra from the Sloan Digital Sky Survey (SDSS) Data Re-
176
+ lease 7 (Abazajian et al. 2009) to lower-dimensional features,
177
+ which are later used for estimation of atmospheric parame-
178
+ ters. Wang et al. (2016a) also used classical (non-convolutional)
179
+ auto-encoders for feature learning on astronomical spectra. They
180
+ compare spectral classification based on their learned features
181
+ with PCA and locally linear embedding (LLE), an alternate
182
+ non-linear dimensionality reduction method. They find that their
183
+ auto-encoder approach performs the best when classifying spec-
184
+ tra among a data set of F, G and K-type stars.
185
+ One problem with the mere use of auto-encoders, especially
186
+ for applications in (astro-)physics, is the entangled mapping of
187
+ concepts into the reduced signal – the latent space. The Varia-
188
+ tional AutoEncoder (VAE: Kingma & Welling 2014) has shown
189
+ to mitigate this effect to some extent. Portillo et al. (2020) used
190
+ a VAE to reconstruct SDSS galaxy spectra resampled to an input
191
+ spectrum with 1000 components. They found that when limiting
192
+ the dimensionality of the latent spaces to 2, 4, 6 or 10 compo-
193
+ nents (β‰₯ 99% compression) the VAEs and traditional autoen-
194
+ coders both reconstructed SDSS galaxy spectra with a lower re-
195
+ construction error than PCA and non-negative matrix factoriza-
196
+ tion (NMF) applied with the same number of components. The
197
+ authors then demonstrated that different galaxy classes occupied
198
+ different areas of the latent space enabling classification.
199
+ Classical auto-encoders are merely composed of fully-
200
+ connected layers and thus suffer from a lack of scalability to
201
+ deeper networks and high-dimensional data – usually the case
202
+ in modern astronomical spectroscopy. We address this drawback
203
+ by incorporating convolutional (and up-convolutional) layers for
204
+ transformation of data to the desired feature space; an idea bor-
205
+ rowed from computer vision. In Zhang et al. (2004) the authors
206
+ map images of people’s faces to low-dimensional manifolds, and
207
+ reconstruct them back. They find a correlation between the face
208
+ pose and the low-dimensional manifold, independently of the
209
+ person in the image. Wang et al. (2016b) shows comprehensive
210
+ experiments on dimensionality reduction using auto-encoders,
211
+ and studies the effects of different latent dimensions using syn-
212
+ thetic and real images. Note that in case of spectra, the networks
213
+ need to be updated to use 1D (up-)convolutional layers, as op-
214
+ posed to 2D in case of images.
215
+ Perhaps the closest to our work, both in terms of the used
216
+ network architecture, as well as the data-driven unsupervised
217
+ concept is Sedaghat et al. (2021), where a convolutional VAE
218
+ Article number, page 2 of 10
219
+
220
+ Sedaghat et al.: Stellar Karaoke
221
+ Fig. 2. A simple auto-encoder trained to reconstruct stellar spectra, can decompose physically meaningful components out of the input, when the
222
+ compression is high enough and the number of convolutional kernels is limited. Blue is the input spectrum and orange is the reconstructed version.
223
+ The region annotated by the red circle indicates a high density of such reconstruction rejections.
224
+ is used to extract knowledge from a large number of HARPS
225
+ spectra, in a fully unsupervised manner. We borrow and use the
226
+ exact same neural architecture for our work. However, we addi-
227
+ tionally utilize the conceptual decomposition of stellar features
228
+ and atmospheric effects into two different spectra; an idea that
229
+ turns out to have been touched as early as in Hadrava (1997),
230
+ however without a data-driven perspective.
231
+ Our Contributions
232
+ β€’ We process a huge number of very high-dimensional spec-
233
+ tra as a whole, letting statistical properties emerge in them,
234
+ allowing to be treated as random processes.
235
+ β€’ We model the telluric components in stellar spectra as inde-
236
+ pendent stars and impose a virtual radial velocity on them to
237
+ achieve statistical whitening/decorrelation.
238
+ β€’ We incorporate a convolutional auto-encoder, that automati-
239
+ cally acts as a source separation tool, rejecting telluric lines
240
+ in a fully unsupervised fashion, with zero explicit modeling.
241
+ 2. Problem Formulation
242
+ We seek to clean adversarial atmospheric effects out of an ar-
243
+ bitrary observed signal, x. The observed signal in our case is a
244
+ stellar spectrum, and so is a function of wavelength, Ξ», letting us
245
+ denote it as x(Ξ»).
246
+ We use the below formulation to model the various phenom-
247
+ ena affecting a stellar spectrum, before it is captured by our sen-
248
+ sors:
249
+ x(Ξ») =
250
+ οΏ½
251
+ s(Ξ») Γ— t(Ξ»)
252
+ οΏ½
253
+ βˆ— h(Ξ») + n(Ξ»)
254
+ (1)
255
+ where s is the stellar spectrum, incorporating line-of-sight ef-
256
+ fects from relative stellar velocity and the interstellar medium. t
257
+ is an imaginary signal representing the telluric lines affecting the
258
+ spectrum. x is the observed signal: a single spectrum that takes
259
+ on different values at each wavelength depending on the flux at
260
+ that wavelength. So we use the notation, x(Ξ»), throughout this
261
+ article.
262
+ We use h to denote what we call the observation transfer
263
+ function, and models the changes the signal goes through dur-
264
+ ing the sensing process, and includes, but is not limited to, the
265
+ line spread function, etc. n models the additive noise which is
266
+ not modeled in the transfer function, h.
267
+ We work with a large ensemble of N observations, mostly
268
+ coming from different sources. We use the subscript, i, to differ-
269
+ entiate between various observations:
270
+ xi(Ξ») =
271
+ οΏ½
272
+ si(Ξ») Γ— ti(Ξ»)
273
+ οΏ½
274
+ βˆ— h(Ξ») + ni(Ξ»)
275
+ i ∈ {1, 2, . . . , N}
276
+ (2)
277
+ Note that the spectrum, s, depends on i, as we work with various
278
+ objects at the same time. The atmospheric conditions are also
279
+ time-variant, an important point that is reflected in dependence
280
+ of t on i. The same is true for noise, n. Without loss of generality,
281
+ and for the sake of simplicity, we assume that h is constant across
282
+ the whole set of observations.
283
+ We seek to eliminate the effect of telluric lines from the
284
+ observed signal, which in this model is equal to extracting
285
+ si(Ξ») βˆ— h(Ξ») for every observation – note that removal of the ob-
286
+ servation effect, h, is not part of the objective here.
287
+ We also model the effect of the radial velocity, v, of the target
288
+ object on the observed spectrum, s as:
289
+ si(λ) = V {˚si(λ), vi}
290
+ = ˚si
291
+ οΏ½
292
+ Ξ»οΏ½1 βˆ’ vi
293
+ c
294
+ οΏ½οΏ½
295
+ (3)
296
+ where ˚si would be the observed spectrum, if the radial velocity
297
+ was zero – i.e. no Doppler shift. We call ˚si the static spectrum
298
+ hereafter. The V{.} operator represents the effect of the radial
299
+ velocity, vi, on the spectrum, as expanded in the second line of
300
+ the above equation, and c is the speed of light. The physical units
301
+ of vi and c can be arbitrarily chosen, as long as they are kept the
302
+ same.
303
+ ti on the other hand, and by definition, do not have any de-
304
+ pendence on vi, and therefore can be modeled as:
305
+ ti(Ξ») = V
306
+ οΏ½
307
+ ˚ti(λ), 0
308
+ οΏ½
309
+ (4)
310
+ 2.1. Continuous vs. Discrete
311
+ All the signals discussed above are of continuous nature up until
312
+ the point they are sensed by the detector. Sensing by the detector
313
+ Article number, page 3 of 10
314
+
315
+ A&A proofs: manuscript no. main
316
+ is a process that involves sampling as one of its steps, converting
317
+ a spectrum into a series of real-valued samples. Therefore, the
318
+ data we work with in our experiments is the discrete representa-
319
+ tion of xi(Ξ»), namely Xl
320
+ i – we use l as the discrete-valued index
321
+ for the sampled pixels.
322
+ However, for the reasons below, keeping the formulation in
323
+ the continuous representation is safe – and clearer. First, the
324
+ signals modeled so far are all the constituent elements of the
325
+ pre-sampling signal, xi(Ξ»), and so the continuous models hold.
326
+ Secondly, the only part of our method which explicitly modifies
327
+ the signal along the wavelength axis, the V {, } operator (sec-
328
+ tion 3.1), does its job by regridding the interpolated version of
329
+ the signal; practically converting the discrete signal back to its
330
+ continuous version, applying the transformation and sampling it
331
+ back again to the discrete space. Therefore mathematical defini-
332
+ tion of the operator in the continuous space is valid.
333
+ 2.2. Signals as Random Processes
334
+ The process of sensing a signal, xi, from an arbitrarily chosen
335
+ object in the sky, which is affected by many non-deterministic
336
+ phenomena along the way, can be seen as one realization of a
337
+ random process. In other words, the set of xi, or their discrete
338
+ representation, Xi, for various i, represent an ensemble of real-
339
+ izations of a random process, {X} 1. Note that in this particular
340
+ application, the index set of the random process is sampled from
341
+ wavelengths, Ξ» – a bit counter-intuitive, as it is usually of a tem-
342
+ poral nature in typical applications.
343
+ Therefore each Xl represents a random variable, with out-
344
+ comes Xl
345
+ iii. Similarly, we can model {S }, { ˚S } and {T} as discrete-
346
+ index random processes of the same type. {S } is then a genera-
347
+ tor of various clean spectra, S i, while { ˚S } generates the static ˚S l
348
+ i
349
+ spectra. As we will see in the next section, this non-deterministic
350
+ view on signals allows us to explain the statistical operations and
351
+ properties of the components more clearly.
352
+ 3. Method
353
+ The method is composed of two key steps:
354
+ a) Increasing the entropy of telluric components across obser-
355
+ vations and,
356
+ b) transforming the spectra into a space where the stellar and
357
+ telluric components are separable, then removing the non-
358
+ dominant one.
359
+ Median Normalization As a pre-processing step, we normalize
360
+ each spectrum in the dataset based on its median, to mitigate the
361
+ effect of different distances in sources, which otherwise results
362
+ in extreme inter-sample flux range variations.
363
+ 3.1. Velocity Whitening
364
+ In the default conditions, the spectroscopic observations are in
365
+ the so-called topocentric reference frame where the telluric lines,
366
+ if they exist, take on the same wavelengths – or pixels locations.
367
+ What it means from a statistical point of view though is this:
368
+ Let us assume, for simplicity, that all the observed objects
369
+ have the same static spectra, ˚si(λ). The resulting spectra, si(λ)
370
+ 1 The term random in this context is not in contradiction with the struc-
371
+ ture in stellar spectra. The structure is encoded in the basic parameters
372
+ of the constituent set of random variables, such as the expected value
373
+ and covariance matrix.
374
+ would then only be different based on their radial velocities,
375
+ vi, which, by definition, are modeled as scaling transformations
376
+ along the wavelength axis (eq. (3)). Note that since vi are out-
377
+ comes of a random variable, si(Ξ») are consequently still highly
378
+ independent and uncorrelated.
379
+ ti on the other hand, are composed of a set of absorption lines
380
+ with different strengths, but all happening at pre-defined specific
381
+ wavelengths. This feature makes them highly correlated across
382
+ multiple observations and easy to fit for any model – fig. 3, left
383
+ column.
384
+ To decrease the existing correlation between realizations of
385
+ the telluric signals, we increase the entropy across ti by random-
386
+ izing them using an emulated radial velocity,
387
+ tβ€²
388
+ i(Ξ») = V
389
+ οΏ½
390
+ ˚ti(Ξ»), vβ€²
391
+ i
392
+ οΏ½
393
+ (5)
394
+ where each vβ€²
395
+ i is a random velocity and is drawn from
396
+ Vβ€² : R β†’ R. The above effect can be implemented by sim-
397
+ ply contracting or expanding the wavelength axis, according to
398
+ eq. (3). Note though that this is a purely artificial phenomenon,
399
+ and although it is chosen to have the same effect as the β€œreal”
400
+ radial velocity of stars, it has no particular meaning for telluric
401
+ lines 2.
402
+ In practice, however, we only have access to the observed
403
+ signal, x, and not its forming components. Hence the proposed
404
+ randomization cannot be applied directly on t alone, and the
405
+ whole observed signal gets affected. But in the below, we show
406
+ that it can still have the desired effect:
407
+ Ξ» βˆ’β†’ Ξ»β€²
408
+ i = Ξ» Γ— (1 βˆ’ vβ€²
409
+ i
410
+ c )
411
+ (6)
412
+ xβ€²
413
+ i(Ξ») = V οΏ½xi(Ξ»), vβ€²
414
+ i
415
+ οΏ½
416
+ = V
417
+ οΏ½οΏ½
418
+ si(Ξ») Γ— ti(Ξ»)
419
+ οΏ½
420
+ βˆ— h(Ξ») + ni(Ξ»), vβ€²
421
+ i
422
+ οΏ½
423
+ = V
424
+ οΏ½οΏ½
425
+ si(Ξ») Γ— ti(Ξ»)
426
+ οΏ½
427
+ βˆ— h(Ξ»), vβ€²
428
+ i
429
+ οΏ½
430
+ + V οΏ½ni(Ξ»), vβ€²
431
+ i
432
+ οΏ½
433
+ (7)
434
+ which, according to the proof provided in appendix A becomes:
435
+ xβ€²
436
+ i(Ξ») = (1 βˆ’ vβ€²
437
+ i
438
+ c )V οΏ½si(Ξ») Γ— ti(Ξ»), vβ€²
439
+ i
440
+ οΏ½ βˆ— V οΏ½h(Ξ»), vβ€²
441
+ i
442
+ οΏ½ + V οΏ½ni(Ξ»), vβ€²
443
+ i
444
+ οΏ½
445
+ = (1 βˆ’ vβ€²
446
+ i
447
+ c )
448
+ οΏ½
449
+ si(Ξ»β€²
450
+ i) Γ— ti(Ξ»β€²
451
+ i)
452
+ οΏ½
453
+ βˆ— h(Ξ»β€²
454
+ i) + ni(Ξ»β€²
455
+ i)
456
+ (8)
457
+ Now since
458
+ οΏ½
459
+ 1 βˆ’
460
+ vβ€²
461
+ i
462
+ c
463
+ οΏ½
464
+ is, for each xβ€²
465
+ i, a constant coefficient and
466
+ is normalized out before being passed to the next step of the
467
+ method, we can see that the emulated randomized velocity has
468
+ founds its way from xi down to ti. In other words, we have
469
+ achieved the required velocity randomization in ti by modifying
470
+ xi. The two main components in the tweaked signal can now be
471
+ rewritten as:
472
+ Telluric:
473
+ Stellar:
474
+ V
475
+ οΏ½
476
+ ˚ti(Ξ»), vβ€²
477
+ i
478
+ οΏ½
479
+ V
480
+ οΏ½
481
+ ˚si(Ξ»), vi + vβ€²
482
+ i
483
+ οΏ½
484
+ So, we have achieved two components which are virutally mov-
485
+ ing independently of each other, when going over various obser-
486
+ vations. Figure 3 illustrates this effect in practice.
487
+ 2 A similar deterministic effect occurs when taking stellar spectra to
488
+ the barycentric frame.
489
+ Article number, page 4 of 10
490
+
491
+ Sedaghat et al.: Stellar Karaoke
492
+ 6265
493
+ 6270
494
+ 6275
495
+ 6280
496
+ topo-centric
497
+ 6265
498
+ 6270
499
+ 6275
500
+ 6280
501
+ ADP.2014-09-16T11:03:35.377
502
+ randomized
503
+ 6265
504
+ 6270
505
+ 6275
506
+ 6280
507
+ 6265
508
+ 6270
509
+ 6275
510
+ 6280
511
+ ADP.2014-09-16T11:03:36.737
512
+ 6265
513
+ 6270
514
+ 6275
515
+ 6280
516
+ Wavelength (Γ…)
517
+ 6265
518
+ 6270
519
+ 6275
520
+ 6280
521
+ Wavelength (Γ…)
522
+ ADP.2014-09-16T11:03:38.307
523
+ Fig. 3. On the left a set of exemplar spectra are depicted. Stellar lines are
524
+ unaligned due to different radial velocities. But telluric lines are aligned,
525
+ even though they may have different shapes due to their inherent time
526
+ dependence (the dotted vertical line indicates the location of a specific
527
+ telluric line in all plots). On the right the same spectra after velocity
528
+ randomization are depicted. Telluric lines are now unaligned too, but
529
+ with a pattern different to that of stellar lines!
530
+ From a statistical point of view, what we achieve with the
531
+ above velocity randomization is a degree of β€œwhitening” of the
532
+ tellurics random process, {T}. More precisely, we decrease the
533
+ mutual correlation between every pair, (T l, T m), pushing the sta-
534
+ tistical behavior of the telluric component toward white noise (Li
535
+ & Zhang 1998; Eldar & Oppenheim 2003; Kessy et al. 2018).
536
+ Note again that (T l, T m), the random variables we try to decor-
537
+ relate, are each defined on a single pixel of the spectrum, and
538
+ their outcomes vary along with different observations. A visual-
539
+ ization of the results of the above whitening process is illustrated
540
+ in section 5.
541
+ 3.2. Deep Feature Space
542
+ We borrow and use the exact architecture of the 1D convolu-
543
+ tional auto-encoder introduced by Sedaghat et al. (2021) – Fig-
544
+ ure 2. The convolutional layers of the encoder transform the in-
545
+ put spectrum down to a pre-defined number of β€œlatent variables”
546
+ at the bottleneck of the network. This low-dimensional repre-
547
+ sentation of the input, also known as the β€œcode”, is the most
548
+ compressed version of the input spectrum. The dimensionality
549
+ of this vector is chosen based on the desired compression rate in
550
+ various experiments. Note though that in the VAE version of our
551
+ networks, which is the case in most of the experiments of this
552
+ work, the latent nodes are implemented probabilistically, each
553
+ being modeled with a pair of scalars: mean and std of a normal
554
+ distribution. On the other side of the bottleneck, the decoder re-
555
+ Fig. 4. Visual illustration of the covariance matrix of the ensemble of
556
+ signals, before and after whitening. On the left, the covariance ma-
557
+ trix of the [6275, 6285]Γ… region for some subset of size 90 of the
558
+ observations is visualized. On the right, the same is done after veloc-
559
+ ity whitening, where vi were sampled from the uniform distribution:
560
+ V ∼ U(βˆ’30km/s, 30km/s). The covariance matrix is closer to the iden-
561
+ tity matrix now, confirming achievement of some degree of whiten-
562
+ ing/decorrelation.
563
+ ceives the compressed code and takes it step-by-step up to the
564
+ same dimension as the original input (218 + 216 for HARPS). We
565
+ keep using the same per-pixel L1 end-to-end loss function, as in-
566
+ troduced in the original work, to achieve acceptable pixel-level
567
+ accuracy.
568
+ This ∼typical architecture has proven to be able to transform
569
+ the input to a space where noise-like components of the signal
570
+ are easily separable. As we show with our experiments, the sta-
571
+ tistical trick developed in the previous section pushes the telluric
572
+ components farther from the stellar features and closer to the
573
+ noise, in the learned feature space, letting them be rejected as
574
+ easily as noise.
575
+ We of course need to constrain the reconstruction abilities of
576
+ the network with a high compression rate as well as a variational
577
+ loss at the bottleneck, not to have too strong of a network capable
578
+ of fitting the two independent components at the same time!
579
+ 4. Data
580
+ We use data from two publicly available colletions: HARPS and
581
+ SDSS. Our main experiments are run on HARPS, while SDSS is
582
+ used as a difficult test-bench.
583
+ 4.1. HARPS
584
+ HARPS (High Accuracy Radial-velocity Planet Searcher; Mayor
585
+ et al. 2003) is an instrument on the 3.6m La Silla Telescope.
586
+ It is a fibre-fed high-resolution echelle spectrograph dedicated
587
+ to the discovery of exoplanets, with a spectral resolution of
588
+ R = 115, 000 and covers the spectral range 378–691nm 3. The
589
+ data used has been instrument-corrected and detector-corrected,
590
+ as well as sky-subtracted.
591
+ Our downloaded dataset initially consisted of 272376 total
592
+ spectra, which after automatic removal of corrupted files, NaNs
593
+ and noise-like ones, was reduced to 267361 β€œstable” ones. This
594
+ collection is mainly composed of stellar spectra. However, there
595
+ are some random contaminant objects too, such as SUN, MOON,
596
+ etc., which we allowed to enter our training set on purpose,
597
+ to increase robustness of the learned features. We trimmed, re-
598
+ gridded and homogenized all spectra before being passed to the
599
+ 3 http://archive.eso.org/wdb/wdb/adp/phase3_main/form
600
+ Article number, page 5 of 10
601
+
602
+ A&A proofs: manuscript no. main
603
+ Fig. 5. Results Qualitative illustration of the results on an exemplar selection of HARPS spectra. Each row depicts one spectrum, with its HARPS
604
+ ID written on the right, while columns focus on different regions of interest. Interpretation hint: reconstruction (orange) should follow the pseudo-
605
+ truth (molecfit: green). The left-most column covers the whole spectrum, as is fed into the network, and illustrates the robustness of the network
606
+ to different characteristics of the spectra (continuum, noise level, etc.). Major stellar lines such as HΞ± can be easily spotted in the less β€˜busy’
607
+ examples. In the middle column we zoom in on a potentially complex region where narrow stellar lines, when existing, can collide with telluric
608
+ lines of similar shapes. E.g. in the second and 4th row there are examples of such cases, where the network rejects the telluric component, while
609
+ still preserving the stellar part very well – thus rejecting the hypothesis that it might be simply rejecting narrow lines by a applying a moving
610
+ average. In the third column, we focus on a region where similarly narrow stellar lines occur in some of the spectra, but not in the others. This way
611
+ we reject the hypothesis that the network might have ’memorized’ the locations of the lines. Note that the network (orange) is even outperforming
612
+ the pseudo-truth (green) in some cases, illustrating a smoother, more robust reconstruction in abrupt changes.
613
+ network in exactly the same way as is done by Sedaghat et al.
614
+ (2021).
615
+ Note that the pipeline the HARPS spectra go through is set
616
+ up such that the spectra are automatically transformed to the
617
+ barycentric reference frame and re-gridded before being stored
618
+ in the archive. The originally captured version, in the topocen-
619
+ tric reference frame, is also not preserved. Therefore we had to
620
+ transform them back to the topocentric frame for our experi-
621
+ Article number, page 6 of 10
622
+
623
+ pseudo-truth
624
+ input
625
+ reconstruction
626
+ Normalized Flux (-)
627
+ ADP.2014-09-16T11:05:10.627
628
+ 0
629
+ 4000
630
+ 5000
631
+ 6000
632
+ 6278
633
+ 6279
634
+ 6280
635
+ 6281
636
+ 6282
637
+ 5565.05567.55570.05572.5
638
+ Normalized Flux (-)
639
+ ADP.2014-09-16T11:03:31.413
640
+ 1
641
+ 4000
642
+ 5000
643
+ 6000
644
+ 6278
645
+ 6279
646
+ 6280
647
+ 6281
648
+ 6282
649
+ 5565.05567.55570.05572.5
650
+ Normalized Flux (-)
651
+ ADP.2014-09-16T11:03:31.673
652
+ 1
653
+ 0
654
+ 4000
655
+ 5000
656
+ 6000
657
+ 6278
658
+ 6279
659
+ 6280
660
+ 6281
661
+ 6282
662
+ 5565.05567.55570.05572.5
663
+ Normalized Flux (-)
664
+ ADP.2014-10-01T10:22:48.507
665
+ 1
666
+ 0
667
+ 4000
668
+ 5000
669
+ 6000
670
+ 6278
671
+ 6279
672
+ 6280
673
+ 6281
674
+ 6282
675
+ 5565.05567.55570.05572.5
676
+ Wavelength (A)
677
+ Wavelength (A)
678
+ Wavelength (A)Sedaghat et al.: Stellar Karaoke
679
+ ments. Although this is touching the core concept of our pre-
680
+ sented method, it turned out to be a safe procedure: transforma-
681
+ tion to the barycentric frame exerts added randomness on the
682
+ radial velocity, which is perfectly compatible with our method.
683
+ In fact, our method has been inspired by observing traces of the
684
+ above-mentioned fact in our initial experiments.
685
+ 4.2. SDSS
686
+ Our second dataset consisted of spectra from the SEGUE (Yanny
687
+ et al. 2009) and SEGUE-2 spectroscopic surveys (Rockosi et al.
688
+ 2022) that were part of the larger Sloan Digital Sky Survey
689
+ (SDSS). The SEGUE surveys both used the 2.5m Sloan Foun-
690
+ dation Telescope (Gunn et al. 2006) located at Apache Point Ob-
691
+ servatory with the two original Sloan Digital Sky Survey fiber
692
+ spectrographs (Smee et al. 2013). The SDSS spectrographs have
693
+ a resolution of R ∼ 1800 and together have 640 fibers (320 each)
694
+ that plug into aluminum "plugplates" for each observation. In
695
+ each plate 32 plugs are reserved for blank sky observations and
696
+ 16 for spectrophotometric standard stars in the field. Each spec-
697
+ trograph has a red and a blue channel that collect data on separate
698
+ CCDs with the blue wavelength range from approximately 3800
699
+ Γ… to 6100 Γ… and the red wavelengths spanning approximately
700
+ 5900 Γ… to 9200 Γ…. Sources in the original SEGUE survey sam-
701
+ pled Milky Way stars at a variety of distances, colors and metal-
702
+ licities while the SEGUE-2 targets focused on stars in the Milky
703
+ Way halo.
704
+ In our experiments we used the ~660,000 uncalibrated spec-
705
+ tra from the red CCD of one of the spectrographs (labeled as ’r1’
706
+ in the SDSS data archive) provided with SDSS Data Release 17
707
+ (Abdurro’uf et al. 2022) and accessible via the DR17 FITS web-
708
+ site4. The uncalibrated spectra consist of multiple 10-30 minute
709
+ exposures of each source. Since the SEGUE surveys imaged
710
+ each source multiple times to create coadded spectra this means
711
+ we have multiple spectra in our dataset for the same source. For
712
+ each plate we exclude the fibers that were intentionally pointed
713
+ at empty patches of sky and labelled "SKY" in the data.
714
+ The uncalibrated spectra used are labelled β€˜spFrameβ€˜ in the
715
+ SDSS data model5 and according to the details of the SDSS spec-
716
+ troscopic pipeline (Stoughton et al. 2002) they are flat-fielded but
717
+ not flux-calibrated. The flux calibration spectra (β€˜spFluxCalibβ€˜
718
+ in the data model) include telluric absorption calculated by the
719
+ spectroscopic pipeline based upon the spectrophotometric stan-
720
+ dard stars that are included in the observation set of each plate.
721
+ These calibration are what we use as a "pseudo-truth" for com-
722
+ parisons of our network results.
723
+ 5. Experiments and Results
724
+ We train and evaluate multiple networks based on various com-
725
+ binations of the below (hyper-)parameters:
726
+ β€’ Latent-space dimensionality
727
+ β€’ Velocity randomization level
728
+ For our main experiments, that include hyper-parameter
729
+ sweeping and controlled tests, we mainy use the HARPS dataset.
730
+ Other datasets are used for comparative testing of the method in
731
+ extreme conditions.
732
+ 4 https://data.sdss.org/sas/dr17/
733
+ 5 https://data.sdss.org/datamodel/
734
+ 5.1. Main Results
735
+ Figure 5 illustrates a few examples of how the method manages
736
+ to reject the telluric lines and preserve the stellar features. The
737
+ example spectra and regions are specifically chosen to rule out
738
+ potential naive hypotheses for how the network rejects telluric
739
+ lines. In other words, the results confirm that the network has
740
+ learned a semantic representation of the constituent components
741
+ and is separating the sources in that feature space, as opposed
742
+ to e.g. simply having β€œmissed” narrower lines (simple averag-
743
+ ing, low-pass filtering). In the caption of fig. 5 a few other such
744
+ hypotheses are elaborated and rejected.
745
+ 5.2. Quantitative Evaluation
746
+ Various (hyper-)parameters, such as the degree of compression
747
+ during dimensionality reduction, or the resolution of the input
748
+ data, may influence the capability of the network in rejecting
749
+ the telluric lines. Precisely, there is always a trade off between
750
+ telluric-rejection and stellar-reconstruction. Therefore, we de-
751
+ velop and incorporate the below mutual metrics to quantify both
752
+ aspects at the same time:
753
+ Qt =
754
+ οΏ½ οΏ½
755
+ Mt οΏ½οΏ½οΏ½ Λ†S βˆ’ ˜G
756
+ οΏ½οΏ½οΏ½
757
+ οΏ½
758
+ οΏ½ Mt
759
+ (9)
760
+ where Qt is a proxy for the quality of rejection of telluric lines.
761
+ Λ†S represents the reconstructed spectrum (i.e. the direct output of
762
+ the network) and ˜G is the pseudo ground truth. ˜G is a corrected
763
+ version of the output of molecfit in case of HARPS data, and a
764
+ modified version of the publicly available calibrated spectra in
765
+ case of SDSS. Mt is a binary mask; a vector of the same size
766
+ of the input, having ones at all pixels containing known telluric
767
+ lines, and zeros everywhere else. Note that the subscript i, index-
768
+ ing each observed spectrum, is omitted in Qt, Λ†S , ˜G and Mt for
769
+ the sake of simplicity, and the summation is calculated over all
770
+ the pixels of each spectrum.
771
+ We similarly define the dual metric for measuring the recon-
772
+ struction (preservation) of the stellar features as follows:
773
+ Qs =
774
+ οΏ½
775
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½Ms
776
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½1 βˆ’
777
+ Λ†S
778
+ ˜G
779
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
780
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
781
+ οΏ½ Ms
782
+ (10)
783
+ where Qs is, conversely, a proxy for the quality of stellar recon-
784
+ struction. Note that both of the above are metrics of a distance
785
+ nature, and hence, the lower the better.
786
+ 5.3. The Effect of Compression
787
+ It is reasonable to expect, and is supported by our experiments
788
+ that, with a lower dimensionality at the information bottleneck
789
+ of the network, the network’s capacity for preserving many con-
790
+ current features during compression diminishes. This results in a
791
+ higher rejection of details in the reconstructed spectra, which can
792
+ be naively interpreted as an improvement in telluric line rejection
793
+ – a decrease in Qt. Note however that stellar reconstruction qual-
794
+ ity can get worse at the same time, and so the two metrics need
795
+ to be considered at the same time.
796
+ Table 1 compares results of various runs with different di-
797
+ mensionalities at the latent space on a fixed subset of HARPS
798
+ spectra. Note how the stellar reconstruction quality starts to de-
799
+ cay by increasing the compression (lower latent dimensions). On
800
+ Article number, page 7 of 10
801
+
802
+ A&A proofs: manuscript no. main
803
+ 8800
804
+ 8900
805
+ 9000
806
+ 9100
807
+ telluric
808
+ 8100
809
+ 8200
810
+ 8300
811
+ 8400
812
+ telluric
813
+ 6400
814
+ 6500
815
+ 6600
816
+ 6700
817
+ 6800
818
+ stellar: H-alpha
819
+ 8400
820
+ 8600
821
+ 8800
822
+ 104_3110_spFrame-r1-00055323_230
823
+ stellar
824
+ 6000
825
+ 7000
826
+ 8000
827
+ 9000
828
+ Normalized Flux (-)
829
+ 8800
830
+ 8900
831
+ 9000
832
+ 9100
833
+ Wavelength (Γ…)
834
+ 8100
835
+ 8200
836
+ 8300
837
+ 8400
838
+ Wavelength (Γ…)
839
+ 6400
840
+ 6500
841
+ 6600
842
+ 6700
843
+ 6800
844
+ Wavelength (Γ…)
845
+ 8400
846
+ 8600
847
+ 8800
848
+ Wavelength (Γ…)
849
+ 104_3114_spFrame-r1-00056165_144
850
+ 6000
851
+ 7000
852
+ 8000
853
+ 9000
854
+ Wavelength (Γ…)
855
+ Normalized Flux (-)
856
+ input
857
+ tellurics profile
858
+ reconstruction
859
+ input
860
+ tellurics profile
861
+ reconstruction
862
+ Fig. 6. Test results on a low-resolution data collection: SDSS. On the left, the whole spectrum is depicted, while the other columns depict various
863
+ regions of interest along the wavelength axis, so the qualitative performance of the method can be assessed on specific stellar and telluric features.
864
+ Note that since pseudo-truth data is not easily obtainable in this case, we over-plot a "tellurics profile" in red, just for visual comparison. It should
865
+ be interpreted differently from the pseudo-truth curves in other figures, as it does follow the telluric lines. Stellar features are preserved very well,
866
+ even in bad signal-to-noise conditions. Some of the wider telluric lines, however, are not fully rejected in this case due to the significantly low
867
+ resolution as compared to HARPS. Please refer to the main text for a more detailed discussion.
868
+ Latent Dim.↓
869
+ Telluric
870
+ Stellar
871
+ 8
872
+ 0.053
873
+ 0.015
874
+ 32
875
+ 0.055
876
+ 0.012
877
+ 128
878
+ 0.063
879
+ 0.014
880
+ 1024
881
+ 0.176
882
+ 0.012
883
+ Table 1. Quality of stellar line reconstruction vs. telluric line rejection
884
+ on HARPS, for different configurations of the network. Latent Dim. is
885
+ the number of dimensions of the code, or the latent representation. A
886
+ lower number is better in both columns.
887
+ the other hand, by increasing the number of latent nodes, the
888
+ network starts to become too powerful, managing to reconstruct
889
+ both components and losing the source separation capability.
890
+ 5.4. The Effect of Velocity-Whitening
891
+ To demonstrate the effect of the whitening step, we compare the
892
+ results of two experiments in controlled conditions. The first ex-
893
+ periment is run on HARPS spectra in topocentric frame, where
894
+ telluric lines are all aligned – zero velocity randomization. Then
895
+ we apply velocity whitening, sampling vi from the uniform dis-
896
+ tribution: V ∼ U(βˆ’30km/s, 30km/s). All other configurations of
897
+ the test are kept fixed. Table 2 shows a clear difference in the
898
+ telluric rejection performance of the two runs.
899
+ HARPS
900
+ Telluric
901
+ Stellar
902
+ Topocentric
903
+ 0.199
904
+ 0.018
905
+ Rand. Velocity
906
+ 0.063
907
+ 0.014
908
+ Table 2. Comparison of two experiments with zero velocity randomiza-
909
+ tion (β€˜Topocentric’) and V ∼ U(βˆ’30km/s, 30km/s) (β€˜Rand. Velocity’).
910
+ 5.5. HARPS vs. SDSS: High- vs. Low-Resolution
911
+ Figure 6 depicts the results of applying our method on SDSS
912
+ spectra. The extremely low pixel resolution we used for this
913
+ dataset (1.66Γ… as opposed to 0.01Γ… for HARPS), makes this
914
+ in practice a stress test for our method, since many of the tel-
915
+ luric lines get merged, appearing as wide artifacts. However, the
916
+ method performs at an acceptable level. E.g. the middle column
917
+ of fig. 6 shows how the method misses the very wide telluric ar-
918
+ tifact while managing to reject the neighboring telluric lines. In
919
+ general, it seems our method can β€œhunt” telluric lines up until
920
+ the point the resolution goes so low that merging/blending of the
921
+ lines converts them to β€œslow” artifacts.
922
+ 6. Conclusions and Future Directions
923
+ We presented a method that, by incorporating a Big Data-
924
+ inspired view at stellar spectra, exploits the statistical indepen-
925
+ dence of the radial velocity of stars with telluric lines in their
926
+ observed spectra, reinforces it using a novel trick, and utilizes
927
+ a fully unsupervised convolutional neural network to reject the
928
+ undesireable part.
929
+ The method is superior to existing, traditional telluric line
930
+ removal tools in terms of preparation effort, performance, and
931
+ accuracy. The fact that it is fully unsupervised, obviates the need
932
+ for any kind of model parameter tuning – which is the case in
933
+ e.g. molecfit, where a list of wavelengths should be manually
934
+ specified to initialize the model. Training the model, in practice,
935
+ calls for merely passing a large number of spectra through the
936
+ network – and it will do the rest.
937
+ Training one network suffices for one whole data collection.
938
+ Once trained, it processes each spectrum in a fraction of a second
939
+ – depending on the size of the spectrum and, consequently, the
940
+ size of the network. But the speed-up does not sacrifice accuracy;
941
+ as seen in section 5, it can even detect and suppress hard-to-
942
+ locate lines which are missed by molecfit – the opposite may
943
+ happen too, though in rarer situations.
944
+ Article number, page 8 of 10
945
+
946
+ Sedaghat et al.: Stellar Karaoke
947
+ Nevertheless, the current version is still a demonstration of
948
+ a research product, showcasing the strengths of a fully unsuper-
949
+ vised approach. But for Stellar Karaoke to become a ready-to-
950
+ use package in every application, more work is required. No-
951
+ tably, the decision of which component to keep and which one
952
+ to reject (stellar vs. telluric), should not be left to the network. It
953
+ can be enforced by a minimal supervision.
954
+ Acknowledgements. Some of the experiments demonstrated in this work have
955
+ been run on compute servers provided by ESO, during NS’s collaboration with
956
+ the ESCAPE project between 2019 and 2021.
957
+ References
958
+ Abazajian, K. N., Adelman-McCarthy, J. K., AgΓΌeros, M. A., et al. 2009, ApJS,
959
+ 182, 543
960
+ Abdurro’uf, Accetta, K., Aerts, C., et al. 2022, ApJS, 259, 35
961
+ Allart, R., Lovis, C., Faria, J., et al. 2022, A&A, 666, A196
962
+ Artigau, Γ‰., Astudillo-Defru, N., Delfosse, X., et al. 2014, in Observatory Oper-
963
+ ations: Strategies, Processes, and Systems V, Vol. 9149, International Society
964
+ for Optics and Photonics, 914905
965
+ Bailer-Jones, C. A., Irwin, M., & Von Hippel, T. 1998, Monthly Notices of the
966
+ Royal Astronomical Society, 298, 361
967
+ Connolly, A. J., Szalay, A. S., Bershady, M. A., Kinney, A. L., & Calzetti, D.
968
+ 1995, AJ, 110, 1071
969
+ Eldar, Y. C. & Oppenheim, A. V. 2003, IEEE Transactions on Information The-
970
+ ory, 49, 1846
971
+ Gullikson, K., Dodson-Robinson, S., & Kraus, A. 2014, AJ, 148, 53
972
+ Gunn, J. E., Siegmund, W. A., Mannery, E. J., et al. 2006, AJ, 131, 2332
973
+ Hadrava, P. 1997, Astronomy and Astrophysics Supplement Series, 122, 581
974
+ Hinton, G. E. & Salakhutdinov, R. R. 2006, Science, 313, 504
975
+ HrudkovΓ‘, M. & Harmanec, P. 2005, A&A, 437, 765
976
+ Jolliffe, I. T. & Cadima, J. 2016, Philosophical Transactions of the Royal Society
977
+ A: Mathematical, Physical and Engineering Sciences, 374, 20150202
978
+ Kessy, A., Lewin, A., & Strimmer, K. 2018, The American Statistician, 72, 309
979
+ Kingma, D. P. & Welling, M. 2014, arXiv:1312.6114 [cs, stat], arXiv: 1312.6114
980
+ Li, G. & Zhang, J. 1998, SankhyΒ―a: The Indian Journal of Statistics, Series A, 119
981
+ Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20
982
+ Portillo, S. K. N., Parejko, J. K., Vergara, J. R., & Connolly, A. J. 2020, AJ, 160,
983
+ 45
984
+ Rockosi, C. M., Lee, Y. S., Morrison, H. L., et al. 2022, ApJS, 259, 60
985
+ Sedaghat, N., Romaniello, M., Carrick, J. E., & Pineau, F.-X. 2021, Monthly
986
+ Notices of the Royal Astronomical Society, 501, 6026
987
+ Smee, S. A., Gunn, J. E., Uomoto, A., et al. 2013, AJ, 146, 32
988
+ Smette, A., Sana, H., Noll, S., et al. 2015, A&A, 576, A77
989
+ Stoughton, C., Lupton, R. H., Bernardi, M., et al. 2002, AJ, 123, 485
990
+ Ulmer-Moll, S., Figueira, P., Neal, J. J., Santos, N. C., & Bonnefoy, M. 2019,
991
+ A&A, 621, A79
992
+ Vacca, W. D., Cushing, M. C., & Rayner, J. T. 2003, PASP, 115, 389
993
+ Wang, K., Guo, P., & Luo, A.-L. 2016a, Monthly Notices of the Royal Astro-
994
+ nomical Society, 465, 4311
995
+ Wang, Y., Yao, H., & Zhao, S. 2016b, Neurocomputing, 184, 232
996
+ Yang, T. & Li, X. 2015, Monthly Notices of the Royal Astronomical Society,
997
+ 452, 158
998
+ Yanny, B., Rockosi, C., Newberg, H. J., et al. 2009, AJ, 137, 4377
999
+ Zhang, C., Wang, J., Zhao, N., & Zhang, D. 2004, Pattern Recognition, 37, 325
1000
+ Article number, page 9 of 10
1001
+
1002
+ A&A proofs: manuscript no. main
1003
+ Appendix A: Wavelength Transformation and
1004
+ Convolution
1005
+ x(Ξ») = z(Ξ») βˆ— h(Ξ»)
1006
+ =
1007
+ οΏ½
1008
+ w
1009
+ z(w)h(Ξ» βˆ’ w)dw
1010
+ (A.1)
1011
+ x(Ξ»β€²
1012
+ i) = x(aΞ»)
1013
+ =
1014
+ οΏ½
1015
+ w
1016
+ z(w)h(aΞ» βˆ’ w)dw
1017
+ (A.2)
1018
+ Let w = au,
1019
+ (A.3)
1020
+ x(Ξ»β€²
1021
+ i) = a
1022
+ οΏ½
1023
+ u
1024
+ z(au)h(aΞ» βˆ’ au)du
1025
+ (A.4)
1026
+ Let z(au) = zβ€²(u), h(au) = hβ€²(u)
1027
+ (A.5)
1028
+ =β‡’ x(Ξ»β€²
1029
+ i) = a
1030
+ οΏ½
1031
+ u
1032
+ zβ€²(u)hβ€²(Ξ» βˆ’ u)du
1033
+ = a zβ€²(Ξ») βˆ— hβ€²(Ξ»)
1034
+ = a z(aΞ») βˆ— h(aΞ»)
1035
+ = a z(Ξ»β€²
1036
+ i) βˆ— h(Ξ»β€²
1037
+ i)
1038
+ (A.6)
1039
+ The same could be shown in a rather shorter way using
1040
+ the Fourier transform. However, to avoid confusion between the
1041
+ terms frequency and wavelength used in two different domains,
1042
+ here we use the direct expansion of the convolution operator.
1043
+ Article number, page 10 of 10
1044
+
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1
+ Observation of anisotropic superfluid density in an artificial crystal
2
+ J. Tao,βˆ— M. Zhao,βˆ— and I. B. Spielman
3
+ Joint Quantum Institute, University of Maryland and National Institute
4
+ of Standards and Technology, College Park, Maryland 20742, USA
5
+ (Dated: January 4, 2023)
6
+ We experimentally and theoretically investigate the anisotropic speed of sound of an atomic
7
+ superfluid (SF) Bose-Einstein condensate in a 1D optical lattice. Because the speed of sound derives
8
+ from the SF density, this implies that the SF density is itself anisotropic. We find that the speed
9
+ of sound is decreased by the optical lattice, and the SF density is concomitantly reduced. This
10
+ reduction is accompanied by the appearance of a normal fluid in the purely Bose condensed phase.
11
+ The reduction in SF density—first predicted [A. J. Leggett, Phys.
12
+ Rev.
13
+ Lett.
14
+ 25 1543–1546
15
+ (1970)] in the context of supersolidityβ€”results from the coexistence of superfluidity and density
16
+ modulations, but is agnostic about the origin of the modulations. We additionally measure the
17
+ moment of inertia of the system in a scissors mode experiment, demonstrating the existence of
18
+ rotational flow. As such we shed light on some supersolid properties using imposed, rather than
19
+ spontaneously formed, density-order.
20
+ Superfluidity and Bose-Einstein condensation (BEC)
21
+ are deeply connected.
22
+ In dilute atomic BECs, the
23
+ superfluid (SF) and condensate densities are generally
24
+ equal [1, 2]. By contrast, SF 4He can be a nearly pure SF,
25
+ with only about 14 % condensate fraction [3], and infinite
26
+ 2D Berezinskii–Kosterlitz–Thouless (BKT) SFs have no
27
+ condensate at all [4, 5]. In 1970 Tony Leggett showed
28
+ that supersolidsβ€”systems spontaneously forming both
29
+ SF and crystalline order (i.e., density modulations)β€”
30
+ exhibit the reverse behavior: SF density far below the
31
+ condensate density [6].
32
+ Here we observe this effect in
33
+ a nearly pure atomic BEC with artificial crystal order
34
+ imprinted by an optical lattice.
35
+ The
36
+ complex-valued
37
+ order
38
+ parameter
39
+ Ο†(r)
40
+ =
41
+ οΏ½
42
+ ρsf exp[iΟ•(r)],
43
+ describing a SF with number den-
44
+ sity ρsf and phase Ο•(r), gives rise to two hallmark
45
+ SF properties: dissipationless supercurrents associated
46
+ with spatial gradients in Ο•(r) and (Bogoliubov [2])
47
+ sound described by traveling waves in Ο•(r).
48
+ Because
49
+ supercurrents arise from phase gradients, they are locally
50
+ irrotational; in liquid 4He, the resulting non-classical
51
+ rotational inertia appears below the SF transition
52
+ temperature Tc.
53
+ Supersolids are more exotic systems
54
+ spontaneously forming crystalline order while exhibiting
55
+ SF transport properties.
56
+ Recent experiments with
57
+ dipolar BECs of Dy and Er are suggestive of these
58
+ properties [7, 8].
59
+ Leggett argued that the modulated
60
+ density ρ(r) of a supersolid leads to an unavoidable
61
+ reduction in ρsf, and derived an upper bound for ρsf [6].
62
+ This reduction results from the 3D density distribution,
63
+ and as such is masked in tight binding descriptions such
64
+ as the Bose-Hubbard model, which makes the unrelated
65
+ prediction of vanishing ρsf at the superfluid to Mott
66
+ insulator transition [9, 10].
67
+ We created an artificial SF crystal by imprinting pe-
68
+ riodic density modulations into an atomic BEC using a
69
+ 1D optical lattice as in Fig. 1(a).
70
+ While these modu-
71
+ lations do not form spontaneously, Leggett’s result still
72
+ applies, making this an ideal system for understanding
73
+ crystalline SFs without the added complexity of spon-
74
+ taneously broken symmetries. We experimentally mea-
75
+ sured an anisotropic speed of sound via Bragg spec-
76
+ troscopy of the phonon mode. This implies the existence
77
+ of an effective anisotropic superfluid densityβ€”which can
78
+ be expressed as a second rank tensor ρsf
79
+ ijβ€”and we find
80
+ that it saturates Leggett’s bound, in agreement with
81
+ Gross-Pitaveskii equation (GPE) simulations.
82
+ We also
83
+ determined an associated anisotropic suppression of the
84
+ moment of inertia in terms of the scissor-mode frequen-
85
+ cies [11, 12].
86
+ Anisotropic superfluidsβ€”Here we consider pure 3D
87
+ BECs whose condensate mode ψ(r) = |ψ(r)| exp[iΟ‘(r)] is
88
+ well described by the Gross-Pitaveskii equation (GPE).
89
+ An optical lattice potential V (r) = (U0/2) cos(2krx) peri-
90
+ odically modulates the condensate density ρ(r) = |ψ(r)|2
91
+ with unit cell (UC) size a = Ο€/kr [Fig. 1(b)-i]. By con-
92
+ trast, the SF order parameter Ο†(r) is a coarse grained
93
+ quantity describing system properties on a scale ≫ a,
94
+ giving the nominally uniform density in Fig. 1(c)-i.
95
+ Even disregarding potential differences in ρsf(r) and
96
+ ρ(r), we argue that Ο†(r) is not simply equal to ψ(r) av-
97
+ eraged over some scale large compared to a. The fun-
98
+ damental origin of this effect can be understood by con-
99
+ sidering a 1D system of size L with periodic boundary
100
+ conditions in which both the condensate phase Ο‘ and SF
101
+ phase Ο• wind by an integer multiple N of 2Ο€ [Fig. 1(b,c)-
102
+ ii], yielding a metastable quantized supercurrent [13].
103
+ To satisfy the steady-state continuity equation, the mi-
104
+ croscopic current J(x) = ρ(x) [β„βˆ‚xΟ‘(x)/m] must be in-
105
+ dependent of x [Fig. 1(b)-ii], however, the periodically
106
+ modulated density ρ(x) > 0 implies the local velocity
107
+ v(x) = β„βˆ‚xΟ‘(x)/m has oscillatory structure and conse-
108
+ quently Ο‘(x) follows a staircase pattern [Fig. 1(b)-iii, iv].
109
+ From macroscopic considerations the superfluid cur-
110
+ rent is J = ρsf [β„βˆ‚xΟ•(x)/m] = 2Ο€Nℏρsf/(mL). Equating
111
+ the currents obtained from considering the condensate
112
+ arXiv:2301.01258v1 [physics.atom-ph] 3 Jan 2023
113
+
114
+ 2
115
+ (a)
116
+ βˆ’1
117
+ 0
118
+ 1
119
+ (b) BEC
120
+ βˆ’1
121
+ 0
122
+ 1
123
+ (c) SF
124
+ Position x/a
125
+ 0.0
126
+ 2.5
127
+ ρ/¯ρ
128
+ i.
129
+ 0
130
+ 1
131
+ 2
132
+ 3
133
+ J (arb.)
134
+ ii.
135
+ 0
136
+ 1
137
+ Ο‘/(2Ο€)
138
+ iii.
139
+ 0
140
+ 5
141
+ v/Β―v
142
+ iv.
143
+ ρsf/¯ρ
144
+ i.
145
+ J (arb.)
146
+ ii.
147
+ Ο•/(2Ο€)
148
+ iii.
149
+ v/Β―v
150
+ iv.
151
+ FIG. 1.
152
+ Concept. (a) A BEC is confined in a harmonic trap
153
+ superimposed with a 1D optical lattice (along ex, green), spa-
154
+ tially modulating the condensate density (red). The dashed
155
+ and dotted lines call out a region of nominally constant mean
156
+ density and the left and right columns indicate the (b) state
157
+ of the condensate and (c) SF in the presence of a current.
158
+ These were computed for a 5Er deep lattice and plot: i. den-
159
+ sity (red), ii. current (green), iii. phase (orange), and iv. local
160
+ velocity (blue). The red dashed line plots the mean density
161
+ ¯ρ.
162
+ mode and the SF order parameter and integrating over
163
+ a unit cell yields Leggett’s equation [6]
164
+ ρsf =a
165
+ οΏ½οΏ½
166
+ UC
167
+ dx
168
+ ρ(x)
169
+ οΏ½βˆ’1
170
+ , as well as Ο• = 1
171
+ a
172
+ οΏ½
173
+ UC
174
+ Ο‘(x)dx. (1)
175
+ This implies that ρsf ≀ ¯ρ, where ¯ρ is the spatial average
176
+ of the condensate density over a UC, and as we discuss
177
+ below the remaining density ρn = ¯ρ βˆ’ ρsf behaves as a
178
+ pseudo-normal fluid. In the more general context where
179
+ the GPE is inapplicable, the Leggett expression for ρsf
180
+ is an upper bound for the SF density in systems with
181
+ crystalline order [6].
182
+ In a 3D system, the current Ji = ρsf
183
+ ij [β„βˆ‚jΟ•/m] derives
184
+ from a SF density tensor. For systems with rectangular
185
+ symmetry [14] ρsf
186
+ ij is diagonal, and the analogs to Eq. (1)
187
+ for each of the three elements use a 1D density integrated
188
+ along the transverse directions. In our experiments this
189
+ implies that the superfluid density is only reduced along
190
+ the direction of the optical lattice, so ρsf
191
+ yy = ρsf
192
+ zz = ¯ρ.
193
+ Experimentβ€”We used 87Rb BECs with N β‰ˆ 2 Γ— 105
194
+ atoms in the |F = 1, mF = 1⟩ hyperfine ground state. A
195
+ 1064 nm trapping laser with an elliptical cross-section,
196
+ traveling along ex provided strong vertical confinement
197
+ with frequency Ο‰z/(2Ο€) = 220 Hz; the in-plane frequen-
198
+ cies, from Ο‰x,y/(2Ο€) = (34, 51) Hz to (56, 36) Hz, were
199
+ optimized for our different experiments. We created a
200
+ 1D optical lattice using a retro-reflected Ξ» = 532 nm
201
+ laser traveling along ex, giving an a = 266 nm lattice pe-
202
+ riod, comparable to the ΞΎ = 170(20) nm minimum heal-
203
+ ing length. The optical lattice was linearly ramped on
204
+ in 100 ms to a final depth ≀ 10 Er, with single pho-
205
+ ton recoil energy and momentum Er = ℏ2k2
206
+ r /(2m), and
207
+ ℏkr = 2πℏ/Ξ» respectively [15]. For Bragg experiments
208
+ the final state was measured using resonant absorption
209
+ imaging after a 15 ms time of flight (TOF); scissors mode
210
+ measurements were performed in-situ using partial trans-
211
+ fer absorption imaging [16].
212
+ Anisotropic speed of soundβ€”The speed of sound for di-
213
+ agonal ρsf
214
+ ij is predicted to result from c2
215
+ i = f sf
216
+ ii /(ΞΊm) in
217
+ terms of the superfluid fractions f sf
218
+ ii = ρsf
219
+ ij/¯ρ, the com-
220
+ pressibility ΞΊ = Β―Οβˆ’1 (βˆ‚Β―Ο/βˆ‚Β΅), and the chemical potential
221
+ Β΅. This reduces to the well-known value c2 = Β΅/m for
222
+ an isotropic homogeneous system (See [17] for the full
223
+ dispersion beyond the linear approximation). The sound
224
+ speed ratio
225
+ c2
226
+ x
227
+ c2y
228
+ = ρsf
229
+ xx
230
+ ρsf
231
+ yy
232
+ = f sf
233
+ xx,
234
+ (2)
235
+ provides direct access to the different components of the
236
+ superfluid density [see [17] for a Josephson sum rule [18]
237
+ argument]. Because the density is y-independent, Eq. (1)
238
+ implies ρsf
239
+ yy = ¯ρ.
240
+ We Bragg-scattered the BEC off a weak sinusoidal po-
241
+ tential with reciprocal lattice vector Ξ΄k slowly moving
242
+ with velocity v by patterning a laser beam with a dig-
243
+ ital micro-mirror device (DMD [19]) and measured the
244
+ scattered fraction p. This results from what are effec-
245
+ tively two interfering laser beams driving two-photon
246
+ transitions with difference-wavevector Ξ΄k and angular fre-
247
+ quency δω = Ξ΄k v. We applied this potential for β‰ˆ 5 ms.
248
+ Bragg transitions ensued when the difference energy and
249
+ momentum were resonant with the BEC’s Bogoliubov
250
+ dispersion, and Fig. 2(a) shows data in the linear regime.
251
+ The width of this spectral feature is limited by our BEC’s
252
+ inhomogeneous density profile; the resonance (vertical
253
+ dashed line) obtained from a Lorentzian fit (solid curve)
254
+ therefore reflects an average speed of sound [20].
255
+ A series of such fits lead to phonon dispersion relations
256
+ with Bragg-lattice period from 2.25 Β΅m to 8.5 Β΅m. Repre-
257
+ sentative dispersions taken along ex and ey are shown in
258
+ Fig. 2(b), and we obtain the phonon speed of sound using
259
+ linear fits. Figure 2(c) summarizes these data showing
260
+ the speed of sound decreasing along the lattice direction
261
+ ex, but slightly increasing along ey (resulting from the
262
+ increased atomic density in the individual lattice sites).
263
+ Finally Fig. 2(d) shows our main result: the normalized
264
+ superfluid density obtained from these data using Eq. (2)
265
+ decreases as a function of U0.
266
+ We compared these data to GPE simulations in two
267
+ ways, we:
268
+ (1) used the Bogoliubov-de Gennes (BdG)
269
+ equations to obtain cx and cy and (2) directly evaluated
270
+ Eq. (1) from the GPE ground state density. The solid
271
+ curves in Fig. 2(c) plot the sound speed obtained from
272
+
273
+ 3
274
+ 0
275
+ 500
276
+ 1000
277
+ δω/2Ο€ (Hz)
278
+ 0.0
279
+ 0.1
280
+ 0.2
281
+ 0.3
282
+ p
283
+ (a)
284
+ 0.0
285
+ 0.2
286
+ 0.4
287
+ Ξ΄k/2Ο€ (Β΅mβˆ’1)
288
+ 0
289
+ 200
290
+ 400
291
+ 600
292
+ 800
293
+ δω/2Ο€ (Hz)
294
+ (b)
295
+ 0
296
+ 2
297
+ 4
298
+ 6
299
+ 8
300
+ 10
301
+ U0/Er
302
+ 0
303
+ 1
304
+ 2
305
+ 3
306
+ c (mm/s)
307
+ cx
308
+ cy
309
+ (c)
310
+ 0
311
+ 2
312
+ 4
313
+ 6
314
+ 8
315
+ 10
316
+ U0/Er
317
+ 0.00
318
+ 0.25
319
+ 0.50
320
+ 0.75
321
+ 1.00
322
+ ρsf
323
+ xx/¯ρ
324
+ (d)
325
+ FIG. 2.
326
+ Bragg spectroscopy. Black and red symbols mark excitations created along ex and ey respectively. (a) Transferred
327
+ population fraction p as a function of frequency difference δω with wavevetor Ξ΄k/2Ο€ = 0.26 Β΅mβˆ’1 and lattice depth U0 = 5.7Er.
328
+ The solid curve is a Lorentzian fit giving the resonance frequency marked by the vertical dashed line. (b) Phonon dispersion
329
+ obtained from Bragg spectra. The bold symbols resulted from (a) and the linear fit (with zero intercept) gives the speed of
330
+ sound. (c) Anisotropic speed of sound. The bold symbols are derived from (b) and the solid curves are from BdG simulations
331
+ (no free parameters [17]). (d) SF density obtained from speed of sound measurements (blue markers, error bars mark single-
332
+ sigma statistical uncertainties). We compare with two models: the red dashed curve plots a homogeneous gas BdG calculation,
333
+ and the solid black curve plots the result of Eq. (1). The simulations used our calibrated experimental parameters.
334
+ solving the 1D BdG [21], and the red dashed curve in (d)
335
+ is the ratio of these speeds. To compare with Leggett’s
336
+ prediction, we found the ground state of the 2D GPE
337
+ for our experimental parameters and evaluated Eq. (1)
338
+ throughout our inhomogeneous system. The black curve
339
+ in Fig. 2 plots the resulting weighted average. Remark-
340
+ ably the BdG results are in near-perfect agreement with
341
+ Leggett’s expression.
342
+ Scissors modeβ€”The single-valued nature of the SF or-
343
+ der parameter greatly affects rotational properties such
344
+ as the moment of inertia I. For highly anisotropic traps,
345
+ the scissors mode [11] describes a fixed density distribu-
346
+ tion pivoting by a small angle ΞΈ about the trap center
347
+ with frequency Ο‰sc. Scissors mode experiments are remi-
348
+ niscent of torsional balance experiments in 4He [22] which
349
+ give access to the non-classical rotational inertia [6].
350
+ It is suggestive to quantify these dynamics in terms of
351
+ the Lagrangian L = I Λ™ΞΈ2/2 βˆ’ V (ΞΈ), for moment of inertia
352
+ 0
353
+ 0.25 0.5 0.75
354
+ 1
355
+ fsf
356
+ xx
357
+ 0.00
358
+ 0.25
359
+ 0.50
360
+ 0.75
361
+ 1.00
362
+ Ο‰sc/Ο‰sc,0
363
+ (a)
364
+ 0
365
+ 2
366
+ 4
367
+ 6
368
+ 8 10
369
+ U0/Er
370
+ 20
371
+ 40
372
+ 60
373
+ Ο‰d/2Ο€ Hz
374
+ Ο‰x,d
375
+ Ο‰y,d
376
+ 0
377
+ 0.25 0.5 0.75
378
+ 1
379
+ fsf
380
+ xx
381
+ βˆ’0.2
382
+ 0.0
383
+ 0.2
384
+ 0.4
385
+ I/Ic
386
+ (54, 36) Hz
387
+ (36, 50) Hz
388
+ (b)
389
+ FIG. 3.
390
+ Moment of inertia from scissors mode.
391
+ (a-inset)
392
+ Measured dipole mode frequencies (markers) along with fits
393
+ (curves) where the frequency at U0 is the only free param-
394
+ eter for each curve. (a) Scissors mode frequency. Blue and
395
+ green correspond to U = 0 trap frequencies (34, 51) Hz and
396
+ (54, 36) Hz respectively. (b) Moment of inertia in units of Ic.
397
+ Symbols are the data computed as described in the text, and
398
+ the solid curves are GPE predictions using I = βˆ‚β„¦βŸ¨Lz⟩, with
399
+ angular frequency Ω.
400
+ I and potential energy V (ΞΈ). For small ΞΈ the potential
401
+ can be expanded as V (ΞΈ) β‰ˆ IΟ‰2
402
+ scΞΈ2/2 with
403
+ I
404
+ Ic
405
+ =
406
+ (Ο‰2
407
+ x βˆ’ Ο‰2
408
+ y)2
409
+ Ο‰2sc(Ο‰2x + Ο‰2y)
410
+ (3)
411
+ in terms of the classical moment of inertia Ic and in agree-
412
+ ment with Ref. [11] for isotropic superfluids. Although
413
+ this interpretation is highly intuitive, it does not survive
414
+ careful consideration.
415
+ The anisotropic superfluid den-
416
+ sity couples radial and azimuthal flow and as a result
417
+ a single parameter Lagrangian is insufficient to describe
418
+ rotational dynamics.
419
+ Instead the superfluid hydrodynamic equations pre-
420
+ dict a moment of inertia scaled by a factor of (f sf
421
+ xxω2
422
+ x βˆ’
423
+ f sf
424
+ yyω2
425
+ y)/(Ο‰2
426
+ x βˆ’ Ο‰2
427
+ y) (see [17]) compared to Eq. (3). There-
428
+ fore we expect Ο‰sc, in conjunction with the superfluid
429
+ density will give I/Ic as a function of lattice depth.
430
+ The inset to Fig. 3(a) plots the dipole mode frequen-
431
+ cies Ο‰x,d and Ο‰y,d for a trap with frequencies (54, 36) Hz.
432
+ The frequency reduction is also related to ρsf via f sf =
433
+ (Ο‰x,d/Ο‰x)2 along the lattice direction [17].
434
+ This ratio
435
+ can also be expressed in terms of an increased effective
436
+ mass that converges to the predictions of single-particle
437
+ band structure [23] when the lattice period falls below
438
+ the healing length; in our case the value computed per-
439
+ turbatively from the GPE differs by about 20 % from the
440
+ band structure prediction. The result of this modeling is
441
+ shown by the solid curves.
442
+ We excited the scissors mode using our DMD to tilt the
443
+ harmonic potential by 50 to 140 mrad for β‰ˆ 1 ms (shorter
444
+ than the trap periods) and let the BEC evolve in the orig-
445
+ inal trap for a variable time. We measured the resulting
446
+ dynamics in-situ and extracted the angle by fitting the re-
447
+ sulting density profile to a rotated Gaussian. Figure 3(a)
448
+ shows scissor mode frequency normalized to the expected
449
+ frequency [24] of Ο‰2
450
+ sc,0 = f sf
451
+ xxω2
452
+ x + f sf
453
+ yyω2
454
+ y for a trap elon-
455
+ gated either along ex [with frequencies (56, 36) Hz, blue]
456
+
457
+ 4
458
+ I/Ic
459
+ I/Ic
460
+ βˆ’20
461
+ 0
462
+ 20
463
+ x (Β΅m)
464
+ βˆ’20
465
+ 0
466
+ 20
467
+ y (Β΅m)
468
+ (a)
469
+ βˆ’20
470
+ 0
471
+ 20
472
+ x (Β΅m)
473
+ βˆ’20
474
+ 0
475
+ 20
476
+ y (Β΅m)
477
+ (b)
478
+ -1.0
479
+ -0.5
480
+ 0.0
481
+ 0.5
482
+ 1.0
483
+ Lz(r) (arb. units)
484
+ βˆ’0.3
485
+ 0.0
486
+ 0.3
487
+ (c)
488
+ 0
489
+ 0.25 0.5 0.75
490
+ 1
491
+ fsf
492
+ xx
493
+ 0.0
494
+ 0.3
495
+ 0.6
496
+ 0
497
+ 1
498
+ (d)
499
+ 0
500
+ 0.25 0.5 0.75
501
+ 1
502
+ fsf
503
+ xx
504
+ 0
505
+ 1
506
+ FIG. 4.
507
+ Moment of inertia in rotating systems computed
508
+ using 2D GPE simulations. The left column indicates sim-
509
+ ulations in which the lattice is static while in the right col-
510
+ umn the lattice co-rotates with the confining potential. (a,b)
511
+ Angular momentum density for trap frequencies 2π×(56,36)
512
+ and U0 = 10Er. (c, d) Total momentum of inertia in traps
513
+ with frequencies 2π×(56,36) (top, green) and 2Ο€ Γ—(36, 56) Hz
514
+ (bottom, blue). Dashed curves plot Isf/Ic and the solid curve
515
+ plots I/Ic.
516
+ or along ey [with frequencies (36, 50) Hz, green]. In both
517
+ cases Ο‰sc is about 5 % in excess of the simple predic-
518
+ tion, perhaps from finite temperature or anharmonicities
519
+ in the ODT.
520
+ We combine these observations in Fig. 3(b) to ob-
521
+ tain I/Ic; the data (symbols) and our 2D GPE simu-
522
+ lations (curves) are in agreement. For traps elongated
523
+ along ex (green) I/Ic unexpectedly changes sign when
524
+ Ο‰x,d = Ο‰y,d. To understand the physical origin of this
525
+ effect we now turn our attention to rotating systems.
526
+ Rotationβ€”Thus far we focused exclusively on the su-
527
+ perfluid density, while avoiding questions about any as-
528
+ sociated normal fluid. We can deduce the existence of a
529
+ normal fluid component by considering two thought ex-
530
+ periments in a 1D ring geometry (with radius R) and
531
+ quantify both in terms of the resulting angular momen-
532
+ tum [25]. In case (i), we consider an Aharonov-Bohm ge-
533
+ ometry and slowly thread the ring with a single quanta
534
+ of magnetic flux (see Ref. [26] for an artificial gauge field
535
+ proposal). The process is equivalent to imprinting a 2Ο€
536
+ phase winding (of the type discussed on page 1), giving
537
+ angular velocity Ω = ℏ/(mR2) and angular momentum
538
+ Lz/ℏ = 2Ο€Rρsf. In case (ii), we consider a complimen-
539
+ tary experiment in which the lattice is very slowly accel-
540
+ erated to a final angular velocity Ω; this is best under-
541
+ stood by transforming into the frame co-rotating with
542
+ the lattice. This leads to a lab frame angular momentum
543
+ Lz/ℏ = 2Ο€R(Β―Οβˆ’Οsf) which we interpret as resulting from
544
+ the normal fluid co-moving with the lattice.
545
+ With this insight we extended our 2D numerical sim-
546
+ ulations to analogous cases for rotating harmonically
547
+ trapped systems where : (i) the lattice is static in the
548
+ lab frame (as in scissors mode experiments) or (ii) it co-
549
+ rotates with the confining potential. In both cases we
550
+ use the coarse graining defined in Eq. (1) to obtain the
551
+ superfluid density and phase. In this way we compute
552
+ the total moment of inertia I from ψ(r, t), the superfluid
553
+ component Isf from Ο†(r, t), and we define the normal
554
+ component as the difference In = I βˆ’ Isf.
555
+ Case (i): as in our 1D thought experiment only the SF
556
+ component responds. Then although βˆ‡Ο• is manifestly
557
+ irrotational, because ρsf
558
+ xx ̸= ρsf
559
+ yy the superfluid current
560
+ can be rotational. In this case, the relative magnitude
561
+ of the co- and counter-rotating contributions vary with
562
+ the lattice depth, leading to regions of negative angular
563
+ momentum density L(r) along the BEC’s semi-minor axis
564
+ [Fig. 4(a)]. The superfluid moment of inertia computed
565
+ from these simulations [Fig. 4(c)] is in full agreement with
566
+ the scissor mode simulation, and as expected for a static
567
+ lattice Isf = I (no normal flow).
568
+ When the lattice is along the semi-minor axis, as pic-
569
+ tured in (a) and the green curve in (c), the counter-
570
+ rotating contribution increases with U0, until the dipole
571
+ mode frequencies along ex and ey invert, after which
572
+ point, I/Ic becomes negative.
573
+ The reverse is the case
574
+ when the lattice is along the semi-major axis and I/Ic
575
+ increases monotonically. This novel observation confirms
576
+ the negative kinetic energy resulting from Λ™ΞΈ.
577
+ Case (ii): In contrast, the angular momentum density
578
+ is strictly positive [Fig. 4(b)] for both lattice orientations
579
+ and I/Ic increases with lattice depth [Fig. 4(d)]. In this
580
+ case the normal fluid to co-rotate with the trap giving
581
+ the current Jn = (βˆ’Οn
582
+ xxy, ρn
583
+ yyx) Λ™ΞΈ. The total I/Ic is then
584
+ the sum of the superfluid [17] and normal contribution
585
+ I
586
+ Ic
587
+ =
588
+ (f sf
589
+ xxω2
590
+ x βˆ’ f sf
591
+ yyω2
592
+ y)2
593
+ (f sf
594
+ xxω2x + f sf
595
+ yyω2y)(ω2x + ω2y) + f n
596
+ xxω2
597
+ x + f n
598
+ yyω2
599
+ y
600
+ Ο‰2x + Ο‰2y
601
+ . (4)
602
+ This result, along with our 2D GPE simulations, are
603
+ plotted in Fig. 4(d). The dashed curve plots the super-
604
+ fluid contribution to Isf/Ic in agreement with the coarse-
605
+ grained GPE (crosses). The solid curve and the triangles
606
+ plot the corresponding total moment of inertia, in excess
607
+ of the SF contribution. This implies the appearance of
608
+ normal fluid flow.
609
+ This agreement confirms that the superfluid contribu-
610
+ tion derives from gradients of the coarse-grained phase
611
+ Ο•, while the normal contribution stems from variations
612
+ of Ο‘ within each lattice site.
613
+ Discussion and outlookβ€”Our inability to obtain I/Ic
614
+ from scissors mode measurements without detailed mod-
615
+ eling reinforces similar conclusions in dipolar gases [27].
616
+ In both cases the simple argument fails because Λ™ΞΈ cou-
617
+ ples to more internal degrees of freedom than Lz alone.
618
+ In this context Ref. [27] concluded that the scissors mode
619
+
620
+ 5
621
+ does yield the moment of inertia when 1D density mod-
622
+ ulations comove with the oscillatory motion: this is con-
623
+ sistent with our findings comparing motion in static and
624
+ rotating lattices. Our GPE simulations indicate that the
625
+ analytical relations generalize to lattices with period in
626
+ excess of the healing length.
627
+ Although we conclude that a normal fluid exists, it is
628
+ inseparable from the optical lattice and lacks any internal
629
+ dynamics of its own, i.e., it is not described by a dynami-
630
+ cal equation of motion. In contrast, both the superstripe
631
+ phase in spin-orbit coupled BECs [28–32] and supersolid
632
+ phases of dipolar gases [33–35], support dynamical den-
633
+ sity modulations. Leggett’s expression applies to both
634
+ of these systems implying a reduced superfluid density,
635
+ which in this case could exhibit dynamics, as expected
636
+ for a system described by a two-fluid model [11, 12].
637
+ This leaves open questions regarding nature the nor-
638
+ mal fluid of spin-orbit coupled systems where an interplay
639
+ between single-particle physics and interactions govern
640
+ supersolid-like properties.
641
+ In addition, ρsf is expected
642
+ to be reduced outside of the superstripe phase [31, 32]
643
+ where the density is uniform (making Leggett’s expres-
644
+ sion inapplicable), but the BEC’s spin vector is spatially
645
+ periodic.
646
+ Note: During the early stages of manuscript prepara-
647
+ tion we become aware of a related work, using a long
648
+ period 1D lattice applied to a homogeneously confined
649
+ 2D BEC.
650
+ The authors thank S. Stringari for suggesting this line
651
+ of investigation and to both S. Stringari and S. Roccuzzo
652
+ for stimulating discussions. In addition W. D. Phillips
653
+ and S. Mukherjee carefully read the manuscript. This
654
+ work was partially supported by the National Institute
655
+ of Standards and Technology, and the National Science
656
+ Foundation through the Physics Frontier Center at the
657
+ Joint Quantum Institute (PHY-1430094) and the Quan-
658
+ tum Leap Challenge Institute for Robust Quantum Sim-
659
+ ulation (OMA-2120757).
660
+ βˆ— These two authors contributed equally
661
+ [1] J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman,
662
+ and E. A. Cornell, Phys. Rev. Lett. 77, 4984 (1996).
663
+ [2] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,
664
+ Rev. Mod. Phys. 71, 463 (1999).
665
+ [3] V. F. Sears, E. C. Svensson, P. Martel,
666
+ and A. D. B.
667
+ Woods, Phys. Rev. Lett. 49, 279 (1982).
668
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+
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1
+ Improvement of Computational Performance of Evolutionary AutoML in a Heterogeneous
2
+ Environment
3
+ Nikolay O. Nikitin, Sergey Teryoshkin, Valerii Pokrovskii, Sergey Pakulin, Denis Nasonov
4
+ ITMO University, Saint-Petersburg, Russia
5
+ Abstract
6
+ Resource-intensive computations are a major factor that limits the effectiveness of automated machine learning solutions. In the
7
+ paper, we propose a modular approach that can be used to increase the quality of evolutionary optimization for modelling pipelines
8
+ with a graph-based structure. It consists of several stages - parallelization, caching and evaluation. Heterogeneous and remote
9
+ resources can be involved in the evaluation stage. The conducted experiments confirm the correctness and effectiveness of the
10
+ proposed approach. The implemented algorithms are available as a part of the open-source framework FEDOT.
11
+ Keywords: AutoML, heterogeneous infrastructure, evolutionary optimization, caching
12
+ 1. Introduction
13
+ Nowadays,
14
+ automated machine learning (AutoML) is
15
+ widely used in science, and industry [16, 33]. The major prob-
16
+ lem of solving real-world tasks with AutoML is the high com-
17
+ putational cost of the search for an optimal modelling pipeline.
18
+ During the evaluation of the candidate pipelines’ quality, many
19
+ machine learning models are trained. This task is very resource-
20
+ intensive, so it can take a considerable amount of time to
21
+ achieve the appropriate result. It can be considered a bottle-
22
+ neck for any existing AutoML solution. This issue raises vari-
23
+ ous problems from different fields: from integration of AutoML
24
+ to business processes [31] to carbon emission and sustainability
25
+ concerns[35].
26
+ There are many approaches for improving computational
27
+ performance that are used in state-of-the-art (SOTA) AutoML
28
+ solutions [8]. First of all, almost all solutions support parallel
29
+ execution. Some of them also support caching of evaluated can-
30
+ didates [22]. Also, the graphics processing unit can be used to
31
+ reduce the training time [15].
32
+ There is a variety of open-source tools that could improve
33
+ the efficiency of certain steps of machine learning pipelines.
34
+ For instance, involving various MLOps tools like MLFlow [38],
35
+ task-specific databases [37] and scaling tools like Ray [18] al-
36
+ lows the effectiveness of ML applications to be notably in-
37
+ creased.
38
+ However, the optimal design of the computational strategy
39
+ depends on the infrastructure and the underlying AutoML al-
40
+ gorithm. The SOTA AutoML solutions are based on differ-
41
+ ent optimization methods: random search, Bayesian optimiza-
42
+ tion, genetic algorithms, and meta-learning [8, 2]. The struc-
43
+ tural patterns used in modelling pipelines can also be different:
44
+ linear pipelines or ensembling techniques (stacking, blending,
45
+ Email address: [email protected] (Nikolay O. Nikitin)
46
+ and boosting) [39]. The most complicated case is a composite
47
+ pipeline represented as a directed acyclic graph [21]. At the
48
+ same time, there is no ready-to-use solution for improving the
49
+ computational performance of an automated open-ended search
50
+ for pipelines in the composite AI field.
51
+ In the paper, we want to propose a adaptive approach
52
+ to reduce the computational cost of AutoML for compos-
53
+ ite pipelines.
54
+ Several techniques are implemented: pipeline
55
+ caching, parallelization of the fitness function evaluation, com-
56
+ putation with hybrid (GPU and CPU) systems, and integration
57
+ with remote distributed systems.
58
+ This approach differs from existing solutions since it can be
59
+ configured for automated machine learning in various computa-
60
+ tional environments (including distributed and heterogeneous).
61
+ Also, the caching procedure can be effectively used for vari-
62
+ ous pipeline designs (linear, weighted ensembles, multi-layer
63
+ ensembles, etc).
64
+ To confirm the effectiveness of the proposed approach in
65
+ empirical way, we conducted a set of numerical experiments us-
66
+ ing set of open datasets of various sizes (described in Table 1).
67
+ The results presented in Section 6 allow us to conclude that a
68
+ larger number of pipelines can be evaluated and better qual-
69
+ ity metrics can be achieved by AutoML using this approach.
70
+ The software implementation is available in the open-source
71
+ AutoML framework FEDOT.
72
+ The paper is organized as follows: Section 2 describes the
73
+ computational strategies used in state-of-the-art AutoML tools.
74
+ Section 3 provides the problem statement for AutoML perfor-
75
+ mance improvement. Section 4 proposes a set of novel im-
76
+ provements for the composite evolutionary AutoML. Section 5
77
+ describes the software implementation of these techniques in an
78
+ open-source framework. Section 6 provides the experimental
79
+ evaluation of the proposed techniques for different case studies.
80
+ Finally, Sec. 7 provides an analysis of the obtained results and
81
+ possible extensions of the research.
82
+ Preprint submitted to Algorithms
83
+ January 13, 2023
84
+ arXiv:2301.05102v1 [cs.LG] 12 Jan 2023
85
+
86
+ 2. Related Works
87
+ There are dozens of open-source AutoML solutions that can
88
+ be used for designing modelling pipelines. The first frame-
89
+ works that became well-known are H2O [15], TPOT [14] and
90
+ Auto-sklearn [4].
91
+ As more novel AutoML solutions, Auto-
92
+ Gluon [3] and LAMA [36] can be noted. Also, there are a lot
93
+ of other AutoML tools with various specific features [17].
94
+ There are different strategies for performance improve-
95
+ ment used in the noted frameworks. In the TPOT framework,
96
+ pipeline caching is implemented [14].
97
+ TPOT-SH [26] uses
98
+ the concept of Successive Halving to explore the search space
99
+ faster, especially for larger datasets.
100
+ Various techniques are
101
+ used to evaluate the pipelines on different subsets of training
102
+ data (e.g. layering [6]).
103
+ A widely-used parallelization tool is the joblib library im-
104
+ plemented in Python. However, there are more advanced frame-
105
+ works for parallelization that can be noted. For example, Ray
106
+ [18] can be used to scale AI and Python applications in dis-
107
+ tributed environments. It provides various instruments for dis-
108
+ tributed data preprocessing, distributed training of ML models
109
+ and scalable hyperparameter tuning.
110
+ Improving the computational performance for evolutionary
111
+ algorithms outside AutoML is also discussed in the literature.
112
+ As an example, parallel GPU-based evaluation of the fitness
113
+ function can be used [24] to solve the expensive problems re-
114
+ lated to big data [23]. There are various techniques that im-
115
+ prove the performance of evolutionary algorithms in concurrent
116
+ mode [7]. The tensor-based computational model can be used
117
+ to achieve cross-platform hardware acceleration [12].
118
+ Also,
119
+ platform-specific open-source solutions are presented in this
120
+ field (e.g. scikit-learn-intelex 1).
121
+ One of the widely used techniques to avoid fitness evalua-
122
+ tion bottlenecks in evolutionary algorithms is caching [29]. Fi-
123
+ nal values of the fitness evaluation can be cached [13] as well
124
+ as partial results [30, 11, 10].
125
+ Moreover, a number of solutions exist that can perform
126
+ remote/distributed training (e.g.
127
+ Auto-sklearn, H2O, TPOT,
128
+ LAMA). These AutoML frameworks use different frameworks
129
+ for distributed computing. Autosklearn and TPOT use Dask2,
130
+ LAMA uses Apache Spark3. H2O uses its own Apache Spark
131
+ modification called Sparkling Water4. Distributed computing
132
+ frameworks allow the processing of large datasets spread over
133
+ the nodes of a cluster system.
134
+ We can conclude that there is a large number of techniques
135
+ and solutions that can reduce the resource consumption for Au-
136
+ toML and EA. However, there is still no well-developed ap-
137
+ proach that can be used to identify graph-based pipelines in
138
+ the heterogeneous computational environment in composite AI
139
+ problems. For this reason, we decided to formulate the problem
140
+ statement specific to composite AutoML and propose possible
141
+ solutions.
142
+ 1https://github.com/intel/scikit-learn-intelex
143
+ 2https://dask.org
144
+ 3https://spark.apache.org
145
+ 4https://h2o.ai/products/h2o-sparkling-water
146
+ 3. Problem Statement
147
+ We want to design multi-task and multi-modal pipelines for
148
+ various tasks using a single flexible instrument. Consequently,
149
+ it becomes necessary to implement the framework’s architec-
150
+ ture more abstractly to separate the pipeline search process
151
+ from the top-level API. The modelling pipeline is represented
152
+ as a directed acyclic graph in this case. Each node (modelling
153
+ or data transformation operation) is described by the operation’s
154
+ name and set of hyper-parameters. If necessary, different data
155
+ sources (tables, time series, images, texts) can be involved in
156
+ the pipeline. Also, metadata is attached to the data flow, making
157
+ it possible to change the task several times during the pipeline
158
+ evaluation (e.g., solve a classification task and then - a regres-
159
+ sion task).
160
+ The drawback of this approach is the increased search space
161
+ that should be explored during the optimization.
162
+ In auto-
163
+ mated modelling, we want to control the balance between open-
164
+ endedness [25] and local search. The simplest way is to apply
165
+ the of direct constraints (e.g. limit to the pipeline size). Also, it
166
+ can be more effective to apply the regularization and sensitivity
167
+ analysis procedures [21] and adaptive optimisation strategies
168
+ to control the convergence of optimisation. At the same time,
169
+ avoiding over-complicated pipelines and reckless spending of a
170
+ limited time budget is also essential. It makes the effectiveness
171
+ of the computational part even more critical for open-ended Au-
172
+ toML.
173
+ It pushes us to compromise between pipeline complexity
174
+ and training time. However, if we can improve computing ef-
175
+ ficiency, the framework will probably be able to build more
176
+ complicated models with higher quality while consuming the
177
+ same training time.
178
+ There are many approaches to improv-
179
+ ing evolutionary algorithms’ computing performance, such as
180
+ parallelization, caching, etc. These approaches can be divided
181
+ into single-machine optimizations and horizontal scaling tech-
182
+ niques. Single-machine optimizations aim to improve comput-
183
+ ing performance only on the machines performing the compu-
184
+ tations. Horizontal scaling allows involving additional servers
185
+ to speed up computing. Both techniques can be used separately
186
+ or combined.
187
+ Evolutionary algorithms’ computational time mainly de-
188
+ pends on the population size and the number of generations.
189
+ Increasing population size leads to an increased probability of
190
+ getting better individuals. A more significant number of gener-
191
+ ations means more attempts to grow better individuals based on
192
+ the best previous generation.
193
+ From a computational point of view, we have several iter-
194
+ ations, each of which requires the results from the previous it-
195
+ eration. So, it is complicated to scale computations over the
196
+ iterations, and the total computation time is Equation 1.
197
+ Ttotal =
198
+ n
199
+ οΏ½
200
+ i=1
201
+ Ti
202
+ (1)
203
+ where n is the number of iterations and Ti - the computa-
204
+ tion time of generation i. Inside one generation, all individuals
205
+ are processed independently from one another, allowing us to
206
+ 2
207
+
208
+ scale these computations according to the available computa-
209
+ tional resources. The population training time can be estimated
210
+ with Equation 2.
211
+ Ttotal =
212
+ n
213
+ οΏ½
214
+ i=1
215
+ argmax
216
+ di
217
+ j∈Di
218
+ (argmin(Ο„(di
219
+ j,
220
+ rj∈Ri/{rjβˆ’1,..,r1}
221
+ Ο‰(Diβˆ’1,..,D1), rj))
222
+ (2)
223
+ where Di is the set of individuals in the population i and di
224
+ j is
225
+ individual j, Ri is the set of available resources on iteration i,
226
+ Ο‰ is a cache function with pre-calculated elements on iterations
227
+ i βˆ’ 1, i βˆ’ 2, ..., 1, and Ο„ is a function that returns the calculation
228
+ time considering caching and evaluation of individuals Di.
229
+ Due to the Equations 1 and 2, we should increase the popu-
230
+ lation size as much as possible. It allows to speed up the algo-
231
+ rithm convergence and improve its result using horizontal scal-
232
+ ing. For this purpose, we can use both remote computing on
233
+ production servers and distributed computing using homoge-
234
+ neous and heterogeneous computing clusters.
235
+ Remote computing allows model training to be delegated to
236
+ a remote infrastructure. This approach is justified if the local
237
+ machine computational resources are insufficient to train a set
238
+ of models in a reasonable amount of time. Remote computing
239
+ may take place on a dedicated computing server or a cluster of
240
+ servers that accepts tasks to train models using REST API, RPC
241
+ or message queues.
242
+ The main challenge in the investigated problem is to pro-
243
+ pose the performance improvement strategy for AutoML that
244
+ is adaptive to various types of computational infrastructures.
245
+ In Figure 1, five classes are noted: shared memory system,
246
+ multi-node cluster with distributed memory, complex homo-
247
+ geneous and heterogeneous supercomputer environments, and
248
+ hybrid systems. The system with structure (a) can execute par-
249
+ allel tasks in a straightforward way. In the systems (b)-(e), the
250
+ remote nodes are involved (homogeneous and heterogeneous).
251
+ For the system (e), the structure is hybrid since various remote
252
+ nodes have different computational performance and connec-
253
+ tions overheads. For this reason, the adaptability of the compu-
254
+ tational strategy is especially important.
255
+ There are various ways can be uses to adapt the compu-
256
+ tational strategy to specific infrastructure.
257
+ For example, the
258
+ empirical performance models [9] can be used to choose the
259
+ optimal infrastructure for evaluating specific pipelines. Sim-
260
+ ple pipelines with low fitting time can be assigned to low-
261
+ performance computational nodes.
262
+ Otherwise, complicated
263
+ pipelines with high fitting time can be assigned to high-
264
+ performance nodes. It makes it necessary to develop a modular
265
+ approach that effectively utilizes all available resources.
266
+ Our main motivation is to develop an approach that can be
267
+ used at the computational layer of AutoML. It should be pos-
268
+ sible to adapt this layer to the specified infrastructure (local or
269
+ remote) in a frame of the same AutoML approach. This solution
270
+ should be high-level, modular and flexible to allow integrating
271
+ it with different AutoML tools. Also, it should support the dif-
272
+ ferent types of pipelines (from simplest linear pipeline to the
273
+ multi-level ensembles).
274
+ 4. Proposed Improvements
275
+ This section is devoted to various aspects of the proposed
276
+ approach for improving the computational performance of evo-
277
+ lutionary AutoML. The high-level scheme of the approach is
278
+ presented in Figure 2. Four main aspects are considered: (1)
279
+ parallelization of the fitness function evaluation; (2) partial
280
+ caching of evaluated individuals; (3) combining CPU and GPU
281
+ to accelerate the processing of individuals (4) integration with
282
+ remote infrastructure for a complex task. Algorithmic-based
283
+ improvements (e.g.
284
+ surrogate-assisted optimization) are not
285
+ considered here.
286
+ The detailed implementation of the proposed approach is
287
+ described in Alg. 1. In this notation, graph represents the struc-
288
+ ture of the composite pipeline. The details of the evolutionary
289
+ optimisation are hidden to make the proposed improvements
290
+ more clear.
291
+ Algorithm 1 High-level pseudocode of the evaluation dispatch-
292
+ ing algorithm implemented in the proposed approach. Paral-
293
+ lelization, caching and evaluation stages are demonstrated for
294
+ processing one generation of the evolutionary algorithm.
295
+ 1: procedure ProcessPopulation
296
+ 2:
297
+ Input:
298
+ inds (set of non-evaluated individuals),
299
+ objective (objective function that calculates the fitness
300
+ of an individual),
301
+ n (number of parallel jobs)
302
+ timer (timer-like object)
303
+ infrastructure (description of setup)
304
+ 3:
305
+ Output: evaluated inds
306
+ 4:
307
+ do in parallel(n)
308
+ 5:
309
+ if timer.enough time( ) then
310
+ 6:
311
+ graph ← inds[i].graph β–· get structure of each ind.
312
+ 7:
313
+ if infrastructure.is remote( ) then
314
+ 8:
315
+ cache ← DistributedCache()
316
+ 9:
317
+ sync cache
318
+ β–· sync cache database
319
+ 10:
320
+ task id ← create task(graph)
321
+ 11:
322
+ wait task id
323
+ 12:
324
+ inds[i].fitness ← request result(graph)
325
+ 13:
326
+ else
327
+ 14:
328
+ prepare graph
329
+ β–· assign CPU and GPU to
330
+ nodes
331
+ 15:
332
+ cache ← LocalCache()
333
+ 16:
334
+ load cache
335
+ β–· init cache database
336
+ 17:
337
+ if cache.exists(graph)( ) then
338
+ 18:
339
+ fit from cache(graph)
340
+ 19:
341
+ inds[i].fitness ← obj(graph, cache)
342
+ 20:
343
+ fit(graph)
344
+ β–· Fit nodes that are not in cache
345
+ 21:
346
+ save cache
347
+ β–· preserve updated cache
348
+ 22:
349
+ if not inds[i].is valid then
350
+ 23:
351
+ delete inds[i]
352
+ β–· for unsuccessful evaluation
353
+ 24:
354
+ else
355
+ 25:
356
+ delete inds[i]
357
+ β–· not enough time, skipping
358
+ 26:
359
+ return inds
360
+ β–· candidates for selection
361
+ 3
362
+
363
+ Figure 1: Different types of computational infrastructures that can be used in AutoML: (a) shared memory (SM) system (b) multi-node cluster with distributed
364
+ memory (c) complex homogeneous supercomputer system with spatially distributed infrastructure (d) supercomputer system with heterogeneous distributed infras-
365
+ tructure.
366
+ Figure 2: Workflow of the proposed approach for the improvement of compu-
367
+ tational performance for composite AutoML
368
+ 4.1. Parallelization
369
+ Parallelizing evolutionary algorithms is not a novel idea.
370
+ There are a lot of papers and open-source solutions devoted
371
+ to this problem. However, parallelization in AutoML has its
372
+ specifics. For example, various computationally efficient strate-
373
+ gies of parallel evolution can be used [9].
374
+ We are considering an evolutionary algorithm for search-
375
+ ing for the best solution in the space of pipelines that can be
376
+ represented as directed acyclic graphs. The classic approach
377
+ to parallelizing evolutionary optimization is evaluating all indi-
378
+ viduals in the population concurrently [9]. It works because of
379
+ the nature of the evolutionary algorithm. There are no depen-
380
+ dencies between individuals in a generation. Other approaches
381
+ suggest dividing populations into isolated parts [32] or using
382
+ co-evolutionary algorithms to divide tasks into subtasks [5].
383
+ The proposed algorithm considers the maximum evaluation
384
+ time length for each pipeline evaluation to resolve the possi-
385
+ ble evaluation time anomalies caused by the stochastic nature
386
+ of data-driven model training. If the training process does not
387
+ converge at least in one cross-validation fold, the time required
388
+ for corresponding fitness evaluation can be increased signifi-
389
+ cantly. So, the individuals that spend excess time on evaluation
390
+ are skipped to preserve the overall performance of the evolu-
391
+ tionary optimizer.
392
+ 4.2. Caching
393
+ The existing caching approaches are aimed at preserving
394
+ and reusing fitted pipelines [22]. However, separate nodes of
395
+ composite pipelines can be cached individually [21]. It makes
396
+ it possible to reuse the fitted models and reduce the fitness func-
397
+ tion’s evaluation time. The optimizer can share the in-memory
398
+ cache across the populations, and individuals [9]. However, it
399
+ raises the problems of memory consumption.
400
+ After analyzing existing solutions, we focused on the rela-
401
+ tional database approach for pipeline caching. More specifi-
402
+ cally, the sqlite3 library was used to implement it. First of all,
403
+ it provides only one output file, which is not guaranteed for
404
+ non-relational databases - e.g. shelve. Secondly, all concurrent
405
+ save-load operations can be fully processed during the parallel
406
+ evaluation of the fitness functions without direct usage of syn-
407
+ chronization primitives, atomic variables and other instruments
408
+ necessary for simultaneous access to data.
409
+ Finally, this approach allows extracting several operations
410
+ simultaneously, which helps to improve the overall perfor-
411
+ mance of caching. Also, the set of operations can be saved
412
+ to the database taking into account the existence of cache items
413
+ with the same primary key.
414
+ The caching procedure for the multi-layer ensemble
415
+ pipelines should take into account that the cached model/-
416
+ operations are suitable only for the specific configuration of
417
+ previous nodes and edges in the modelling pipelines.
418
+ So,
419
+ the key contains the recursive description of the structure
420
+ 4
421
+
422
+ a)Shared memory
423
+ b) Cluster with
424
+ distributed memory
425
+ system (SM)
426
+ c)Homogeneoussupercomputer
427
+ environment
428
+ Core
429
+ Core
430
+ Core
431
+ Main
432
+ 1
433
+ N
434
+ Node
435
+ Sheduler
436
+ Node
437
+ Node
438
+ Node
439
+ 1
440
+ N2
441
+ Node
442
+ Node
443
+ Node
444
+ 1
445
+ ..
446
+ N
447
+ Node
448
+ Node
449
+ Node
450
+ Node
451
+ Node
452
+ Node
453
+ 1
454
+ N1
455
+ d) Heterogeneous
456
+ 1
457
+ N3
458
+ supercomputerenvironment
459
+ e) Hybrid environment
460
+ Sheduler
461
+ SM
462
+ System
463
+ Low-speedchannel
464
+ High-speed channel
465
+ Low-speed channel Mid speed channel High-speed channel
466
+ SM
467
+ Node
468
+ Node
469
+ Node
470
+ SM
471
+ Clust.
472
+ Clust.
473
+ Embed.
474
+ Embed.
475
+ Syst.
476
+ 1
477
+ N
478
+ Syst.
479
+ system
480
+ systemCandidate
481
+ ML pipeline
482
+ Population K
483
+ 0000
484
+ N
485
+ Individuals
486
+ 不
487
+ Parallelization
488
+ Evaluation
489
+ Caching
490
+ Local
491
+ Remote
492
+ CPU
493
+ GPUof previous nodes and edges.
494
+ Also, the identifier of cross-
495
+ validation fold is specified. The following notation is used:
496
+ (/[node name] [hparams];)/[node name] [hparams]...”, where
497
+ / denotes the beginning of the node name and round brackets
498
+ represent the nested edges. The caching details are presented in
499
+ Figure 3.
500
+ Figure 3: Interaction between operations’ cache and the modelling pipeline that
501
+ should be fitted
502
+ 4.3. GPU
503
+ Evaluating ML models with GPUs is a well-developed fea-
504
+ ture in many solutions. For example, the RAPIDS library [34]
505
+ contains the CuML module that allows training classification,
506
+ regression and clustering models with GPUs. To adapt this so-
507
+ lution to composite pipelines, we should consider a setup in
508
+ which only a part of the nodes can be evaluated with GPUs. In
509
+ this situation, the pipeline should be fitted in a heterogeneous
510
+ way.
511
+ The proposed approach makes it possible to use both CPUs
512
+ and GPUs for fitting by separating the ML model type and its
513
+ implementation. The same model (e.g. random forest) can have
514
+ several implementations (CPU-based and GPU-based).
515
+ Figure 4 shows an example of a computationally hetero-
516
+ geneous composite model structure. Data transfer between the
517
+ GPU-based nodes (yellow) is performed within the video mem-
518
+ ory, and the models themselves in the nodes are trained on
519
+ graphics processing units (GPUs). Other nodes are executed
520
+ on CPUs.
521
+ Due to the multiple software limitations set by RAPIDS li-
522
+ braries (CUDA-compatible GPU driver, restricted set of sup-
523
+ ported operation systems), it is practical to conduct the compu-
524
+ tations within Docker-based containers.
525
+ 4.4. Remote evaluation
526
+ Remote evaluation can be integrated into the evolutionary
527
+ optimiser in various ways. Both dataset folds and population
528
+ parts can be distributed across several computational nodes to
529
+ satisfy time or memory limits. Since evolutionary algorithms
530
+ do not always require processing large datasets, we have fo-
531
+ cused on the parallelism aspect of remote computing. The pro-
532
+ posed implementation relies on Kubernetes. The REST API
533
+ Figure 4: The structure of a pipeline that can be evaluated in a heterogeneous
534
+ (CPU - blue and GPU - yellow) way
535
+ service inside the Kubernetes cluster is used to run computa-
536
+ tions via HTTP requests. The client implements a wrapper for
537
+ requests.
538
+ During the population training, the evaluator uses the
539
+ client’s methods to process individuals on the Kubernetes clus-
540
+ ter. Then, after starting processing all individuals, the evaluator
541
+ waits for computations to be completed via client methods. The
542
+ run request contains the container image, resources limit for the
543
+ container, mount paths and model parameters. The REST API
544
+ service creates the requested container and keeps monitoring it.
545
+ The client uses requests to the REST API service to get actual
546
+ containers’ statuses to see if it is still running, completed or
547
+ failed.
548
+ Finally, the client downloads the fitted pipeline when the
549
+ training is completed. We wrap the result into a compressed
550
+ archive to reduce the amount of data transferred over the net-
551
+ work. Then, the files are sent to the client. This process scheme
552
+ is presented in Figure 5.
553
+ Figure 5: Communication between AutoML and remote cluster
554
+ This way, we can divide the population training process into
555
+ three stages: (1) requests to evaluate individuals, (2) computing
556
+ and waiting for the completion and (3) fetching the results.
557
+ 5. Software Implementation
558
+ The proposed approach can be used as a part of the archi-
559
+ tecture that includes:
560
+ 5
561
+
562
+ Preprocessor cache
563
+ Nodes cache
564
+ data_description 1: fitted preprocessor 1
565
+ node uid 1: fitted node 1
566
+ data description 2: fitted preprocessor 2
567
+ node uid 2: fitted node 2
568
+ Data fold 1
569
+ preprocessor
570
+ Node 1
571
+ uid=node 4/
572
+ evaluation
573
+ (node1;node2;node3)
574
+ Fitness
575
+ Data
576
+ Data fold 2
577
+ Node 2
578
+ Node 4
579
+ Node 3
580
+ Data fold 3
581
+ Pipeline
582
+ 8
583
+ Evo. opt
584
+ 88
585
+ N
586
+ N+1
587
+ pop.
588
+ pop.Decision
589
+ Tree
590
+ (GPU)
591
+ SVC
592
+ (GPU)
593
+ Random
594
+ Scaling
595
+ Forest
596
+ (GPU)
597
+ Logit
598
+ (CPU)
599
+ GPU memory
600
+ (CPU)
601
+ XGBoost
602
+ (CPU)LocalAutoML
603
+ Kubernetes
604
+ RemoteEvaluatol
605
+ Client
606
+ AutoML
607
+ Create individual
608
+ Individual-1
609
+ create task
610
+ RESTAPI
611
+ AutoML
612
+ Wait until ready
613
+ get task status
614
+ Service
615
+ Individual-2
616
+ AutoML
617
+ Fetch individual
618
+ download result
619
+ Individual-Nβ€’ The model repository block, which provides storage and
620
+ selection of various implementations of predictive mod-
621
+ els and data processing blocks. One model can contain
622
+ several implementations (e.g., for CPU and GPU);
623
+ β€’ The block of the generative design of composite mod-
624
+ els, which implements the creation of models with spec-
625
+ ified properties by evolutionary algorithms. The proper-
626
+ ties of the models are determined by the target function
627
+ passed to the optimizer. If there is more than one tar-
628
+ get function specified (as an example, the training time
629
+ and modelling error can be used together as objectives
630
+ for AutoML), then the multi-criteria formulation of the
631
+ optimization problem is implemented, where the result
632
+ of the model design is a Pareto front containing various
633
+ compromising solutions. The genotype is represented in
634
+ graph form, and the crossing and mutation operators are
635
+ implemented accordingly.
636
+ β€’ The pipeline execution block on a given computational
637
+ infrastructure. It allows individual pipeline execution on
638
+ the given computational nodes.
639
+ This architecture is implemented in the core of the open-
640
+ source FEDOT framework. Different aspects of its implemen-
641
+ tation are already detailed in a series of papers: [19] describes
642
+ the main schemes and the implementation of the evolutionary
643
+ operators, [28] is devoted to the multi-objective modification of
644
+ this approach, and [20] provides an extended description of the
645
+ various aspects of the evolutionary design for composite mod-
646
+ elling pipelines. The tuning strategy of the pipeline hyperpa-
647
+ rameters is based on Bayesian optimization.
648
+ Custom models can be put inside this node. Search space
649
+ for hyperparameters and initial approximations for the models
650
+ should be specified manually if necessary. It makes it possi-
651
+ ble to involve the infrastructure-specific implementation of the
652
+ model in AutoML.
653
+ The example below demonstrates the AutoML workflow
654
+ from input data processing to obtaining prediction.
655
+ api = Fedot(problem=’classification ’,
656
+ seed =42, timeout =30, preset=’gpu’)
657
+ api.fit(features=x_train , target=
658
+ y_train)
659
+ predictions = api.predict(features=
660
+ x_test)
661
+ Figure 6 provides the UML class diagram for the imple-
662
+ mentation of various evaluation strategies that allow combining
663
+ CPU- and GPU-based nodes in a single modelling pipeline. A
664
+ high-level modelling method (e.g. Support Vector Classifica-
665
+ tion) can be implemented using different algorithms: a CPU-
666
+ optimised implementation of SVC can be obtained from the
667
+ scikit-learn library [27]. In contrast, a GPU-optimised imple-
668
+ mentation is available in the CUML library. The proposed ar-
669
+ chitecture makes it possible to hide these details inside the spe-
670
+ cific modelling pipeline and use the same optimisation logic for
671
+ different implementations of the algorithms.
672
+ Figure 6: The class diagram for the implementation of a modelling pipeline that
673
+ consists of several operations. The Operation class represents a high-level mod-
674
+ elling strategy that is used inside the operation. EvaluationStrategy is a base
675
+ class for the algorithmic implementation of this strategy. SklearnEvaluation-
676
+ Strategy represents the implementation obtained from the scikit-learn library
677
+ and CumlEvaluationStrategy represents the implementation from the CUML
678
+ library.
679
+ The optimizer operates on individual models as a black box
680
+ with input, output and fit/predict methods. The following build-
681
+ ing blocks can be used for pipelines: models (Bernoulli Naive
682
+ Bayes classifier, logistic regression, multilayer perceptron, ran-
683
+ dom forest, gradient boosting, k-nearest classifier, QDA, LDA,
684
+ decision tree) and data transformation operations (scaling, nor-
685
+ malization, polynomial features transformation, principal com-
686
+ ponent analysis, independent component analysis, isolation for-
687
+ est, resampling).
688
+ 6. Experimental Studies
689
+ We conducted a series of experiments to confirm the cor-
690
+ rectness and effectiveness of the proposed approach.
691
+ It can
692
+ be divided into experiments with local and remote infrastruc-
693
+ ture. As benchmarks, various classification datasets from the
694
+ OpenML base [1] and synthetic datasets were used (the full list
695
+ is presented in Table 1). A description of the computational
696
+ infrastructure is provided for each experiment.
697
+ The following methodology was used for experimental
698
+ studies: each experiment started with dividing samples into two
699
+ groups: β€˜learning’ and β€˜validation’ samples in the ratio 70% to
700
+ 30% to avoid data leaks. Then, the learning sample was trans-
701
+ ferred to the evolutionary optimizer. During the optimisation,
702
+ the 5-fold cross-validation procedure was applied to estimate
703
+ the values of the fitness function.
704
+ The experiment is repeated three times for each dataset to
705
+ take the stochasticity of the optimizer into account. The quality
706
+ metrics are averaged over these iterations.
707
+ 6.1. Local infrastructure
708
+ For experiments with the local infrastructure, we configured
709
+ a server based on Xeon Cascadelake (2900MHz) with 12 cores
710
+ 6
711
+
712
+ EvaluationStrategy
713
+ operation_type
714
+ fito
715
+ Operation
716
+ predictO
717
+ operation_type
718
+ A
719
+ strategy
720
+ define_strategy0
721
+ exectute_strategy0
722
+ SklearnEvaluationStrategy
723
+ operation_implementation
724
+ fito
725
+ predicto
726
+ CUMLEvaluationStrategy
727
+ Pipeline
728
+ operation_implementation
729
+ fito
730
+ predict0Table 1: The properties of OpenML datasets that were used during the exper-
731
+ iments. The random forest model is used as a baseline for the training time
732
+ estimation.
733
+ Dataset
734
+ name
735
+ Rows,
736
+ 10Λ†3
737
+ Feat.
738
+ Total
739
+ elem.,
740
+ 10Λ†3
741
+ Base.
742
+ train.
743
+ time,
744
+ sec
745
+ Num.
746
+ of
747
+ clas
748
+ ses
749
+ adult
750
+ 49
751
+ 14
752
+ 684
753
+ 12.5
754
+ 2
755
+ amazon
756
+ employee
757
+ access
758
+ 33
759
+ 9
760
+ 295
761
+ 1.5
762
+ 2
763
+ australian
764
+ 0.69
765
+ 15
766
+ 10
767
+ 0.2
768
+ 2
769
+ bank-
770
+ marketing
771
+ 45
772
+ 17
773
+ 769
774
+ 6.9
775
+ 2
776
+ blood-
777
+ transfusion
778
+ 0.75
779
+ 5
780
+ 4
781
+ 0.1
782
+ 2
783
+ car
784
+ 1,8
785
+ 7
786
+ 12
787
+ 0.5
788
+ 4
789
+ cnae-9
790
+ 1,1
791
+ 857
792
+ 926
793
+ 14.7
794
+ 9
795
+ jungle chess
796
+ 2pcs
797
+ 45
798
+ 7
799
+ 314
800
+ 48.0
801
+ 3
802
+ numerai28
803
+ 96
804
+ 22
805
+ 2119
806
+ 8.4
807
+ 2
808
+ phoneme
809
+ 54
810
+ 6
811
+ 32
812
+ 0.12
813
+ 2
814
+ sylvine
815
+ 51
816
+ 21
817
+ 108
818
+ 0.5
819
+ 2
820
+ volkert
821
+ 58
822
+ 181
823
+ 10554
824
+ 128.4
825
+ 10
826
+ synthetic
827
+ blobs
828
+ 100
829
+ 10
830
+ 1000
831
+ 6.2
832
+ 2
833
+ synthetic
834
+ moons
835
+ 1
836
+ 2
837
+ 2
838
+ 0.12
839
+ 2
840
+ and 24Gb memory.
841
+ As our approach claims to increase the number of evaluated
842
+ pipelines during fitting due to caching, it will be correct to com-
843
+ pare this metric with and without the caching option. For that
844
+ reason, we created the benchmark considering different compu-
845
+ tational setups for AutoML. It utilizes a dataset for classifica-
846
+ tion present using the FEDOT framework as a test bench.
847
+ In Figure 7 the comparison of cache-based and cache-free
848
+ configurations is provided. For the first one, both the pipelines
849
+ cache and data preprocessing cache are activated. The number
850
+ of parallel jobs used during optimization is one.
851
+ During evolutionary optimization, a lot of candidate solu-
852
+ tions (pipelines) are evaluated. We repeated the experiment for
853
+ different timeouts that limit the execution time for the entire Au-
854
+ toML run since they affect the number of evaluated pipelines.
855
+ Also, an additional time limit is applied to the entire pipeline (to
856
+ process the fit time anomalies for large pipelines). It is specified
857
+ as 1/4 of the total timeout.
858
+ Because of the stochastic nature of the optimization-based
859
+ experiments, each run was repeated three times, and the ob-
860
+ tained metrics were averaged.
861
+ The results presented in Figure 8 are obtained with the
862
+ n jobs hyperparameter value equal to 12.
863
+ Table 2 summarises the averaged metrics of the experiments
864
+ with single-process and multi-process caching. The average
865
+ performance was increased by 14 %, which empirically con-
866
+ Figure 7: The dependence between the number of pipelines and the usage of
867
+ cache (averaged for ten runs). Single-processing is used.
868
+ Figure 8: The dependence between the number of pipelines and the usage of
869
+ cache (averaged for ten runs). 8 parallel jobs are used.
870
+ firms the effectiveness of the proposed approach.
871
+ The next stage of the experiment is devoted to the analysis
872
+ of the evolutionary algorithm’s performance in multiprocessing
873
+ mode. We compared algorithm performance with the number
874
+ of processes equal to 1 and 8 with a timeout set to 10 minutes.
875
+ The optimization of the pipeline structure was repeated three
876
+ times with no seed and with five cross validation folds to take
877
+ stochasticity into account.
878
+ The dependency of correctly evaluated pipelines on a speci-
879
+ fied number of jobs for a single dataset is presented in Figure 9.
880
+ It can be seen that near-linear improvement in parallel speedup
881
+ is achieved. Figure 10 demonstrates the dependency of the best
882
+ fitness calculated using cross-validation on the timestamp from
883
+ the configuration. Launches with 8 processes find a better solu-
884
+ tion faster than launches with one process.
885
+ Table 3 summarises the averaged results of experiments in
886
+ single-process and multiprocessing modes. The fitness score
887
+ calculated using cross-validation increases linearly with the
888
+ number of evaluated pipelines. It confirms the effectiveness of
889
+ the local parallelization of evolutionary AutoML.
890
+ 7
891
+
892
+ 350
893
+ without cache
894
+ with cache
895
+ 5
896
+ actual time for optimization in minutes
897
+ 300
898
+ correctly evaluated pipelines
899
+ 4
900
+ 250
901
+ 200
902
+ 3
903
+ 150
904
+ 2
905
+ 100
906
+ 1
907
+ 50
908
+ 1.0
909
+ 1.5
910
+ 2.0
911
+ 2.5
912
+ 3.0
913
+ 3.5
914
+ 4.0
915
+ 4.5
916
+ 5.0
917
+ timeout in minuteswithout cache
918
+ with cache
919
+ 5
920
+ 1200
921
+ actual time for optimization in minutes
922
+ correctly evaluated pipelines
923
+ 4
924
+ 1000
925
+ 800
926
+ 3
927
+ 600
928
+ 2
929
+ 400
930
+ 1
931
+ 1.0
932
+ 1.5
933
+ 2.0
934
+ 2.5
935
+ 3.0
936
+ 3.5
937
+ 4.0
938
+ 4.5
939
+ 5.0
940
+ timeout in minutesTable 2: The results of experiments with caching of pipeline nodes and data preprocessing operations. The first column indicates whether the cache has been used,
941
+ and the second column represents the number of parallel processes. The next three columns represent different metric values (the number of evaluated pipelines,
942
+ ROC AUC for validation sample, and ROC AUC for cross-validation of training sample).
943
+ Dataset
944
+ Configuration
945
+ Pipelines count
946
+ ROC-AUC final
947
+ ROC-AUC cross-validation
948
+ Cache
949
+ Number of processes
950
+ adult
951
+ on
952
+ 1
953
+ 27
954
+ 0,92
955
+ 0,9117
956
+ off
957
+ 23
958
+ 0,9213
959
+ 0,913
960
+ on
961
+ 8
962
+ 190
963
+ 0,921
964
+ 0,9131
965
+ off
966
+ 170
967
+ 0,922
968
+ 0,9137
969
+ amazon employee access
970
+ on
971
+ 1
972
+ 85
973
+ 0,8447
974
+ 0,8346
975
+ off
976
+ 78
977
+ 0,8497
978
+ 0,8376
979
+ on
980
+ 8
981
+ 416
982
+ 0,8507
983
+ 0,8356
984
+ off
985
+ 369
986
+ 0,849
987
+ 0,8398
988
+ australian
989
+ on
990
+ 1
991
+ 879
992
+ 0,9313
993
+ 0,9432
994
+ off
995
+ 838
996
+ 0,9283
997
+ 0,9401
998
+ on
999
+ 8
1000
+ 6354
1001
+ 0,928
1002
+ 0,9411
1003
+ off
1004
+ 6199
1005
+ 0,934
1006
+ 0,9442
1007
+ bank-marketing
1008
+ on
1009
+ 1
1010
+ 38
1011
+ 0,93
1012
+ 0,931
1013
+ off
1014
+ 30
1015
+ 0,9313
1016
+ 0,93
1017
+ on
1018
+ 8
1019
+ 205
1020
+ 0,931
1021
+ 0,932
1022
+ off
1023
+ 211
1024
+ 0,932
1025
+ 0,931
1026
+ blood-transfusion
1027
+ -service-center
1028
+ on
1029
+ 1
1030
+ 2175
1031
+ 0,748
1032
+ 0,75
1033
+ off
1034
+ 2064
1035
+ 0,7383
1036
+ 0,759
1037
+ on
1038
+ 8
1039
+ 13943
1040
+ 0,745
1041
+ 0,761
1042
+ off
1043
+ 13834
1044
+ 0,749
1045
+ 0,7659
1046
+ car
1047
+ on
1048
+ 1
1049
+ 812
1050
+ 0,921
1051
+ 0,933
1052
+ off
1053
+ 728
1054
+ 0,9233
1055
+ 0,9319
1056
+ on
1057
+ 8
1058
+ 4856
1059
+ 0,922
1060
+ 0,935
1061
+ off
1062
+ 4608
1063
+ 0,92
1064
+ 0,934
1065
+ cnae-9
1066
+ on
1067
+ 1
1068
+ 214
1069
+ 0,995
1070
+ 0,9939
1071
+ off
1072
+ 195
1073
+ 0,995
1074
+ 0,9939
1075
+ on
1076
+ 8
1077
+ 1100
1078
+ 0,995
1079
+ 0,9942
1080
+ off
1081
+ 1161
1082
+ 0,995
1083
+ 0,9953
1084
+ jungle chess 2pcs raw
1085
+ endgame complete
1086
+ on
1087
+ 1
1088
+ 30
1089
+ 0,9671
1090
+ 0,9637
1091
+ off
1092
+ 44
1093
+ 0,9667
1094
+ 0,9627
1095
+ on
1096
+ 8
1097
+ 89
1098
+ 0,969
1099
+ 0,9631
1100
+ off
1101
+ 99
1102
+ 0,9713
1103
+ 0,9649
1104
+ numerai28
1105
+ on
1106
+ 1
1107
+ 3
1108
+ 0.508
1109
+ 0,51
1110
+ off
1111
+ 6
1112
+ 0,511
1113
+ 0,5182
1114
+ on
1115
+ 8
1116
+ 20
1117
+ 0,527
1118
+ 0,528
1119
+ off
1120
+ 23
1121
+ 0,5273
1122
+ 0,528
1123
+ phoneme
1124
+ on
1125
+ 1
1126
+ 354
1127
+ 0,9599
1128
+ 0,951
1129
+ off
1130
+ 325
1131
+ 0,9597
1132
+ 0,9515
1133
+ on
1134
+ 8
1135
+ 1954
1136
+ 0,9631
1137
+ 0,955
1138
+ off
1139
+ 1885
1140
+ 0,963
1141
+ 0,9547
1142
+ sylvine
1143
+ on
1144
+ 1
1145
+ 221
1146
+ 0,9852
1147
+ 0,9809
1148
+ off
1149
+ 206
1150
+ 0,9853
1151
+ 0,9806
1152
+ on
1153
+ 8
1154
+ 932
1155
+ 0,9878
1156
+ 0,981
1157
+ off
1158
+ 827
1159
+ 0,9877
1160
+ 0,9829
1161
+ volkert
1162
+ on
1163
+ 1
1164
+ 4
1165
+ 0,932
1166
+ 0,9298
1167
+ off
1168
+ 6
1169
+ 0,9393
1170
+ 0,9344
1171
+ on
1172
+ 8
1173
+ 20
1174
+ 0,9313
1175
+ 0,934
1176
+ off
1177
+ 21
1178
+ 0,9317
1179
+ 0,9273
1180
+ synthetic blobs
1181
+ on
1182
+ 1
1183
+ 31
1184
+ 1
1185
+ 1
1186
+ off
1187
+ 27
1188
+ 1
1189
+ 1
1190
+ on
1191
+ 8
1192
+ 235
1193
+ 1
1194
+ 1
1195
+ off
1196
+ 224
1197
+ 1
1198
+ 1
1199
+ synthetic moons
1200
+ on
1201
+ 1
1202
+ 1124
1203
+ 1
1204
+ 1
1205
+ off
1206
+ 1026
1207
+ 1
1208
+ 1
1209
+ on
1210
+ 8
1211
+ 12356
1212
+ 1
1213
+ 1
1214
+ off
1215
+ 12227
1216
+ 1
1217
+ 1
1218
+ 8
1219
+
1220
+ Table 3: The results of experiments with parallelization of evolution. The first two rows for each dataset (1 and 8 jobs) represent the results obtained with a fit time
1221
+ limit for pipelines, while the ”without limit” row contain the results obtained without limits with 8 jobs.
1222
+ Dataset
1223
+ Number of processes
1224
+ Pipelines
1225
+ count
1226
+ ROC-AUC
1227
+ final
1228
+ ROC-AUC
1229
+ cross-validation
1230
+ adult
1231
+ 1 (with limit)
1232
+ 23
1233
+ 0,9213
1234
+ 0,913
1235
+ 8 (with limit)
1236
+ 170
1237
+ 0,922
1238
+ 0,9137
1239
+ 8 (without limit)
1240
+ 126
1241
+ 0,9217
1242
+ 0,9141
1243
+ amazon employee access
1244
+ 1 (with limit)
1245
+ 78
1246
+ 0,8497
1247
+ 0,8376
1248
+ 8 (with limit)
1249
+ 369
1250
+ 0,849
1251
+ 0,8398
1252
+ 8 (without limit)
1253
+ 329
1254
+ 0,849
1255
+ 0,8387
1256
+ australian
1257
+ 1 (with limit)
1258
+ 838
1259
+ 0,9283
1260
+ 0,9401
1261
+ 8 (with limit)
1262
+ 6199
1263
+ 0,934
1264
+ 0,9442
1265
+ 8 (without limit)
1266
+ 10009
1267
+ 0,9307
1268
+ 0,9444
1269
+ bank-marketing
1270
+ 1 (with limit)
1271
+ 30
1272
+ 0,9313
1273
+ 0,93
1274
+ 8 (with limit)
1275
+ 211
1276
+ 0,932
1277
+ 0,931
1278
+ 8 (without limit)
1279
+ 184
1280
+ 0,9313
1281
+ 0,93
1282
+ blood-transfusion-service-center
1283
+ 1 (with limit)
1284
+ 2064
1285
+ 0,7383
1286
+ 0,759
1287
+ 8 (with limit)
1288
+ 13834
1289
+ 0,749
1290
+ 0,7659
1291
+ 8 (without limit)
1292
+ 8329
1293
+ 0,716
1294
+ 0,7658
1295
+ car
1296
+ 1 (with limit)
1297
+ 728
1298
+ 0,9233
1299
+ 0,9319
1300
+ 8 (with limit)
1301
+ 4608
1302
+ 0,92
1303
+ 0,934
1304
+ 8 (without limit)
1305
+ 4612
1306
+ 0,925
1307
+ 0,9345
1308
+ cnae-9
1309
+ 1 (with limit)
1310
+ 195
1311
+ 0,995
1312
+ 0,9939
1313
+ 8 (with limit)
1314
+ 1161
1315
+ 0,995
1316
+ 0,9953
1317
+ 8 (without limit)
1318
+ 1168
1319
+ 0,995
1320
+ 0,9949
1321
+ jungle chess 2pcs raw endgame complete
1322
+ 1 (with limit)
1323
+ 44
1324
+ 0,9667
1325
+ 0,9627
1326
+ 8 (with limit)
1327
+ 99
1328
+ 0,9713
1329
+ 0,9649
1330
+ 8 (without limit)
1331
+ 144
1332
+ 0,9723
1333
+ 0,9661
1334
+ numerai28
1335
+ 1 (with limit)
1336
+ 6
1337
+ 0,511
1338
+ 0,5182
1339
+ 8 (with limit)
1340
+ 23
1341
+ 0,5273
1342
+ 0,528
1343
+ 8 (without limit)
1344
+ 22
1345
+ 0,5253
1346
+ 0,5266
1347
+ phoneme
1348
+ 1 (with limit)
1349
+ 325
1350
+ 0,9597
1351
+ 0,9515
1352
+ 8 (with limit)
1353
+ 1885
1354
+ 0,963
1355
+ 0,9547
1356
+ 8 (without limit)
1357
+ 1917
1358
+ 0,963
1359
+ 0,9536
1360
+ sylvine
1361
+ 1 (with limit)
1362
+ 206
1363
+ 0,9853
1364
+ 0,9806
1365
+ 8 (with limit)
1366
+ 827
1367
+ 0,9877
1368
+ 0,9829
1369
+ 8 (without limit)
1370
+ 816
1371
+ 0,986
1372
+ 0,9821
1373
+ volkert
1374
+ 1 (with limit)
1375
+ 6
1376
+ 0,9373
1377
+ 0,9329
1378
+ 8 (with limit)
1379
+ 21
1380
+ 0,9393
1381
+ 0,9344
1382
+ 8 (without limit)
1383
+ 21
1384
+ 0,9317
1385
+ 0,9273
1386
+ synthetic blobs
1387
+ 1 (with limit)
1388
+ 8
1389
+ 1
1390
+ 1
1391
+ 8 (with limit)
1392
+ 37
1393
+ 1
1394
+ 1
1395
+ 8 (without limit)
1396
+ 30
1397
+ 1
1398
+ 1
1399
+ synthetic moons
1400
+ 1 (with limit)
1401
+ 1026
1402
+ 1
1403
+ 1
1404
+ 8 (with limit)
1405
+ 12227
1406
+ 1
1407
+ 1
1408
+ 8 (without limit)
1409
+ 12569
1410
+ 1
1411
+ 1
1412
+ 9
1413
+
1414
+ Figure 9: The detailed analysis of dependency of number of pipelines from the
1415
+ number of jobs (dataset blood-transfusion-service-center)
1416
+ Figure 10: The dependency of the best fitness values on the optimization time.
1417
+ The intervals represent the stochasticity of the optimization runs.
1418
+ 6.2. Heterogeneous infrastructure
1419
+ In the next series of experiments, we aim to estimate the
1420
+ efficiency of heterogeneous infrastructure for large datasets.
1421
+ Computation experiments were performed in a supercomputer
1422
+ environment configured based on two DGX-1 clusters. Each
1423
+ cluster contains eight Tesla V100 graphics cards and 128 GB of
1424
+ video memory. The number of graphics cores is 40960.
1425
+ The first experiment compares AutoML performance for the
1426
+ CPU-only and the hybrid infrastructures for various tasks. The
1427
+ aim of the experiment is to estimate the decreasing of fitting
1428
+ time after involvement of GPU-based nodes.
1429
+ To reduce the
1430
+ computational complexity of experiments, we decided not to
1431
+ use the full set of datasets from Table 1. In this experiment, four
1432
+ synthetic binary classification datasets with 10 features and dif-
1433
+ ferent number of rows (10000, 100000, 200000 and and 300000
1434
+ rows).
1435
+ Both single-model (SVC) and multi-model pipelines
1436
+ (consisting of SVC, Logistic Regression, and Random Forest)
1437
+ are considered to estimate the overhead for data flow transfer
1438
+ between models in the pipeline. The results are presented in
1439
+ Table 4: The training time of the pipelines on synthetic data under different
1440
+ conditions (single SVC classifier and composite pipeline with several models
1441
+ are considered). Averaged efficiency estimations are presented for homoge-
1442
+ neous (one server with multi-core CPU) and heterogeneous (CPU and GPU)
1443
+ computing environments.
1444
+ Rows, 103
1445
+ Fitting time, sec
1446
+ Improvement,
1447
+ %
1448
+ Single
1449
+ model
1450
+ Comp.
1451
+ pipeline
1452
+ Single
1453
+ model
1454
+ Comp.
1455
+ pipeline
1456
+ CPU
1457
+ CPU+
1458
+ GPU
1459
+ CPU
1460
+ CPU+
1461
+ GPU
1462
+ 10
1463
+ 0.3
1464
+ 2.2
1465
+ 0.7
1466
+ 2.4
1467
+ -
1468
+ -
1469
+ 100
1470
+ 11.4
1471
+ 1.9
1472
+ 8.3
1473
+ 1.6
1474
+ 91
1475
+ 88
1476
+ 200
1477
+ 39.3
1478
+ 2.8
1479
+ 21
1480
+ 3.2
1481
+ 94
1482
+ 85
1483
+ 300
1484
+ 76.0
1485
+ 4.2
1486
+ 37.8
1487
+ 5.5
1488
+ 95
1489
+ 86
1490
+ Table 4.
1491
+ The results confirmed that the overhead could exceed the
1492
+ performance gain for a small amount of data. However, the
1493
+ proposed hybrid approach to pipeline evaluation is reasonably
1494
+ practical for large datasets.
1495
+ 6.3. Remote infrastructure
1496
+ The next series of experiments uses a homogeneous cluster
1497
+ of 20 nodes under Kubernetes control. Each node has 40 CPU
1498
+ cores and 256 Gb RAM. We have trained populations of 50,
1499
+ 100 and 200 individuals. Each population has been trained four
1500
+ times, then the estimated time values were averaged. The aim
1501
+ of this experiment is to analyze the structure of computing time
1502
+ for remote evaluation and confirm that remote evaluation can
1503
+ be viable for large datasets regardless of existing overheads. To
1504
+ make the results of the experiment more compact, we focused
1505
+ on an analysis of a single synthetic binary classification dataset
1506
+ with 300000 rows and 10 features.
1507
+ Figure 11a presents training time depending on population
1508
+ size without results fetching. The evaluation of each individual
1509
+ has no CPU and system memory limits. Also, we have drawn
1510
+ a linear fit time as reference line. The initial point for this line
1511
+ is the time for the population size of 50. We found that training
1512
+ time increases almost linearly with the increase in population.
1513
+ To explain the near-linear time growth, we can consider the
1514
+ operations that took the most time during the computing and
1515
+ the overheads they have. Since requests overheads are nearly 0
1516
+ seconds, they are not presented in Figure 12.
1517
+ Figure 12 shows that computing time and overheads for
1518
+ each individual are the same and are independent of popula-
1519
+ tion size. It means that the performance bottleneck is not on the
1520
+ cluster side. The fit stages timeline (including results fetching)
1521
+ is presented in Figure 13.
1522
+ This timeline shows that the most time is spent on results
1523
+ fetching. Result fetching consists of zip-file downloading over
1524
+ the network, unpacking and then deserializing the model. Since
1525
+ the results are fetched concurrently, a large number of individu-
1526
+ als lead to a high network and drive load on the local machine.
1527
+ Moreover, even though the overhead for the request to run
1528
+ one individual is less than 1 sec, a large number of requests also
1529
+ 10
1530
+
1531
+ 8 process
1532
+ 0.855
1533
+ 1process
1534
+ 0.850
1535
+ 0.845
1536
+ score
1537
+ ROC-AUC
1538
+ 0.840
1539
+ 0.835
1540
+ 0.830
1541
+ 0.825
1542
+ 0
1543
+ 500
1544
+ 1000
1545
+ 1500
1546
+ 2000
1547
+ 2500
1548
+ 3000
1549
+ Time in secondsDependency ofthe bestfitness from numberof jobs
1550
+ 14000
1551
+ 8
1552
+ minutes
1553
+ 12000
1554
+ 7
1555
+ correctly evaluated pipelines
1556
+ 6
1557
+ actual time for optimization in
1558
+ 10000
1559
+ 8000
1560
+ 4
1561
+ 6000
1562
+ m
1563
+ 4000
1564
+ 2
1565
+ 1
1566
+ 2000
1567
+ 1
1568
+ 2
1569
+ 3
1570
+ 4
1571
+ 5
1572
+ 6
1573
+ 7
1574
+ 8
1575
+ numberof jobs(a) Without limits (fast calculations)
1576
+ (b) CPU limit = 0.2 core (heavy calculations emulation)
1577
+ Figure 11: The dependence of the total fit time on individual numbers in different computational setups. The orange line represents linear acceleration; the blue line
1578
+ represents the observed values of fit time.
1579
+ Figure 12: Overheads and computing time during experiments with remote infrastructure
1580
+ Figure 13: The explanation of remote training timeline with remote infrastruc-
1581
+ ture.
1582
+ consume a significant amount of time. The time range between
1583
+ the last request and the last completed individual is less than 20
1584
+ seconds, and it falls within the 75-percentile of computing time.
1585
+ It means that the computing cluster is underutilized because it
1586
+ is not running the individuals in parallel as well as expected
1587
+ because it is waiting for requests from the client.
1588
+ To sum up, it is not reasonable to use remote training if we
1589
+ have lightweight and fast computations. Overheads in the form
1590
+ of requests and results fetching will be significantly larger than
1591
+ the payload. To prove this assumption, we have repeated the
1592
+ same experiment, but we have artificially introduced CPU limit-
1593
+ ing for each individual (up to 0.2 CPU core) to emulate ”heavy”
1594
+ computing. The results for heavy-weight tasks are presented in
1595
+ Figure 11b.
1596
+ We can conclude that remote computing provides a signif-
1597
+ icant speedup for expensive computations. However, the over-
1598
+ head for small datasets should be taken into account.
1599
+ 7. Conclusions and Discussions
1600
+ In the paper, we propose a modular approach that improves
1601
+ the efficiency of evolutionary AutoML in a heterogeneous en-
1602
+ 11
1603
+
1604
+ remote fit time
1605
+ linear fit timeremote fit time
1606
+ linear fit timeδΈ»vironment. The proposed approach differs from existing solu-
1607
+ tions since it can be configured for automated machine learning
1608
+ in various computational environments. It makes it possible
1609
+ to parallelize and distribute the computational tasks across hy-
1610
+ brid and/or remote computational systems. Also, caching al-
1611
+ gorithms are implemented to increase the optimization perfor-
1612
+ mance for composite pipelines.
1613
+ The AutoML-based experimental setup consisted of (1) the
1614
+ estimation of parallel speedup for a different number of pro-
1615
+ cesses; (2) an analysis of the efficiency of the cache; (3) an anal-
1616
+ ysis of the GPU computations efficiency; (4) optimisation runs
1617
+ with remote infrastructure involved. The experiments confirm
1618
+ the proposed approach’s efficiency. It allows achieving signifi-
1619
+ cant improvements in the number of evaluated individuals and
1620
+ in the fitness function.
1621
+ There are several ways to improve remote computing per-
1622
+ formance aimed at different bottlenecks that can be used sepa-
1623
+ rately or combined:
1624
+ 1. Efficient cluster resources utilization requires a custom
1625
+ scheduler and additional plugins for batch workload such
1626
+ as Volcano5.
1627
+ 2. Refuse to request to run each individual. Better to use
1628
+ one request to run a batch of individuals. This way, the
1629
+ number of requests will be reduced to one independent
1630
+ request for the all population. Also, we can apply specu-
1631
+ lative computing mode when the number of rest individ-
1632
+ uals is small;
1633
+ 3. Provide a cluster file system mount on the local machine.
1634
+ This will reduce the number of requests for downloading
1635
+ results, and the client will also skip zip file unpacking.
1636
+ Instead, the client will read the results from the mounted
1637
+ file system. If it is impossible, then we have to implement
1638
+ not only batch run requests but also batch download re-
1639
+ quests;
1640
+ 4. Perform model validation using remote infrastructure
1641
+ too.
1642
+ This way, we also have to provide a validation
1643
+ dataset to the remote system. Remote computing will
1644
+ validate individuals and save the score. It will make it
1645
+ unnecessary to fetch the trained models, and the calcu-
1646
+ lated score will be enough for further decisions;
1647
+ 5. Heterogeneous environment.
1648
+ We can use a heteroge-
1649
+ neous environment not only on the cluster layer but on
1650
+ the client-server layer. For example, the client can per-
1651
+ form lightweight calculations locally, heavy calculations
1652
+ at the same time will be sent to the cluster, and the heavi-
1653
+ est calculations may be sent to the most powerful cluster
1654
+ nodes (e.g. special GPU nodes).
1655
+ Another direction of improvement is the support of large
1656
+ dataset processing. It can be based on the implementation of
1657
+ the distributed evaluation of different folds of the data set. The
1658
+ caching system can also be implemented in a distributed way.
1659
+ 5https://volcano.sh
1660
+ 8. Code and Data Availability
1661
+ The software implementation of all described methods and
1662
+ algorithms is available in the open repository https://gith
1663
+ ub.com/ITMO-NSS-team/fedot-performance-improve
1664
+ ment-benchmark.
1665
+ References
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+
DdE4T4oBgHgl3EQfew2P/content/tmp_files/load_file.txt ADDED
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1
+ UNIFYSPEECH: A UNIFIED FRAMEWORK FOR ZERO-
2
+ SHOT TEXT-TO-SPEECH AND VOICE CONVERSION
3
+ Haogeng Liu1, Tao Wang1, Ruibo Fu2, Jiangyan Yi2, Zhengqi Wen2
4
+ Chinese Academy of Science
5
+ Beijing, China
6
+ Jianhua Tao1
7
+ Department of Automation, Tsinghua University
8
+ School of Artificial Intelligence, University of Chinese Academy of Sciences, China
9
+ Beijing, China
10
+ ABSTRACT
11
+ Text-to-speech (TTS) and voice conversion (VC) are two different tasks both aim-
12
+ ing at generating high quality speaking voice according to different input modality.
13
+ Due to their similarity, this paper proposes UnifySpeech, which brings TTS and
14
+ VC into a unified framework for the first time. The model is based on the assump-
15
+ tion that speech can be decoupled into three independent components: content
16
+ information, speaker information, prosody information. Both TTS and VC can be
17
+ regarded as mining these three parts of information from the input and completing
18
+ the reconstruction of speech. For TTS, the speech content information is derived
19
+ from the text, while in VC it’s derived from the source speech, so all the remaining
20
+ units are shared except for the speech content extraction module in the two tasks.
21
+ We applied vector quantization and domain constrain to bridge the gap between
22
+ the content domains of TTS and VC. Objective and subjective evaluation shows
23
+ that by combining the two task, TTS obtains better speaker modeling ability while
24
+ VC gets hold of impressive speech content decoupling capability.
25
+ Index Terms: decoupling, zero-shot learning, text-to-speech, voice conversion,
26
+ vector quantization
27
+ 1
28
+ INTRODUCTION
29
+ Cloning the voice of the target speaker is an attractive technology, which can be applied to various
30
+ scenes (Sisman et al., 2020), such as entertainment creation, personalized mobile assistants, security
31
+ field, etc. The most ideal voice cloning operation is to give only one relatively short speech of the
32
+ unseen target speaker as a reference and then any speech of the target speaker can be synthesized,
33
+ which is called zero-shot voice clone. In the speech research community, text-to-speech (TTS) and
34
+ voice conversion (VC) are two mainstream ways to realize voice clone (Sisman et al., 2020). There-
35
+ fore, a variety of techniques for zero-shot TTS and VC have been proposed recently (Gorodetskii &
36
+ Ozhiganov, 2022; Tang et al., 2021).
37
+ However, although TTS and VC techniques are two important ways of voice clone with same output
38
+ form, the research of TTS and VC is more or less independent. There isn’t much interaction between
39
+ them. But they are both speech synthesis task. In terms of speech generation, we categorize the
40
+ information of the target speaker’s speech into three kinds of information: (1) speech content, the
41
+ characters of phonemes or phonetic posteriorgram (PPG) in voice conversion, represents the content
42
+ of the speech. (2) speaker information, which represents the characteristics of speakers, is related
43
+ to the speaker’s articulation organ. (3) prosody information, which covers the intonation, stress,
44
+ and rhythm of speech. According to FastSpeech2 (Ren et al., 2020), pitch, energy and duration
45
+ information can reflected them certainly. As TTS extracts speech content directly from phonemes,
46
+ it is easier to obtain content information irrelevant to speaker than VC. As VC encounters more
47
+ speakers, it’s possible to obtain more robust speaker modeling ability. So, by integrating TTS and
48
+ 1
49
+ arXiv:2301.03801v1 [cs.SD] 10 Jan 2023
50
+
51
+ VC into a unified framework and combining their training data, it can help the model learn these
52
+ three kinds of information better.
53
+ Unfortunately, investigating TTS and VC in the same framework is challenging, as the speech
54
+ content are extracted from different modality. Specifically, the speech content in TTS is obtained
55
+ through phoneme information while the phonemes and speech in TTS are unequal sequences, need-
56
+ ing attention mechanism to align them. However, the attention mechanism is often affected by the
57
+ speaker’s information, so it is impossible to learn the speech content representation completely ir-
58
+ relevant to the speaker. While in VC, the source speech and target speech are aligned in speech
59
+ content, so the speech content can be extracted directly from the source speech, which is different
60
+ from the TTS. In contrast to speech content information, speaker information and prosody infor-
61
+ mation can be modeled using the same network in TTS and VC. Therefore, the difficulty here is
62
+ to keep speech content for TTS and VC consistent. With the development of speech synthesis, the
63
+ recently proposed Adaspeech2 (Yan et al., 2021) can combine text information with speech infor-
64
+ mation, which can effectively decouple the speech content from the input. The unified framework
65
+ becomes possible.
66
+ Overall, the main contributions of this paper are:
67
+ β€’ We propose UnifySpeech, a unified framework for TTS and VC. VC enables unlabeled data
68
+ to join training process, making TTS encounters more speakers. TTS enhance the ability
69
+ for voice content decoupling in VC. Thus, both pipeline benefits from the other one.
70
+ β€’ We apply vector quantization and domain constrain to bridge the gap between the content
71
+ domains of TTS and VC. Ablation experiment shows this method’s effectiveness.
72
+ β€’ We perform extensive experiments: zero-shot TTS, zero-shot VC. Results proves that
73
+ jointly trained TTS outperformes StyleSpeech (Min et al., 2021)and jointly trained VC
74
+ gains better speech decoupling ability.
75
+ Demos for this paper are available at https://liuhaogeng.github.io/UnifySpeech/.
76
+ 2
77
+ BACKGROUND
78
+ In this section, we will briefly review the background of this work, including neural TTS and VC
79
+ models, and the zero-shot learning for TTS and VC tasks.
80
+ 2.1
81
+ TEXT-TO-SPEECH TASK
82
+ TTS task is to model the mapping between text and speech, which is a modeling problem between
83
+ two unequal length sequences. According to the alignment mechanism (Battenberg et al., 2020)
84
+ between text and speech in the model, the end-to-end TTS model can be divided into two categories:
85
+ 1) Using a neural network to learn the alignment information between text and speech, such as
86
+ local sensitive attention in Tacotron (Wang et al., 2017b). Various improvements to the attention
87
+ mechanism have been proposed. In addition, inspired by CTC (Kim et al., 2017) in ASR, glow-
88
+ TTS (Kim et al., 2020), VITS (Kim et al., 2021) can automatically learn the alignment information.
89
+ 2) By introducing the duration information (Ren et al., 2019) of phonemes as prior knowledge, the
90
+ text is upsampled to achieve alignment (McAuliffe et al., 2017). Since the upsampled information
91
+ based on text is independent of the speaker, the speech content can be well separated from the
92
+ speech. Therefore, we introduce the duration information to build the TTS model in this paper.
93
+ 2.2
94
+ VOICE CONVERSION TASK
95
+ Voice conversion can be seen as two steps (Sisman et al., 2020).
96
+ Firstly, extract the speaker-
97
+ independent speech content information from the source speech, and then embed the target speaker
98
+ information to the speech content to reconstruct the speech of the target speaker. According to the
99
+ way of extracting speech content, the VC model can be divided into two categories: (1) text-based
100
+ approach. (2) Information bottleneck approach. The first approach requires an additional pre-trained
101
+ ASR model. Since the ASR is trained in a supervised manner, it demands a lot of paired text and
102
+ speech. Additionally, pipeline modeling is easy to accumulate errors and affects the performance of
103
+ 2
104
+
105
+ the system. Therefore, a lot of research work is focused on the latter. By adding some restrictions
106
+ to the information bottleneck, different kinds of information can be decoupled. However, if the in-
107
+ formation bottleneck can not be decoupled well, the performance of the model will be significantly
108
+ reduced.
109
+ 2.3
110
+ ZERO SHOT LEARNING FOR TTS AND VC
111
+ The research of zero-shot learning based on TTS and VC focused on how to extract effective speaker
112
+ information and then embed it into TTS or VC model for joint training or segmented training.
113
+ Typical speaker features include i-vector (Wang et al., 2017a), d-vector (Variani et al., 2014), x-
114
+ vector (Snyder et al., 2018) and so on. In addition, the modules that extract speaker-style informa-
115
+ tion through the specially designed network structure, such as GST (Wang et al., 2018), VAE (Van
116
+ Den Oord et al., 2017), can also achieve good results.
117
+ 3
118
+ UNIFYSPEECH
119
+ In this section, we introduce the details of UnifySpeech for zero-shot TTS and VC tasks. We first
120
+ give the key idea of UnifySpeech: speech factorization, and then introduce the formulation of Uni-
121
+ fySpeech. Finally, we will describe the model structure of UnifySpeech.
122
+ Figure 1: Structure of UnifySpeech.
123
+ 3.1
124
+ SPEECH REPRESENTATION DISENTANGLEMENT
125
+ The core of the controllable and migratable speech generation task is to decouple the components of
126
+ the generated speech first, and then control and transfer each component. Although some end-to-end
127
+ models can directly model the relationship between text and speech (TTS) or speech-to-speech task
128
+ (VC), due to the mutual coupling of various components of the end-to-end model, it brings great
129
+ difficulties to the transfer learning of the model. Therefore, we first decouple the speech generation
130
+ task into three independent components and then input them into the decoder to synthesize speech.
131
+ This idea is also the main architecture of UnifySpeech. The three components and their sources will
132
+ be described in detail below.
133
+ Speech Content: To generate intelligible speech signals, it is important to model accurate speech
134
+ content information. Speech content is linguistic information, which is irrelevant to the speaker.
135
+ Due to the different types of input signals of TTS and VC, the ways of extracting speech content are
136
+ different. For TTS, the source of speech content is text. Firstly, to learn the context information of
137
+ the text, a text encoder is used to encode the text to obtain the context representation. The context
138
+ representation is up-sampled according to the phoneme’s duration information to obtain the speech
139
+ content information. For VC, since the source speech is aligned with the target speech, we directly
140
+ use a content encoder to extract speech content information from the source speech.
141
+ Speaker: The speaker information includes the speaker’s characteristics, such as the timbre, volume,
142
+ etc. We can extract the speaker information from the given speech of the target speaker. This is
143
+ 3
144
+
145
+ Decoder
146
+ Decoder
147
+ Pitch
148
+ Pitch
149
+ Pitch predictor
150
+ Pitch predictor
151
+ Prosody information Speaker embedding
152
+ Speech content
153
+ Speech content
154
+ Speaker embedding Prosody information
155
+ VQ
156
+ VQ
157
+ Prosody encoder
158
+ Speaker encoder
159
+ Speaker encoder
160
+ Prosody encoder
161
+ Text encoder
162
+ Content encoder
163
+ Pitch
164
+ Reference mel
165
+ Phonemes
166
+ Speech
167
+ Reference mel
168
+ Pitchcommon for TTS and VC tasks, so the speaker extraction network can be shared in the two tasks.
169
+ Since the speaker extraction network can directly extract information from speech without text, so a
170
+ large number of data without text annotation in the VC task can be used for training, which can help
171
+ to improve the transfer learning ability of the model.
172
+ Prosody: The prosody information represents how the speaker says the content information. It
173
+ is independent of the speaker information and related to the way of expression. Since the pitch
174
+ information can reflect the rhythm of speech, therefore, the pitch information is used to extract
175
+ prosody information. The prosody information, like speaker information, can be shared by both TTS
176
+ and VC. In addition, in the training process, we can obtain pitch information from the ground truth
177
+ speech, but there is no ground truth in the inference stage. Therefore, a pitch prediction module (Ren
178
+ et al., 2020) is proposed in the training stage, which takes the speech content information and speaker
179
+ information as the input to predict the pitch information.
180
+ 3.2
181
+ SPEECH CONTENT WITH VECTOR QUANTIZATION
182
+ Since there are different ways to obtain the speech content in TTS and VC, it is very easy to de-
183
+ viate between the two domains. If this deviation occurs, it will cause devastating damage to some
184
+ downstream shared modules (such as pitch predictor and decoder). Therefore, to ensure that the
185
+ consistency of the two speech content domains as close as possible, we reconstruct them from two
186
+ aspects:
187
+ β€’ First, we use the shared codebook to quantify the continuous feature space.
188
+ β€’ Second, we use the labeled data to narrow two discrete feature spaces.
189
+ The detailed process is described below. Suppose that the vector obtained by the text encoder in
190
+ TTS pipeline is Cp = (C1
191
+ p, C2
192
+ p, Β· Β· Β· , CT
193
+ p ) with length T, the vector obtained by the content encoder
194
+ in VC pipeline is Cs = (C1
195
+ s, C2
196
+ s, Β· Β· Β· , CT
197
+ β€²
198
+ s ) with length T
199
+ β€². It should be noted that we add a length
200
+ regulator module in the text encoder to solve the problem of length mismatch between the text and
201
+ speech sequence, which is introduced in FastSpeech (Ren et al., 2019). Therefore, if text and speech
202
+ are paired, T
203
+ β€² = T. The vector Cp and Cs is a sequence of continuous vector in Eq. Due to the
204
+ large representation range of continuous features, Cp and Cs are difficult to match. We borrow
205
+ the discretization method for latent variables from Vector Quantized Variational AutoEncoder (Van
206
+ Den Oord et al., 2017). Specifically, for each time step t, the continuous latent representations Ct
207
+ p in
208
+ Cp can be mapped into C
209
+ t
210
+ p by finding the nearest pre-defined discretized embedding in the dictionary
211
+ as:
212
+ C
213
+ t
214
+ p = ek,
215
+ k = argminj
216
+ οΏ½οΏ½Ct
217
+ p βˆ’ ej
218
+ οΏ½οΏ½
219
+ 2
220
+ (1)
221
+ where ej is the j-th embedding in the codebook dictionary, and j ∈ 1, 2, · · · , V . Since selecting
222
+ the entry with the minimum distance will cause the operation to be non-differentiable, the straight-
223
+ through gradient estimator can be used to approximate the gradient, which can be expressed as:
224
+ Β―ht = ht + ev βˆ’ sg (ht) ,
225
+ v = arg min
226
+ k
227
+ βˆ₯ht βˆ’ ekβˆ₯2
228
+ (2)
229
+ where sg(Β·) is the stop-gradient (Van Den Oord et al., 2017) operation that treats its input as constant
230
+ during back-propagation.
231
+ After vector quantization, the quantized sequence Cp
232
+ =
233
+ (C
234
+ 1
235
+ p, C
236
+ 2
237
+ p, Β· Β· Β· , C
238
+ T
239
+ p ) and Cs
240
+ =
241
+ (C
242
+ 1
243
+ s, C
244
+ 2
245
+ s, Β· Β· Β· , C
246
+ T
247
+ s ) can be obtained. It should be noted that when Cp and Cs are quantized into
248
+ Cp and Cs, they share the same codebook e. The advantage of this is that since the speech content
249
+ features Cp in the TTS pipeline are independent of the speaker, sharing the same codebook can
250
+ help learn the speech content features Cs independent of the speaker in the VC pipeline, which is
251
+ essential for the VC task.
252
+ Although both pipeline use the same codebook for coding, there is no guarantee that there is no
253
+ deviation between the two fields after coding. Therefore, to further eliminate the deviation between
254
+ the two domains, we use the labeled speech in the TTS pipeline to supervise the training of the two
255
+ domains. Specifically, for paired text and speech data, we constrain the feature distance between the
256
+ quantized sequence Cp and Cs:
257
+ 4
258
+
259
+ Lpair =
260
+ οΏ½οΏ½Cp βˆ’ Cs
261
+ οΏ½οΏ½2
262
+ 2
263
+ (3)
264
+ In this way, we can efficiently minimize the domain discrepancy by using limited labeled data.
265
+ Figure 2: Structure of vector quantized operation.
266
+ 3.3
267
+ UNIFYSPEECH PIPELINE
268
+ An overview of our proposed UnifySpeech architecture is illustrated in Fig. 2. It consists of a
269
+ sequence-to-sequence TTS, and a sequence-to-sequence VC. The key idea is to share most of the
270
+ module parameters (speaker encoder, prosody encoder, decoder and pitch predictor) and map the
271
+ speech content in TTS and VC to the same space. As mentioned above, the UnifySpeech allows
272
+ us to train the model on the concatenation of both the labeled and unlabeled data. For supervised
273
+ training with labeled data, both models can be trained independently by minimizing the loss between
274
+ their predicted speech and the ground truth. For unsupervised training with unlabeled data, the VC
275
+ pipeline can be trained, and the parameters are shared with TTS.
276
+ To further clarify the training process, we unrolled the framework as follows.
277
+ 3.3.1
278
+ TTS PIPELINE
279
+ Denote the text and speech sequence pair (x, y, F0) ∈ D, where D is the paired text and speech
280
+ corpus. Each element in the text sequence x represents a phoneme or character, while each element
281
+ in the speech sequence y represents a frame of speech. F0 is the pitch information of y. The
282
+ representation obtained after three encoders are speech content C, speaker S and prosody P.
283
+ Then, the three parts are added and input into a decoder to obtain the predicted speech yβ€². In addition,
284
+ to obtain the pitch information in the interference stage, we use the content information and speaker
285
+ information to predict the F0. These processes can be expressed as:
286
+ yβ€² = decoder(C + S + P)
287
+ (4)
288
+ F0β€² = pitch predictor(C + S)
289
+ (5)
290
+ Therefore, the reconstruction loss in TTS process includes two parts:
291
+ LV C
292
+ rec = MSE(y, yβ€²) + MSE(F0, F0β€²)
293
+ (6)
294
+ 5
295
+
296
+ C
297
+ Look up min
298
+ Look up min
299
+ distance
300
+ distance
301
+ vector
302
+ vector
303
+ 1
304
+ V
305
+ L2
306
+ L2
307
+ distance
308
+ ev
309
+ distance
310
+ Ct
311
+ Cs
312
+ Text encoder
313
+ Content encoderIn addition, we use the content encoder in the VC pipeline to extract the content representation Cs
314
+ for the training speech in TTS, and close the distance between the two domains by calculating the
315
+ distance loss of Cs and Cp, which is explained in Sec. 3.2.
316
+ Lpair =
317
+ οΏ½οΏ½Cp βˆ’ Cs
318
+ οΏ½οΏ½2
319
+ 2
320
+ (7)
321
+ The loss of TTS pipeline can be expressed as:
322
+ LT T S = LT T S
323
+ rec
324
+ + Lpair
325
+ (8)
326
+ where MSE denotes the mean squared errors.
327
+ 3.3.2
328
+ VC PIPELINE
329
+ Denote all the unlabeled or labeled speech (y, F0) ∈ Y . We first extract three information from
330
+ (y, F0), which are speech content Cs, speaker Ss and prosody Ps.
331
+ Then, similar to the TTS pipeline, the shared decoder and pitch predictor module is used to predict
332
+ the speech signal yβ€² and F0β€², which is similar to the Eq.1 and Eq.2.
333
+ The loss of VC pipeline only includes reconstruction loss, which can be expressed as:
334
+ LV C = MSE(Y, Y β€²) + MSE(F0, F0β€²)
335
+ (9)
336
+ where MSE denotes the mean squared errors.
337
+ 3.3.3
338
+ TRAINING PROCESS
339
+ With such a unified framework, TTS and VC can learn from each other through joint training. The
340
+ details of the algorithm can be found below.
341
+ Procedure 1 UnifySpeech training algorithm
342
+ 1: Input: Paired speech and text dataset (x, y), speech only dataset y
343
+ β€²
344
+ 2: repeat
345
+ 3:
346
+ # A. TTS pipeline with speech-text data pairs
347
+ 1.
348
+ Sample paired speech and text (x, y)
349
+ 2.
350
+ Extract speech content information from x for domain loss
351
+ 3.
352
+ Generate the predict speech y, pitch F0 and speech content from text
353
+ 4.
354
+ Calculate the loss for TTS LT T S
355
+ rec
356
+ 4:
357
+ # B. VC pipeline with speech-only data
358
+ 1.
359
+ Sample paired speech and text (x, y)
360
+ 2.
361
+ Extract speech content information from y for domain loss
362
+ 3.
363
+ Calculate the domain loss for the two pipeline Lpair
364
+ 4.
365
+ Sample speech y
366
+ β€² in speech only dataset
367
+ 5.
368
+ Generate the predict speech y, pitch F0 and speech content from speech y
369
+ β€²
370
+ 6.
371
+ Calculate the loss for VC LV C
372
+ 5:
373
+ # C. Loss combination:
374
+ 1.
375
+ Combine all loss (LT T S
376
+ rec , Lpair, LV C) into a single loss variable
377
+ 2.
378
+ Calculate TTS and VC parameters gradient
379
+ 3.
380
+ Update TTS and VC parameters with gradient descent optimization
381
+ 6: until convergence
382
+ 4
383
+ EXPERIMENTS AND RESULTS ANALYSIS
384
+ In this section, we conduct experiments to evaluate our proposed methods. The experiments are
385
+ carried out from two aspects: zero-shot TTS, zero-shot VC.
386
+ 6
387
+
388
+ 4.1
389
+ DATASETS
390
+ Two datasets are used to simulate labeled data and unlabeled data, respectively. VCTK dataset, an
391
+ English language dataset containing 44 hours of speech and 109 speakers is used as labeled data.
392
+ Each speaker has approximately 400 sentences. LibriTTS (Zen et al., 2019) are used as unlabeled
393
+ data, which consists of 585 hours of speech data from 2484 speakers. We only use the speech data in
394
+ LibriTTS and discard the text for unsupervised training. In this way, it can simulate the scene where
395
+ a large number of speech that we can obtain are unlabeled. We use a 16-bit, 22050 Hz sampling rate
396
+ for all experiments. The 80-dim Mel spectrogram is extracted with Hann windowing, frame shift
397
+ of 12.5-ms, frame length of 50-ms, and 1024-point Fourier transform. In this experiment, we use
398
+ hifigan (Kong et al., 2020) as vocoder.
399
+ 4.2
400
+ MODEL DETAILS
401
+ Figure 3: Structure of each module in UnifySpeech. FFT (Ren et al., 2019) means feed-forward
402
+ Transformer.
403
+ The detail of each module in our proposed method is shown in the Fig. 3. Specifically, to make
404
+ the sequence of speech content extracted from the text encoder and content encoder equal, a length
405
+ regulator is added to the text encoder, which is inspired by the FastSpeech (Ren et al., 2019). The
406
+ structure of the duration predictor is the same as that in FastSpeech. The structure of the decoder
407
+ and content encoder is similar, but the dimensions of input and output are opposite. For the prosody
408
+ encoder, we first quantize F0 of each frame to 32 possible values and encode them into a learnable
409
+ embedding vector according to the value. And we change the output of the pitch predictor into a
410
+ distribution. By this way, pitch prediction becomes a classification task, reducing the difficulty of
411
+ frame-level pitch prediction. We use the speaker encoder in StyleSpeech and fuse features with Style
412
+ Adaptive Layer Norm(SALN) (Min et al., 2021) method to make the comparision fairly.
413
+ The number of feed-forward Transformer (Vaswani et al., 2017) (FFT) blocks in the text encoder is
414
+ 4 and it is 6 in the decoder module. In each FFT block, the dimension of hidden states is 256. The
415
+ kernel sizes of all the 1D-convolution are set to 3. The dropout rate is set to 0.5. The dimension of the
416
+ last linear layer in the decoder is 256. The dimension of last linear layer in encoders (text encoder,
417
+ pitch encoder, content encoder) is 256. An Adam optimizer (Kingma & Ba, 2014) is used to update
418
+ the parameters. The initial learning rate is 0.001 and the learning rate decreased exponentially.
419
+ 7
420
+
421
+ Linear layer
422
+ Linear layer
423
+ Duration
424
+ Length regulator
425
+ predictor
426
+ m block FFT
427
+ 不
428
+ 不
429
+ LN + Dropout
430
+ n block FFT
431
+ 不
432
+ Speaker
433
+ Speaker
434
+ Embedding
435
+ ConvlD + ReLU
436
+ Phoneme
437
+ Embedding
438
+ Embedding
439
+ Linear layer
440
+ Linear layer
441
+ Pooling
442
+ δΈͺ
443
+ LN + Dropout
444
+ LN + Dropout
445
+ FFT
446
+ ConvlD + ReLU
447
+ ConvlD + ReLU
448
+ LN + Dropout
449
+ LN + Dropout
450
+ 不
451
+ Pitch Embedding
452
+ ConvlD + ReLU
453
+ ConvlD + ReLU
454
+ Randomly select a clip
455
+ LSTM
456
+ Quantify to [0, 32]
457
+ from target speaker4.3
458
+ ZERO-SHOT TTS
459
+ We first carried out zero-shot TTS task. We choose four speakers from VCTK that are not used
460
+ during training process as target speakers. For each speaker, we randomly select about 20 sentences
461
+ to be our target. Then we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek,
462
+ 1993) (Mel-cepstrum distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR
463
+ (correlation factor of F0) between synthesized speech and ground truth speech as the objective met-
464
+ rics.
465
+ Table 1: Objective evaluation results for zero-shot TTS.
466
+ Model
467
+ F0 RMSE (Hz)
468
+ MCD (db)
469
+ V/UV
470
+ F0 CORR
471
+ UnifySpeech
472
+ 17.84
473
+ 2.51
474
+ 16.9%
475
+ 0.93
476
+ StyleSpeech
477
+ 19.02
478
+ 2.63
479
+ 18.06%
480
+ 0.92
481
+ Subjective evaluation was also conducted to compare the speech’s quality and similarity. We choose
482
+ mean opinion scores(MOS) for naturalness and similarity mean opinion scores(SMOS) for similar-
483
+ ity. Both metrics are rated in 1-to-5 scale and reported with the 95% confidence intervals (CI).
484
+ Table 2: Mean opinion score (MOS) of the models. With VC means the model is jointly trained.
485
+ Model
486
+ MOS
487
+ SMOS
488
+ GT
489
+ 4.32 Β± 0.15
490
+ βˆ’
491
+ GT mel + Vocoder
492
+ 4.09 Β± 0.15
493
+ βˆ’
494
+ StyleSpeech
495
+ 3.52 Β± 0.13
496
+ 3.82 Β± 0.13
497
+ UnifySpeech-TTS (with VC)
498
+ 3.76 Β± 0.12
499
+ 3.95 Β± 0.13
500
+ To better show our method’s effectiveness, we demonstrate the t-SNE projection of the speaker
501
+ embedding vectors from speakers in both VCTK and LibriTTS. (Van der Maaten & Hinton, 2008)
502
+ Fig. 4 shows the speaker visualization. For the seen speakers (x) and unseen speakers (o), the
503
+ corresponding speaker embedding form a cluster and distinct from others. The boundary between
504
+ different speakers is clear. This shows that the speaker encoder performs well.
505
+ Figure 4: Speaker visualization of generated speeches,where the circle and triangle indicate unseen
506
+ speaker,while x and square indicate seen speaker.
507
+ 4.4
508
+ ZERO-SHOT VC
509
+ We carried out zero-shot VC task, using unseen speakers voice to be the reference speech. As we
510
+ are lack of parallel corpus, we only conduct subjective evaluation. But VC and TTS shares the same
511
+ 8
512
+
513
+ 30
514
+ p225
515
+ p226
516
+ p227
517
+ 20
518
+ p228
519
+ p279
520
+ X
521
+ p274
522
+ 10
523
+ p307
524
+ p311
525
+ 8012
526
+ 0
527
+ οΏ₯5181
528
+ 3885
529
+ 2961
530
+ -10
531
+ 1089
532
+ 7021
533
+ 20
534
+ KX
535
+ -20
536
+ 15
537
+ -10
538
+ -5
539
+ 0
540
+ 5
541
+ 10
542
+ 15speaker encoder, Fig. 4 in zero-shot TTS can also be a reference.
543
+ Just as in zero-shot TTS task, we choose mean opinion scores(MOS) for naturalness and similarity
544
+ mean opinion scores(SMOS) for similarity. Both metrics are rated in 1-to-5 scale and reported with
545
+ the 95% confidence intervals (CI).
546
+ Table 3: Mean opinion score (MOS) of the VC models. With TTS means the model is jointly
547
+ trained.
548
+ Model
549
+ MOS
550
+ SMOS
551
+ GT
552
+ 4.32 Β± 0.15
553
+ βˆ’
554
+ GT mel + Vocoder
555
+ 4.09 Β± 0.15
556
+ βˆ’
557
+ UnifySpeech-VC(without TTS)
558
+ 3.63 Β± 0.13
559
+ 1.31 Β± 0.06
560
+ UnifySpeech-VC(with TTS)
561
+ 3.58 Β± 0.12
562
+ 3.31 Β± 0.13
563
+ It can be found that when VC pipeline is trained alone, its performance is poor. In other words, it
564
+ doesn’t have the ability to discriminate. But jointly training with TTS improves its speech decou-
565
+ pling ability, indirectly improving the speaker modeling ability, which we will analysis later.
566
+ 4.5
567
+ ABLATION EXPERIMENT
568
+ To figure out whether jointly training is effective, we separately train the TTS and VC parts, carrying
569
+ out objective evaluation on them. We also remove the VQ parts and test the model. For TTS,
570
+ we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek, 1993) (Mel-cepstrum
571
+ distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR (correlation factor of F0)
572
+ between synthesized speech and ground truth speech as the objective metrics.
573
+ Table 4: Objective evaluation results for zero-shot TTS.
574
+ Model
575
+ F0 RMSE (Hz)
576
+ MCD (db)
577
+ V/UV
578
+ F0 CORR
579
+ UnifySpeech-TTS(without VC)
580
+ 19.31
581
+ 2.55
582
+ 16.9%
583
+ 0.91
584
+ UnifySpeech-TTS-novq(with VC)
585
+ 19.41
586
+ 2.58
587
+ 17.2%
588
+ 0.92
589
+ UnifySpeech-TTS(with VC)
590
+ 17.84
591
+ 2.51
592
+ 16.9%
593
+ 0.93
594
+ For VC, we separately calculate the Average Cosine Similarity (ACS) (Lei et al., 2022) for the
595
+ embedding from same (S-ACS) and different speakers (D-ACS). And then their ratio is used as an
596
+ evaluation metric.
597
+ Table 5: Ratio of ACS from same and different speaker. Unseen or seen means whether the speaker
598
+ is from test set. With or without TTS means whether the VC pipeline is trained with TTS pipeline.
599
+ Model
600
+ Sβˆ’ACS
601
+ Dβˆ’ACS (unseen)
602
+ Sβˆ’ACS
603
+ Dβˆ’ACS (seen)
604
+ UnifySpeech-VC (with TTS)
605
+ 2.8
606
+ 6.0
607
+ UnifySpeech-VC (without TTS)
608
+ 1.0
609
+ 1.0
610
+ To validate whether VQ can bridge the gap between the speech content parts in TTS and VC, we
611
+ caculate the L2 distance between the phoneme representation from TTS and VC. As in VC there
612
+ are many frames represent same phoneme, so we choose their clustering center as the correspond-
613
+ ing phoneme representation. We randomly select 4 sentences from validation set to carry out the
614
+ experiment.
615
+ Table 6: L2 distance between the phoneme representation in TTS and VC. S1 means sentence1.
616
+ Model
617
+ S1
618
+ S2
619
+ S3
620
+ S4
621
+ average
622
+ UnifySpeech-VC(without VQ)
623
+ 0.464
624
+ 0.476
625
+ 0.474
626
+ 0.553
627
+ 0.492
628
+ UnifySpeech-VC(with VQ)
629
+ 0.152
630
+ 0.205
631
+ 0.208
632
+ 0.205
633
+ 0.193
634
+ 9
635
+
636
+ 4.6
637
+ RESULT ANALYSIS
638
+ Above results show that jointly training actually improves TTS’s speaker modeling ability and VC’s
639
+ speech decoupling ability. And VQ enhances the consistency of representation of the same content
640
+ from phonemes and speeches, ensuring the model’s correctly working.
641
+ For TTS, sharing modules with VC enables unlabeled data to participate in its training process.
642
+ Along with the richer speaker style pattern, the speaker style modeling capability has been enhanced.
643
+ For VC, when trained alone, it doesn’t have the speaker modeling ability. As the self-supervised
644
+ training process aims at reducing the reconstruction loss of source audio, the reference audio of the
645
+ target speaker isn’t crucial in the process. Only the content encoder and the decoder are enough for
646
+ the process, that’s possibly why the speaker embeddings are all similar, though they are from differ-
647
+ ent speakers. When jointly trained, the text loss plays a role of regularization factor, resulting that the
648
+ content encoder just extracting the content information from the source speech (speaker information
649
+ is reserved and others are discarded). This makes the reference speech with rich speaker information
650
+ become indispensable for the reconstruction process. Thus the model’s speaker modeling ability has
651
+ been improved.
652
+ 5
653
+ CONCLUSIONS
654
+ In this paper, we propose UnifySpeech, a unified framework for TTS and VC. Both task benefits
655
+ from the other one. Due to training with large amounts of unlabeled data, their few-shot modeling
656
+ ability makes progress as well as the synthesized speech’s quality. In the future, further improving
657
+ the synthesized speech’s quality and making the generated speech’s style more similar to target
658
+ speaker will be our endeavor.
659
+ REFERENCES
660
+ Eric Battenberg, RJ Skerry-Ryan, Soroosh Mariooryad, Daisy Stanton, David Kao, Matt Shannon,
661
+ and Tom Bagby. Location-relative attention mechanisms for robust long-form speech synthesis. In
662
+ ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing
663
+ (ICASSP), pp. 6194–6198. IEEE, 2020.
664
+ Artem Gorodetskii and Ivan Ozhiganov. Zero-shot long-form voice cloning with dynamic convolu-
665
+ tion attention. arXiv preprint arXiv:2201.10375, 2022.
666
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+ signal processing (ICASSP), pp. 4835–4839. IEEE, 2017.
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+ Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint
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+ arXiv:1412.6980, 2014.
677
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+ Robert Kubichek. Mel-cepstral distance measure for objective speech quality assessment. In Pro-
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+
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1
+ filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf,len=465
2
+ page_content='UNIFYSPEECH: A UNIFIED FRAMEWORK FOR ZERO- SHOT TEXT-TO-SPEECH AND VOICE CONVERSION Haogeng Liu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
3
+ page_content=' Tao Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
4
+ page_content=' Ruibo Fu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
5
+ page_content=' Jiangyan Yi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
6
+ page_content=' Zhengqi Wen2 Chinese Academy of Science Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
7
+ page_content=' China Jianhua Tao1 Department of Automation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
8
+ page_content=' Tsinghua University School of Artificial Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
9
+ page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
10
+ page_content=' China Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
11
+ page_content=' China ABSTRACT Text-to-speech (TTS) and voice conversion (VC) are two different tasks both aim- ing at generating high quality speaking voice according to different input modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
12
+ page_content=' Due to their similarity, this paper proposes UnifySpeech, which brings TTS and VC into a unified framework for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
13
+ page_content=' The model is based on the assump- tion that speech can be decoupled into three independent components: content information, speaker information, prosody information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
14
+ page_content=' Both TTS and VC can be regarded as mining these three parts of information from the input and completing the reconstruction of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
15
+ page_content=' For TTS, the speech content information is derived from the text, while in VC it’s derived from the source speech, so all the remaining units are shared except for the speech content extraction module in the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
16
+ page_content=' We applied vector quantization and domain constrain to bridge the gap between the content domains of TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
17
+ page_content=' Objective and subjective evaluation shows that by combining the two task, TTS obtains better speaker modeling ability while VC gets hold of impressive speech content decoupling capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
18
+ page_content=' Index Terms: decoupling, zero-shot learning, text-to-speech, voice conversion, vector quantization 1 INTRODUCTION Cloning the voice of the target speaker is an attractive technology, which can be applied to various scenes (Sisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
19
+ page_content=', 2020), such as entertainment creation, personalized mobile assistants, security field, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
20
+ page_content=' The most ideal voice cloning operation is to give only one relatively short speech of the unseen target speaker as a reference and then any speech of the target speaker can be synthesized, which is called zero-shot voice clone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
21
+ page_content=' In the speech research community, text-to-speech (TTS) and voice conversion (VC) are two mainstream ways to realize voice clone (Sisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
22
+ page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
23
+ page_content=' There- fore, a variety of techniques for zero-shot TTS and VC have been proposed recently (Gorodetskii & Ozhiganov, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
24
+ page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
25
+ page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
26
+ page_content=' However, although TTS and VC techniques are two important ways of voice clone with same output form, the research of TTS and VC is more or less independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
27
+ page_content=' There isn’t much interaction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
28
+ page_content=' But they are both speech synthesis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
29
+ page_content=' In terms of speech generation, we categorize the information of the target speaker’s speech into three kinds of information: (1) speech content, the characters of phonemes or phonetic posteriorgram (PPG) in voice conversion, represents the content of the speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
30
+ page_content=' (2) speaker information, which represents the characteristics of speakers, is related to the speaker’s articulation organ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
31
+ page_content=' (3) prosody information, which covers the intonation, stress, and rhythm of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
32
+ page_content=' According to FastSpeech2 (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
33
+ page_content=', 2020), pitch, energy and duration information can reflected them certainly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
34
+ page_content=' As TTS extracts speech content directly from phonemes, it is easier to obtain content information irrelevant to speaker than VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
35
+ page_content=' As VC encounters more speakers, it’s possible to obtain more robust speaker modeling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
36
+ page_content=' So, by integrating TTS and 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
37
+ page_content='03801v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
38
+ page_content='SD] 10 Jan 2023 VC into a unified framework and combining their training data, it can help the model learn these three kinds of information better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
39
+ page_content=' Unfortunately, investigating TTS and VC in the same framework is challenging, as the speech content are extracted from different modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
40
+ page_content=' Specifically, the speech content in TTS is obtained through phoneme information while the phonemes and speech in TTS are unequal sequences, need- ing attention mechanism to align them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
41
+ page_content=' However, the attention mechanism is often affected by the speaker’s information, so it is impossible to learn the speech content representation completely ir- relevant to the speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
42
+ page_content=' While in VC, the source speech and target speech are aligned in speech content, so the speech content can be extracted directly from the source speech, which is different from the TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
43
+ page_content=' In contrast to speech content information, speaker information and prosody infor- mation can be modeled using the same network in TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
44
+ page_content=' Therefore, the difficulty here is to keep speech content for TTS and VC consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
45
+ page_content=' With the development of speech synthesis, the recently proposed Adaspeech2 (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
46
+ page_content=', 2021) can combine text information with speech infor- mation, which can effectively decouple the speech content from the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
47
+ page_content=' The unified framework becomes possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
48
+ page_content=' Overall, the main contributions of this paper are: We propose UnifySpeech, a unified framework for TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
49
+ page_content=' VC enables unlabeled data to join training process, making TTS encounters more speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
50
+ page_content=' TTS enhance the ability for voice content decoupling in VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
51
+ page_content=' Thus, both pipeline benefits from the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
52
+ page_content=' We apply vector quantization and domain constrain to bridge the gap between the content domains of TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
53
+ page_content=' Ablation experiment shows this method’s effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
54
+ page_content=' We perform extensive experiments: zero-shot TTS, zero-shot VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
55
+ page_content=' Results proves that jointly trained TTS outperformes StyleSpeech (Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
56
+ page_content=', 2021)and jointly trained VC gains better speech decoupling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
57
+ page_content=' Demos for this paper are available at https://liuhaogeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
58
+ page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
59
+ page_content='io/UnifySpeech/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
60
+ page_content=' 2 BACKGROUND In this section, we will briefly review the background of this work, including neural TTS and VC models, and the zero-shot learning for TTS and VC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
61
+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
62
+ page_content='1 TEXT-TO-SPEECH TASK TTS task is to model the mapping between text and speech, which is a modeling problem between two unequal length sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
63
+ page_content=' According to the alignment mechanism (Battenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
64
+ page_content=', 2020) between text and speech in the model, the end-to-end TTS model can be divided into two categories: 1) Using a neural network to learn the alignment information between text and speech, such as local sensitive attention in Tacotron (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
65
+ page_content=', 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
66
+ page_content=' Various improvements to the attention mechanism have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
67
+ page_content=' In addition, inspired by CTC (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
68
+ page_content=', 2017) in ASR, glow- TTS (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
69
+ page_content=', 2020), VITS (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
70
+ page_content=', 2021) can automatically learn the alignment information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
71
+ page_content=' 2) By introducing the duration information (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
72
+ page_content=', 2019) of phonemes as prior knowledge, the text is upsampled to achieve alignment (McAuliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
73
+ page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
74
+ page_content=' Since the upsampled information based on text is independent of the speaker, the speech content can be well separated from the speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
75
+ page_content=' Therefore, we introduce the duration information to build the TTS model in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
76
+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='2 VOICE CONVERSION TASK Voice conversion can be seen as two steps (Sisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
79
+ page_content=' Firstly, extract the speaker- independent speech content information from the source speech, and then embed the target speaker information to the speech content to reconstruct the speech of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' According to the way of extracting speech content, the VC model can be divided into two categories: (1) text-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' (2) Information bottleneck approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The first approach requires an additional pre-trained ASR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Since the ASR is trained in a supervised manner, it demands a lot of paired text and speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Additionally, pipeline modeling is easy to accumulate errors and affects the performance of 2 the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Therefore, a lot of research work is focused on the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' By adding some restrictions to the information bottleneck, different kinds of information can be decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' However, if the in- formation bottleneck can not be decoupled well, the performance of the model will be significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='3 ZERO SHOT LEARNING FOR TTS AND VC The research of zero-shot learning based on TTS and VC focused on how to extract effective speaker information and then embed it into TTS or VC model for joint training or segmented training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Typical speaker features include i-vector (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2017a), d-vector (Variani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2014), x- vector (Snyder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2018) and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' In addition, the modules that extract speaker-style informa- tion through the specially designed network structure, such as GST (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2018), VAE (Van Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2017), can also achieve good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 3 UNIFYSPEECH In this section, we introduce the details of UnifySpeech for zero-shot TTS and VC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We first give the key idea of UnifySpeech: speech factorization, and then introduce the formulation of Uni- fySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Finally, we will describe the model structure of UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Figure 1: Structure of UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='1 SPEECH REPRESENTATION DISENTANGLEMENT The core of the controllable and migratable speech generation task is to decouple the components of the generated speech first, and then control and transfer each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Although some end-to-end models can directly model the relationship between text and speech (TTS) or speech-to-speech task (VC), due to the mutual coupling of various components of the end-to-end model, it brings great difficulties to the transfer learning of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Therefore, we first decouple the speech generation task into three independent components and then input them into the decoder to synthesize speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' This idea is also the main architecture of UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The three components and their sources will be described in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Speech Content: To generate intelligible speech signals, it is important to model accurate speech content information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Speech content is linguistic information, which is irrelevant to the speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Due to the different types of input signals of TTS and VC, the ways of extracting speech content are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' For TTS, the source of speech content is text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Firstly, to learn the context information of the text, a text encoder is used to encode the text to obtain the context representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The context representation is up-sampled according to the phoneme’s duration information to obtain the speech content information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' For VC, since the source speech is aligned with the target speech, we directly use a content encoder to extract speech content information from the source speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Speaker: The speaker information includes the speaker’s characteristics, such as the timbre, volume, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We can extract the speaker information from the given speech of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' This is 3 Decoder Decoder Pitch Pitch Pitch predictor Pitch predictor Prosody information Speaker embedding Speech content Speech content Speaker embedding Prosody information VQ VQ Prosody encoder Speaker encoder Speaker encoder Prosody encoder Text encoder Content encoder Pitch Reference mel Phonemes Speech Reference mel Pitchcommon for TTS and VC tasks, so the speaker extraction network can be shared in the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Since the speaker extraction network can directly extract information from speech without text, so a large number of data without text annotation in the VC task can be used for training, which can help to improve the transfer learning ability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Prosody: The prosody information represents how the speaker says the content information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' It is independent of the speaker information and related to the way of expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Since the pitch information can reflect the rhythm of speech, therefore, the pitch information is used to extract prosody information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The prosody information, like speaker information, can be shared by both TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' In addition, in the training process, we can obtain pitch information from the ground truth speech, but there is no ground truth in the inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Therefore, a pitch prediction module (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2020) is proposed in the training stage, which takes the speech content information and speaker information as the input to predict the pitch information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='2 SPEECH CONTENT WITH VECTOR QUANTIZATION Since there are different ways to obtain the speech content in TTS and VC, it is very easy to de- viate between the two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' If this deviation occurs, it will cause devastating damage to some downstream shared modules (such as pitch predictor and decoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Therefore, to ensure that the consistency of the two speech content domains as close as possible, we reconstruct them from two aspects: First, we use the shared codebook to quantify the continuous feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Second, we use the labeled data to narrow two discrete feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The detailed process is described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Suppose that the vector obtained by the text encoder in TTS pipeline is Cp = (C1 p, C2 p, Β· Β· Β· , CT p ) with length T, the vector obtained by the content encoder in VC pipeline is Cs = (C1 s, C2 s, Β· Β· Β· , CT β€² s ) with length T β€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' It should be noted that we add a length regulator module in the text encoder to solve the problem of length mismatch between the text and speech sequence, which is introduced in FastSpeech (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Therefore, if text and speech are paired, T β€² = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The vector Cp and Cs is a sequence of continuous vector in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Due to the large representation range of continuous features, Cp and Cs are difficult to match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We borrow the discretization method for latent variables from Vector Quantized Variational AutoEncoder (Van Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Specifically, for each time step t, the continuous latent representations Ct p in Cp can be mapped into C t p by finding the nearest pre-defined discretized embedding in the dictionary as: C t p = ek, k = argminj οΏ½οΏ½Ct p βˆ’ ej οΏ½οΏ½ 2 (1) where ej is the j-th embedding in the codebook dictionary, and j ∈ 1, 2, Β· Β· Β· , V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Since selecting the entry with the minimum distance will cause the operation to be non-differentiable, the straight- through gradient estimator can be used to approximate the gradient, which can be expressed as: Β―ht = ht + ev βˆ’ sg (ht) , v = arg min k βˆ₯ht βˆ’ ekβˆ₯2 (2) where sg(Β·) is the stop-gradient (Van Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2017) operation that treats its input as constant during back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' After vector quantization, the quantized sequence Cp = (C 1 p, C 2 p, Β· Β· Β· , C T p ) and Cs = (C 1 s, C 2 s, Β· Β· Β· , C T s ) can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' It should be noted that when Cp and Cs are quantized into Cp and Cs, they share the same codebook e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The advantage of this is that since the speech content features Cp in the TTS pipeline are independent of the speaker, sharing the same codebook can help learn the speech content features Cs independent of the speaker in the VC pipeline, which is essential for the VC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Although both pipeline use the same codebook for coding, there is no guarantee that there is no deviation between the two fields after coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Therefore, to further eliminate the deviation between the two domains, we use the labeled speech in the TTS pipeline to supervise the training of the two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Specifically, for paired text and speech data, we constrain the feature distance between the quantized sequence Cp and Cs: 4 Lpair = οΏ½οΏ½Cp βˆ’ Cs οΏ½οΏ½2 2 (3) In this way, we can efficiently minimize the domain discrepancy by using limited labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Figure 2: Structure of vector quantized operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='3 UNIFYSPEECH PIPELINE An overview of our proposed UnifySpeech architecture is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' It consists of a sequence-to-sequence TTS, and a sequence-to-sequence VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The key idea is to share most of the module parameters (speaker encoder, prosody encoder, decoder and pitch predictor) and map the speech content in TTS and VC to the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' As mentioned above, the UnifySpeech allows us to train the model on the concatenation of both the labeled and unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' For supervised training with labeled data, both models can be trained independently by minimizing the loss between their predicted speech and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' For unsupervised training with unlabeled data, the VC pipeline can be trained, and the parameters are shared with TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' To further clarify the training process, we unrolled the framework as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='1 TTS PIPELINE Denote the text and speech sequence pair (x, y, F0) ∈ D, where D is the paired text and speech corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Each element in the text sequence x represents a phoneme or character, while each element in the speech sequence y represents a frame of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' F0 is the pitch information of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The representation obtained after three encoders are speech content C, speaker S and prosody P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Then, the three parts are added and input into a decoder to obtain the predicted speech yβ€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' In addition, to obtain the pitch information in the interference stage, we use the content information and speaker information to predict the F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' These processes can be expressed as: yβ€² = decoder(C + S + P) (4) F0β€² = pitch predictor(C + S) (5) Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' the reconstruction loss in TTS process includes two parts: LV C rec = MSE(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' yβ€²) + MSE(F0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' F0β€²) (6) 5 C Look up min Look up min distance distance vector vector 1 V L2 L2 distance ev distance Ct Cs Text encoder Content encoderIn addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' we use the content encoder in the VC pipeline to extract the content representation Cs for the training speech in TTS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' and close the distance between the two domains by calculating the distance loss of Cs and Cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' which is explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Lpair = οΏ½οΏ½Cp βˆ’ Cs οΏ½οΏ½2 2 (7) The loss of TTS pipeline can be expressed as: LT T S = LT T S rec + Lpair (8) where MSE denotes the mean squared errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='2 VC PIPELINE Denote all the unlabeled or labeled speech (y, F0) ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We first extract three information from (y, F0), which are speech content Cs, speaker Ss and prosody Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Then, similar to the TTS pipeline, the shared decoder and pitch predictor module is used to predict the speech signal yβ€² and F0β€², which is similar to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The loss of VC pipeline only includes reconstruction loss, which can be expressed as: LV C = MSE(Y, Y β€²) + MSE(F0, F0β€²) (9) where MSE denotes the mean squared errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='3 TRAINING PROCESS With such a unified framework, TTS and VC can learn from each other through joint training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The details of the algorithm can be found below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Procedure 1 UnifySpeech training algorithm 1: Input: Paired speech and text dataset (x, y), speech only dataset y β€² 2: repeat 3: # A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' TTS pipeline with speech-text data pairs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Sample paired speech and text (x, y) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Extract speech content information from x for domain loss 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Generate the predict speech y, pitch F0 and speech content from text 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Calculate the loss for TTS LT T S rec 4: # B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' VC pipeline with speech-only data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Sample paired speech and text (x, y) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Extract speech content information from y for domain loss 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Calculate the domain loss for the two pipeline Lpair 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Sample speech y β€² in speech only dataset 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Generate the predict speech y, pitch F0 and speech content from speech y β€² 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Calculate the loss for VC LV C 5: # C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Loss combination: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Combine all loss (LT T S rec , Lpair, LV C) into a single loss variable 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Calculate TTS and VC parameters gradient 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Update TTS and VC parameters with gradient descent optimization 6: until convergence 4 EXPERIMENTS AND RESULTS ANALYSIS In this section, we conduct experiments to evaluate our proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The experiments are carried out from two aspects: zero-shot TTS, zero-shot VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='1 DATASETS Two datasets are used to simulate labeled data and unlabeled data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' VCTK dataset, an English language dataset containing 44 hours of speech and 109 speakers is used as labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Each speaker has approximately 400 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' LibriTTS (Zen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2019) are used as unlabeled data, which consists of 585 hours of speech data from 2484 speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We only use the speech data in LibriTTS and discard the text for unsupervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' In this way, it can simulate the scene where a large number of speech that we can obtain are unlabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We use a 16-bit, 22050 Hz sampling rate for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The 80-dim Mel spectrogram is extracted with Hann windowing, frame shift of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='5-ms, frame length of 50-ms, and 1024-point Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' In this experiment, we use hifigan (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2020) as vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='2 MODEL DETAILS Figure 3: Structure of each module in UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' FFT (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2019) means feed-forward Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The detail of each module in our proposed method is shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Specifically, to make the sequence of speech content extracted from the text encoder and content encoder equal, a length regulator is added to the text encoder, which is inspired by the FastSpeech (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The structure of the duration predictor is the same as that in FastSpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The structure of the decoder and content encoder is similar, but the dimensions of input and output are opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' For the prosody encoder, we first quantize F0 of each frame to 32 possible values and encode them into a learnable embedding vector according to the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' And we change the output of the pitch predictor into a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' By this way, pitch prediction becomes a classification task, reducing the difficulty of frame-level pitch prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We use the speaker encoder in StyleSpeech and fuse features with Style Adaptive Layer Norm(SALN) (Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2021) method to make the comparision fairly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The number of feed-forward Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=', 2017) (FFT) blocks in the text encoder is 4 and it is 6 in the decoder module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' In each FFT block, the dimension of hidden states is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The kernel sizes of all the 1D-convolution are set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The dropout rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The dimension of the last linear layer in the decoder is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The dimension of last linear layer in encoders (text encoder, pitch encoder, content encoder) is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' An Adam optimizer (Kingma & Ba, 2014) is used to update the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The initial learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='001 and the learning rate decreased exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 7 Linear layer Linear layer Duration Length regulator predictor m block FFT 不 不 LN + Dropout n block FFT 不 Speaker Speaker Embedding ConvlD + ReLU Phoneme Embedding Embedding Linear layer Linear layer Pooling δΈͺ LN + Dropout LN + Dropout FFT ConvlD + ReLU ConvlD + ReLU LN + Dropout LN + Dropout 不 Pitch Embedding ConvlD + ReLU ConvlD + ReLU Randomly select a clip LSTM Quantify to [0, 32] from target speaker4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='3 ZERO-SHOT TTS We first carried out zero-shot TTS task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We choose four speakers from VCTK that are not used during training process as target speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' For each speaker, we randomly select about 20 sentences to be our target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Then we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek, 1993) (Mel-cepstrum distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR (correlation factor of F0) between synthesized speech and ground truth speech as the objective met- rics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Table 1: Objective evaluation results for zero-shot TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Model F0 RMSE (Hz) MCD (db) V/UV F0 CORR UnifySpeech 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='51 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='93 StyleSpeech 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='63 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='06% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='92 Subjective evaluation was also conducted to compare the speech’s quality and similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We choose mean opinion scores(MOS) for naturalness and similarity mean opinion scores(SMOS) for similar- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Both metrics are rated in 1-to-5 scale and reported with the 95% confidence intervals (CI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Table 2: Mean opinion score (MOS) of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' With VC means the model is jointly trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Model MOS SMOS GT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='32 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='15 βˆ’ GT mel + Vocoder 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
267
+ page_content='09 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
268
+ page_content='15 βˆ’ StyleSpeech 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
269
+ page_content='52 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
270
+ page_content='13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
271
+ page_content='82 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='13 UnifySpeech-TTS (with VC) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
273
+ page_content='76 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
275
+ page_content='95 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='13 To better show our method’s effectiveness, we demonstrate the t-SNE projection of the speaker embedding vectors from speakers in both VCTK and LibriTTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' (Van der Maaten & Hinton, 2008) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 4 shows the speaker visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' For the seen speakers (x) and unseen speakers (o), the corresponding speaker embedding form a cluster and distinct from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' The boundary between different speakers is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' This shows that the speaker encoder performs well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Figure 4: Speaker visualization of generated speeches,where the circle and triangle indicate unseen speaker,while x and square indicate seen speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='4 ZERO-SHOT VC We carried out zero-shot VC task, using unseen speakers voice to be the reference speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' As we are lack of parallel corpus, we only conduct subjective evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' But VC and TTS shares the same 8 30 p225 p226 p227 20 p228 p279 X p274 10 p307 p311 8012 0 οΏ₯5181 3885 2961 10 1089 7021 20 KX 20 15 10 5 0 5 10 15speaker encoder, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 4 in zero-shot TTS can also be a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Just as in zero-shot TTS task, we choose mean opinion scores(MOS) for naturalness and similarity mean opinion scores(SMOS) for similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Both metrics are rated in 1-to-5 scale and reported with the 95% confidence intervals (CI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Table 3: Mean opinion score (MOS) of the VC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
291
+ page_content=' With TTS means the model is jointly trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Model MOS SMOS GT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
293
+ page_content='32 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
294
+ page_content='15 βˆ’ GT mel + Vocoder 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
295
+ page_content='09 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
296
+ page_content='15 βˆ’ UnifySpeech-VC(without TTS) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
297
+ page_content='63 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='31 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='06 UnifySpeech-VC(with TTS) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='58 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='31 Β± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='13 It can be found that when VC pipeline is trained alone, its performance is poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' In other words, it doesn’t have the ability to discriminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' But jointly training with TTS improves its speech decou- pling ability, indirectly improving the speaker modeling ability, which we will analysis later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='5 ABLATION EXPERIMENT To figure out whether jointly training is effective, we separately train the TTS and VC parts, carrying out objective evaluation on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We also remove the VQ parts and test the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' For TTS, we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek, 1993) (Mel-cepstrum distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR (correlation factor of F0) between synthesized speech and ground truth speech as the objective metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Table 4: Objective evaluation results for zero-shot TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Model F0 RMSE (Hz) MCD (db) V/UV F0 CORR UnifySpeech-TTS(without VC) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='55 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='91 UnifySpeech-TTS-novq(with VC) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='58 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='92 UnifySpeech-TTS(with VC) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='51 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='93 For VC, we separately calculate the Average Cosine Similarity (ACS) (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
325
+ page_content=', 2022) for the embedding from same (S-ACS) and different speakers (D-ACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' And then their ratio is used as an evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Table 5: Ratio of ACS from same and different speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Unseen or seen means whether the speaker is from test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' With or without TTS means whether the VC pipeline is trained with TTS pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Model Sβˆ’ACS Dβˆ’ACS (unseen) Sβˆ’ACS Dβˆ’ACS (seen) UnifySpeech-VC (with TTS) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='0 UnifySpeech-VC (without TTS) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='0 To validate whether VQ can bridge the gap between the speech content parts in TTS and VC, we caculate the L2 distance between the phoneme representation from TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' As in VC there are many frames represent same phoneme, so we choose their clustering center as the correspond- ing phoneme representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' We randomly select 4 sentences from validation set to carry out the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Table 6: L2 distance between the phoneme representation in TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
338
+ page_content=' S1 means sentence1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
339
+ page_content=' Model S1 S2 S3 S4 average UnifySpeech-VC(without VQ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
340
+ page_content='464 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
341
+ page_content='476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
342
+ page_content='474 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
343
+ page_content='553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
344
+ page_content='492 UnifySpeech-VC(with VQ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
345
+ page_content='152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
346
+ page_content='205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
347
+ page_content='208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
348
+ page_content='205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
349
+ page_content='193 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content='6 RESULT ANALYSIS Above results show that jointly training actually improves TTS’s speaker modeling ability and VC’s speech decoupling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' And VQ enhances the consistency of representation of the same content from phonemes and speeches, ensuring the model’s correctly working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
352
+ page_content=' For TTS, sharing modules with VC enables unlabeled data to participate in its training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
353
+ page_content=' Along with the richer speaker style pattern, the speaker style modeling capability has been enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
354
+ page_content=' For VC, when trained alone, it doesn’t have the speaker modeling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' As the self-supervised training process aims at reducing the reconstruction loss of source audio, the reference audio of the target speaker isn’t crucial in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Only the content encoder and the decoder are enough for the process, that’s possibly why the speaker embeddings are all similar, though they are from differ- ent speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' When jointly trained, the text loss plays a role of regularization factor, resulting that the content encoder just extracting the content information from the source speech (speaker information is reserved and others are discarded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' This makes the reference speech with rich speaker information become indispensable for the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Thus the model’s speaker modeling ability has been improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 5 CONCLUSIONS In this paper, we propose UnifySpeech, a unified framework for TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Both task benefits from the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Due to training with large amounts of unlabeled data, their few-shot modeling ability makes progress as well as the synthesized speech’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' In the future, further improving the synthesized speech’s quality and making the generated speech’s style more similar to target speaker will be our endeavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' REFERENCES Eric Battenberg, RJ Skerry-Ryan, Soroosh Mariooryad, Daisy Stanton, David Kao, Matt Shannon, and Tom Bagby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' Location-relative attention mechanisms for robust long-form speech synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+ page_content=' 6194–6198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
368
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+ filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf,len=382
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+ page_content='Towards an AI-enabled Connected Industry: AGV Communication and Sensor Measurement Datasets Rodrigo HernangΒ΄omezβˆ—, Alexandros PalaiosΒ§, Cara WatermannΒ§, Daniel SchΒ¨aufeleβˆ—, Philipp GeuerΒ§, Rafail Ismayilovβˆ—, Mohammad Parvini†, Anton Krause†, Martin Kasparickβˆ—, Thomas NeugebauerΒΆ, Oscar D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Ramos-Cantor‑, Hugues Tchouankem‑, Jose Leon CalvoΒ§, Bo Chenβˆ—βˆ—, SΕ‚awomir StaΒ΄nczakβˆ—βˆ₯, Gerhard Fettweis† βˆ—Fraunhofer Heinrich Hertz Institute, Germany, {firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='lastname}@hhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='fraunhofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='de Β§Ericsson Research, Germany, {alex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='palaios, cara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='watermann, philipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
9
+ page_content='geuer}@ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
10
+ page_content='com †Vodafone Chair, Technische UniversitΒ¨at Dresden, Germany, {mohammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
11
+ page_content='parvini, anton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
12
+ page_content='krause, gerhard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
13
+ page_content='fettweis}@tu-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
14
+ page_content='de ‑Corporate Research, Robert Bosch GmbH, Germany, {oscardario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
15
+ page_content='ramoscantor, huguesnarcisse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
16
+ page_content='tchouankem}@de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
17
+ page_content='bosch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
18
+ page_content='com βˆ₯Network Information Theory Group, Technische UniversitΒ¨at Berlin, Germany ΒΆGΒ¨otting KG, Germany, neugebauer@goetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
19
+ page_content='de βˆ—βˆ—Enway GmbH, Germany, bo@enway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
20
+ page_content='ai Abstractβ€”This paper presents two wireless measurement cam- paigns in industrial testbeds: industrial Vehicle-to-vehicle (iV2V) and industrial Vehicle-to-infrastructure plus Sensor (iV2I+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
21
+ page_content=' De- tailed information about the two captured datasets is provided as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
22
+ page_content=' iV2V covers sidelink communication scenarios between Automated Guided Vehicles (AGVs), while iV2I+ is conducted at an industrial setting where an autonomous cleaning robot is connected to a private cellular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
23
+ page_content=' The combination of dif- ferent communication technologies, together with a common mea- surement methodology, provides insights that can be exploited by Machine Learning (ML) for tasks such as fingerprinting, line-of-sight detection, prediction of quality of service or link selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
24
+ page_content=' Moreover, the datasets are labelled and pre-filtered for fast on-boarding and applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
25
+ page_content=' The corresponding testbeds and measurements are also presented in detail for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
26
+ page_content=' Index Termsβ€”Measurement data, QoS prediction, AGV, drive tests, V2X, campus networks, wireless communications I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
27
+ page_content=' INTRODUCTION It is anticipated that the next generation of wireless commu- nication systems (5G and beyond) will bring about an upsurge in the number of new services and applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
28
+ page_content=' each of which demanding for a specific Quality of Service (QoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
29
+ page_content=' In parallel, there is a resurgence of interest in promoting the concept of predictive Quality of Service (pQoS), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
30
+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
31
+ page_content=', QoS estimation for a given time instance in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
32
+ page_content=' This can be done in different prediction horizons, ranging from milliseconds to hours or even days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
33
+ page_content=' pQoS can pave the way to satisfy a very demanding set of QoS requirements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
34
+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
35
+ page_content=', very low latency, minimum Signal-to-Noise Ratio (SNR), delay, packet error rate, or huge Uplink (UL) or Downlink (DL) throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
36
+ page_content=' pQoS can be particularly important for wireless networks in the industrial domain, where communication needs to be highly reliable due to, among other reasons, its integration into This work was supported by the Federal Ministry of Education and Re- search (BMBF) of the Federal Republic of Germany as part of the AI4Mobile project (16KIS1170K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
37
+ page_content=' The authors alone are responsible for the content of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
38
+ page_content=' control loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
39
+ page_content=' Wireless links are especially relevant in mobile setups, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
40
+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
41
+ page_content=', with one or more Automated Guided Vehicles (AGVs) connected in a Vehicle-to-vehicle (V2V), Vehicle-to- infrastructure (V2I) or Vehicle-to-everything (V2X) manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
42
+ page_content=' In this regard, some datasets are available for automotive sce- narios to train and test Machine Learning (ML) algorithms and thus enhance such schemes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
43
+ page_content=' However, the availability of datasets from industrial and indoor measurement campaigns, such as [2], is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
44
+ page_content=' With proper knowledge of the upcoming QoS conditions, pQoS can facilitate the proper operation of industrial applications to guarantee human-machine safe interaction or robot cooperation to fulfill a common task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
45
+ page_content=' Other use cases may include tele-operated driving, high- density platooning, and High Definition (HD) map collecting and sharing for optimal route selection [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
46
+ page_content=' In this manner, we can see a growing tendency toward applying deep learning algorithms for pQoS applications, such as [5]–[8], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
47
+ page_content=' A consolidated overview of the ML- enabled throughput prediction scenarios is presented in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
48
+ page_content=' In the same vein, [6] investigates a ML-model to predict the throughput in a non-standalone 5G network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
49
+ page_content=' ML has been a very active research area in the past few years and there is ample literature around it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
50
+ page_content=' however, its real- world implementation or validation has remained elusive for industrial communication due to its high dependency on avail- able datasets to test, validate and generalize the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
51
+ page_content=' Therefore, creating a reference dataset from experimental testbeds or practical simulations is paramount to evaluate the underlying theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
52
+ page_content=' In this paper, we will describe the industrial Vehicle- to-vehicle (iV2V) dataset and the industrial Vehicle-to- infrastructure plus Sensor (iV2I+) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
53
+ page_content=' These two datasets aim to pave the way for future advancement in the experi- mentation of mobile networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
54
+ page_content=' The measurement campaigns that were conducted here are part of a bigger measurement framework and procedure that is described in detail in [9] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
55
+ page_content='03364v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
56
+ page_content='NI] 20 Dec 2022 with some first results being reported in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
57
+ page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
58
+ page_content=' In Section II and Section III, we describe the iV2V and iV2I+ testbed and datasets and we elaborate on their details and components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
59
+ page_content=' We conclude with an overview of possible future research directions in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
60
+ page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
61
+ page_content=' THE IV2V TESTBED AND DATASET In this section, we present the first of the two collected datasets and the considerations that have been taken into account for its measurement campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
62
+ page_content=' We give a brief in- troduction to the sidelink technology, continue with a detailed description of the testbed, and finally describe the processing and resulting data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
63
+ page_content=' Figure 1a depicts one of the AGVs, carrying the measure- ment and communication hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
64
+ page_content=' The AGVs communicate directly in a V2V manner, using the sidelink technology as introduced by 3rd Generation Partnership Project (3GPP) in Release 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
65
+ page_content=' In the sidelink setup, every AGV acts both as transmitter and sender (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
66
+ page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
67
+ page_content=' Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
68
+ page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
69
+ page_content=' Testbed Components 1) Sidelink: Sidelink has been standardized in 3GPP during 4G and 5G mobile networks to define a framework where communication is possible with and without network coverage and with varying degrees of interaction between the devices and the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
70
+ page_content=' Two modes of resource allocation are defined [11]: Network-based resource allocation (Mode 1 in 5G sidelink and Mode 3 in 4G sidelink): This mode is only available when all the devices are in network coverage, and the network selects the resources and other transmit parameters used by the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
71
+ page_content=' Autonomous resource allocation (Mode 2 in 5G sidelink and Mode 4 in 4G sidelink): This mode offers a com- pletely decentralized solution in which the User Equip- ments (UEs) autonomously select the resources and other transmit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
72
+ page_content=' Overall, network-based resource allocation can outperform the autonomous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
73
+ page_content=' This is due to the network controlling the resources to be used by each of the UEs involving UL signaling from the UE to the network to obtain a grant for transmissions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
74
+ page_content=' On the other hand, autonomous resource allocation is mainly useful when there is no possibility of having network coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
75
+ page_content=' The basic operation of autonomous resource selection involves the device performing sensing within a pre-configured resource pool, detecting which resources are not in use by other devices with higher-priority traffic, and choosing some of these free resources for its transmissions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
76
+ page_content=' The autonomous resource allocation is more prone to collisions while also suffering from hidden node and half duplex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
77
+ page_content=' Solutions have been considered in 3GPP to mitigate these issues [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
78
+ page_content=' For the measurement campaign, we use a full stack, software-based, standard-compliant and open implementation of the 3GPP Release 14 PC5 Mode 4 standard [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
79
+ page_content=' The plat- form allows research concepts and standard features to be val- idated in hardware testbeds and it provides interfaces and tools for recording measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
80
+ page_content=' Changes and adjustments are possible at every layer, which allows a realistic verification of new features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
81
+ page_content=' The sidelink software (all layers incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
82
+ page_content=' baseband processing) can be run on standard general purpose computing hardware in connection with suitable Software Defined Radio (SDR) hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
83
+ page_content=' We opted for a full stack implementation, thus providing a standard based IP to IP (one to all) interface for any application, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
84
+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
85
+ page_content=', all protocols on OSI layer 3 and higher can be transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
86
+ page_content=' The hardware setup is shown in Figure 1c (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
87
+ page_content=' [14] for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
88
+ page_content=' 2) Localization: Precise position of the communicating devices is required to link the environmental conditions with the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
89
+ page_content=' For that purpose, the position information provided by the AGVs, carrying the communication entities, was recorded during the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
90
+ page_content=' Two types of local- ization methods are used by different AGVs in the testbed, namely, marker/track-based, and Simultaneous Localization and Mapping (SLAM)-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
91
+ page_content=' In the former method, the AGVs follow a track on the floor with help of an onboard camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
92
+ page_content=' Additionally, Radio-frequency identification (RFID) tags are placed on the track to provide the exact position information to the AGVs, when they pass over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Between the RFID tags, the AGVs estimate their position by using odometry, which describes a method of estimating the position and orientation of a mobile system using data from its propulsion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Wheel-driven systems use the measurement of the wheel rota- tions for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
95
+ page_content=' In combination with dead reckoning, odometry is a basic navigation method for ground-based vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
96
+ page_content=' Since the AGVs do not leave the track, the transversal error is in the order of few mm, while the longitudinal error depends on the separation distance between the RFID tags and the positional accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' For our testbed, the longitudinal error was in the order of few cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' In the latter localization method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
100
+ page_content=', SLAM-based, the AGVs are equipped with a laser scanner to detect and estimate the distance to landmarks (reference points) in the testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
101
+ page_content=' These landmarks are also defined in the map of the AGV system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
102
+ page_content=' Hence, the AGVs can estimate their position in the map through a combination of information from several landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
103
+ page_content=' In the testbed, we achieve a position accuracy in the order of few cm with the SLAM-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
104
+ page_content=' The reported AGV position was timestamped during the measurements, so that a combination with other measured data is possible during post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
105
+ page_content=' Unless otherwise stated, the measurements scenarios presented below consider that the SLAM-based AGVs were static and the marker/track-based AGVs were moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
106
+ page_content=' Since the moving AGV is guided by an optical line, the real lateral position is better than Β±2 mm (3Οƒ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
107
+ page_content=' The along track (longitudinal) error while passing an RFID tag has a timing uncertainty of up to 30 ms, which gives an error depending on the actual speed (up to 30 mm at 1m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
108
+ page_content=' Dead reckoning results in additional errors being displayed due to the route (a) AGV testbed in industrial environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
109
+ page_content=' (b) Schematic illustration of sidelink mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
110
+ page_content=' (c) Sidelink SDR platform with all components inte- grated in 19 inch case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 1: iV2V testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
113
+ page_content=' and steering angle sensors not being perfectly adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The longitudinal error increases with the length of the unsupported route driven without an RFID tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
115
+ page_content=' The repeatability of the position information is less than Β±2 mm transversally and less than +2 cm longitudinally at the driven speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
116
+ page_content=' 3) Time Synchronization: To enable accurate evaluation of network latency and other QoS properties, a proper time synchronization is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The time was synchronized across sidelink devices by running NTP over Ethernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
118
+ page_content=' The error is typically in the order of several Β΅s with a worst case error of 1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
119
+ page_content=' However, precisely quantifying this is difficult without specialized measurement equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 4) Controlled Packet Generation: To ensure a highly pre- cise packet generation, we used a network packet generator tool based on Real-time User Datagram Protocol (UDP) Data Emitter (RUDE) & Collector for RUDE (CRUDE) which is able to produce heterogeneous UDP network traffic for realistic network workloads [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' It consists of two main modules: RUDE generates traffic to the network, which is then received and logged by the other module of the network with CRUDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
122
+ page_content=' We extended the packet generator tool by enabling the capability to log all channel information for successfully received packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
123
+ page_content=' 5) Automatic Gain Control: In order to be able to assess the received signal quality, which is an elementary quantity for assessing the QoS, the function of the Automatic Gain Control (AGC) in a receiver must be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
124
+ page_content=' AGC is a feature in an RF receiver signal path that is used to keep the received signal magnitude at a suitable level for subsequent signal processing so that signals are not clipped and the receive path with good sensitivity is operated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
125
+ page_content=' The function of the AGC is technically realized by controllable amplifiers in the received signal path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
126
+ page_content=' For this purpose, the signal level is determined in the signal processing during special preambles at the beginning of a defined data frame and regulated within a specified range by setting the AGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
127
+ page_content=' A criterion for the regulation can be the evaluation of the preamble of the Orthogonal Frequency- Division Multiplexing (OFDM)-based signal obtained from the I/Q samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
128
+ page_content=' However, the corresponding values at the antenna input, namely Received Signal Strength Indicator (RSSI) and Reference Signal Received Power (RSRP), are of interest for signal evaluation and a corresponding indication of comparable level values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
129
+ page_content=' With knowledge of the amplification and attenuation of individual components in the front-end and the AGC setting, these can be determined from the measured values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
131
+ page_content=' from the total gain between the antenna input and the evaluation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
132
+ page_content=' The AGC calculations have already been performed before the output and the values supplied for pre- processing are the correct values and can be used as is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
134
+ page_content=' Measurement Scenarios We collected data for roughly 10 hours over the course of two days to acquire almost 50 GB communication data between up to three industrial AGVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
135
+ page_content=' A schematic of the test area and its surroundings is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
137
+ page_content=' The dotted gray line depicts the track used by AGV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
138
+ page_content=' Several obstacles, depicted in different blue tones in the figure, were located within the test area to achieve different radio propagation conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
139
+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
140
+ page_content=', Line-of-Sight (LOS) and Non-Line-of-Sight (NLOS) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
141
+ page_content=' The obstacles were rearranged during the measurement campaign to create two scenarios, A and B, with different N/LOS characteristics, as marked in light blue in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
143
+ page_content=' The Dataset 1) Captured Sidelink Data: For each scenario illustrated in Figure 2, we capture the sidelink channel parameters for every transmitter/receiver pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
144
+ page_content=' The selected sidelink channel parameters and their description are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
145
+ page_content=' The parameters in the table are obtained/estimated from the De- modulation Reference Signal (DMRS) of the Physical Sidelink Shared Channel (PSSCH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
146
+ page_content=' The AGV localization data is provided as x and y coordi- nates in a local coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
147
+ page_content=' 2) Dataset Pre-processing: In this section, we describe the pre-processing of captured sidelink data, and we present a dataset constructed with the pre-processed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
148
+ page_content=' The dataset is constructed in a tabular format where each row represents a sample and the columns contain the value of the mea- sured sidelink channel parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
149
+ page_content=' Note that the side channel RF Frontend 3) ise Shecker SAt BasebandPC SDR46cm X Wall Height ~3,10 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
150
+ page_content='5m 22m 65cm 1m 123cm (0,0) AGV 2 170cm AGV 3 263cm 251cm 2 3 AGV 1 1 AGV Test Track Glass door/window Foam wall Metallic wall Foam wall in scenario B only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
151
+ page_content=' Replaced by metallic wall in scenario A Foam wall in scenario A only 2 3 1 AGV 1 moves along the test track Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
152
+ page_content=' 2: Illustration of measurement scenarios A & B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
153
+ page_content=' TABLE I: Selected iV2V Data Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
154
+ page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='Parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
157
+ page_content='SNR [dB] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
158
+ page_content='Derived from noise and power estimations of DMRS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
159
+ page_content='RSRP [dBm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
160
+ page_content='Average energy per carrier/RE for DMRS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
161
+ page_content='RSSI [dBm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
162
+ page_content='Signal power over the whole band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
163
+ page_content='Noise Power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
164
+ page_content='Estimated on DMRS in decoded subframe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
165
+ page_content='Time [sec] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
166
+ page_content='Receive time of first IQ-Sample of decoded subframe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
167
+ page_content='Frame Number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
168
+ page_content='System frame number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
169
+ page_content='Subframe Number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
170
+ page_content='System subframe number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
171
+ page_content='UHD Rx Gain [dB] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
172
+ page_content='Receive antenna gain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
173
+ page_content='SCI FRL N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
174
+ page_content='Starting subchannel of decoded PSSCH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
175
+ page_content='SCI FRL L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
176
+ page_content='Number of used subchannels for PSSCH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
177
+ page_content='RLC SN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
178
+ page_content='Sequence number of radio link control header ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
179
+ page_content='Location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
180
+ page_content='Local x and y coordinates of AGV 1 on the track ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
181
+ page_content='parameters and AGV location are measured independently ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
182
+ page_content='and simultaneously in different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
183
+ page_content=' With this setting, each measuring device embeds its own timestamp into the measured parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
184
+ page_content=' Since the processing of the received signal in different devices requires different lengths of time, the embedded timestamps in these devices also have some differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
185
+ page_content=' We align the timestamps between the location data and the sidelink data as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Given the timestamp tloc n corresponding to the measured location Ln = [xn, yn] of the AGV at nth time step, we find the timestamp tsl n from the measured sidelink such that οΏ½οΏ½tloc n βˆ’ tsl n οΏ½οΏ½ ≀ Ξ³, where Ξ³ denotes the alignment tolerance, and we use Ξ³ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='005s to construct the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' In addition, we present indicators Mn ∈ Z and Sn ∈ Z, where Mn indicates the measurement scenario (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=', the place- ment of obstacles), and Sn denotes the source of the received sidelink signal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=', AGV1 receives a signal from AGV2 or from AGV3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' With the parameters described in Table I, the nth row of the tabular dataset is denoted by Rn, and it contains the parameters as Rn = οΏ½ tloc n , tsl n, οΏ½οΏ½tloc n βˆ’ tsl n οΏ½οΏ½ , Ln, Pn, Sn, Mn οΏ½ , where Pn ∈ RK denotes the K measured parameters of sidelink channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' THE IV2I+ TESTBED AND DATASET A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Testbed Components The testbed for the measurements is located in an industrial co-working space in Berlin, with a layout as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The hall had a gateway, which allowed the AGV to drive outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 1) The AGV: The AGV used in the testbed is an au- tonomous cleaning robot from the company Enway, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' They are specially designed for use under the operating conditions of the manufacturing industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The sweeper has all the necessary navigation data saved on a digital map, and drives over the cleaning area autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Thanks to high-performance sensors and control software from Enway, the AGV navigates the environment completely independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Using a combination of laser distance mea- surement and cameras, the robot captures the environment in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' This 360-degree view enables very safe navigation between people, complex production lines, and overhanging systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The AGV immediately detects obstacles that suddenly appear along the route, and drives around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Additional equipment such as floor markings, QR codes, or magnetic tracks are not required for navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' In the event that the AGV encounters an unsolvable situation, the robot stops and reports automatically to Enway headquarters via a data connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The remote team monitors every movement of the device around the clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The specialists can end autonomous journeys at any time, and can take control from a distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Collisions with people, production systems, vehicles, and stored goods are thus avoided at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The machine can also be navigated manually by the operating personnel on site, if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Because the AGV is a retrofitted ride- on sweeper, it can be controlled from the driver’s seat in the traditional sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Ongoing use of the autonomous sweeper can be monitored, controlled, and then evaluated using mobile devices such as smartphones, laptops, or tablets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The software completely logs the cleaning trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 2) Cellular Network: The mobile network used for the measurements in this test bed corresponded to a standardized 4G campus network with TDD medium access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The bandwidth was 20 MHz in the frequency band between 3700 and 3800 MHz approved by the Federal Network Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' A correspond- ing frequency assignment was applied for the period of the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The hardware consisted of a server running the LTE core and a radio base station with integrated antennas connected to the server via Gbit LAN and powered via Power over Ethernet (PoE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The location of the base station is marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 3c with a yellow circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' A Mini-PC with the Linux operating system was used as the UE to carry out QoS-relevant measurements on the AGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' A Quectel RM500Q-GL card was used as the radio device, which was connected to the Mini-PC via USB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' External antennas were connected to the radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The server provides a service interface to which the appli- cations required for the measurements can be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The stationary applications can communicate with mobile applica- tions running in the Mini-PC via the service interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' This (a) The Autonomous Cleaning Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' (b) Architecture of the iV2I+ Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' (c) Map of the Environment as captured by the LIDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Walls and AGV tracking route are shown with red and black, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The yellow circle is the location of the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 3: iV2I+ testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' way, the location-dependent data rate and latency parameters relevant for evaluating the QoS can be determined at the Mini- PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 3) Time Synchronization: The AGV and the server were time synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 3b visualizes the communication set- up, where the dotted arrow shows the wireless connection and the solid line the cable connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Both the server and the Mini-PC on the AGV were connected to a GPS receiver, allowing accurate time synchronization by using Pulse-per- Second (PPS) signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The maximum error is typically in Β΅s- range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' However, the AGV only had consistent GPS reception at the start point and the outdoor area, which could lead to inaccuracies the ms-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 4) LTE Modem Access: We used Mobile Insight, an open- source cross-platform application for mobile network moni- toring and analytics to capture mobile network data at the Mini-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' It collects mobile network information across several cellular protocols, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Radio Resource Control (RRC) or EPS Mobility Management (EMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' During the measurement campaign, the available information was logged every 40 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Additionally, a Python script that accesses the modem via a virtual serial interface, a few radio parameters (RSSI, RSRP, Reference Signal Received Quality (RSRQ), Signal- to-Interference-plus-Noise Ratio (SINR)) were logged every 200 ms by the modem and written to a file with a time stamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The data can then be linked to other data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' the location, via the common time stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 5) Controlled Packet Generation: We used the application iperf3 on the Mini-PC and the server to generate UDP traffic in either UL or DL direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Apart from the generated iperf3 log files, tcpdump was used both on the Mini-PC and server to capture all incoming and outgoing packets at the respective network interfaces, allowing a more detailed evaluation on a packet level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Measurement Scenarios For the measurement campaign, one AGV, equipped with the sensors and measurement devices described in the prior section, was driving through the testbed area over the course of three days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Overall, 16 hours of data were collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' UL and DL communication was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Two types of packet flows were established to generate high and medium throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' In DL, high throughput measurements were con- ducted with a throughput target of 80 Mbps, in UL with 25 Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Approximately twice as many high throughput measurements as medium throughput measurements were col- lected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The route contained a diverse set of radio conditions, namely: LOS and NLOS situations, coverage loss as well as indoor and outdoor measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The measured path is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The Dataset For each scenario, data from the described network compo- nents and the sensors from the AGV was collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' A subset of the captured data is presented in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' TABLE II: Selected iV2I+ Data Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Parameter Description SINR [dB] Derived from noise and power estimations of DMRS RSRP [dBm] Average energy per carrier/RE for DMRS RSSI [dBm] Signal power over the whole band Throughput Acquired throughput in respective link direction Ping [ms] Time in ms until a ping reply was received Jitter Delay variation measured over 1s Odometry Fused position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' orientation and speed of the AGV Map static elevation Single pre-computed map of the whole area Near/far map obstacles 36 m2/400 m2 obstacle map around the AGV LIDAR 3D point cloud with obstacles 1) AGV-Sensor Data: The AGV delivers a series of sensor data via its Robot Operating System (ROS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' including the last fourth rows in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Except for Light Detection and Rang- ing (LIDAR), all ROS topics shown here are obtained through MENWAY IENWAY 21178 ITENWAY 100% AUTONOMOUS | 100% ELECTRIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='sensor fusion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=', through techniques such as Extended Kalman Filter (EKF)) and provide the relevant information from a wireless perspective: position, orientation and speed of the AGV and location of walls and obstacles in different formats (2D, 3D, offline and online, near and far).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' The raw input for the sensor fusion comes from sources including pure wheel odometry, drive commands, an Inertial Measurement Unit (IMU) and the already mentioned LIDAR, all of them also available in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Pre-Processing Similar to what was described in Section II-C2 the data was pre-processed to further simplify the work with the collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Moreover, and since the position of the base station is known and fixed, the distance and clearance of the wireless link can be easily inferred from the sensor data and is provided as part of the pre-processed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' We have merged the GPS logs, the LTE stack measurements and the throughput measurement together with the sensor- based link distance and link clearance into a single dataframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' As these data streams have different sampling frequencies, we re-sampled as needed to 1 second before the final merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
282
+ page_content=' CONCLUSION In this paper, we have describe industrial Vehicle-to-vehicle (iV2V) and industrial Vehicle-to-infrastructure plus Sensor (iV2I+), two testbeds for wireless communications in indus- trial settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
283
+ page_content=' We have provided detailed information about the components of the testbeds, together with the initial concept and captured scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
284
+ page_content=' As mentioned before, the described datasets are publicly available for ML research, what we consider a valuable contribution to the available industrial datasets both in terms of size and quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
285
+ page_content=' iV2V and iV2I+ contain extensive and complete data that we believe to be highly useful to answer questions regarding the use and generalisation of ML for mobile use cases in industrial environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
286
+ page_content=' Indeed, both datasets can be used to train and evaluate methods for pQoS, which is a crucial enabler for high reliability in wireless industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
287
+ page_content=' Moreover, pQoS in general, and our data in particular can be used as an ingredient for ML algorithms that optimize the network itself, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
288
+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
289
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290
+ page_content=' This type of new network capabilities will be an important part of the evolution of wireless com- munications, such as the 6G cellular evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
291
+ page_content=' Finally, the addition of AGV sensor data and localization opens the gate to advanced techniques like fingerprinting or channel charting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+ page_content=' Accessed on: 2022-10-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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1
+ Adaptive Dynamic Global Illumination
2
+ SAYANTAN DATTA, McGill Unviversity, Canada
3
+ NEGAR GOLI, Huawei/AMD, Canada
4
+ JERRY ZHANG, Huawei, Canada
5
+ 𝑑 = start
6
+ 𝑑 = start + 4𝑠
7
+ 𝑑 = start + 8𝑠
8
+ Direct only
9
+ Direct + Indirect
10
+ Tunnel interior (D + I)
11
+ SSIM/MSE:
12
+ 0.989/0.007
13
+ SSIM/MSE:
14
+ 0.974/0.008
15
+ SSIM/MSE:
16
+ 0.970/0.008
17
+ SSIM/MSE:
18
+ 0.957/0.008
19
+ SSIM/MSE:
20
+ 0.942/0.009
21
+ SSIM/MSE:
22
+ 0.939/0.010
23
+ SSIM/MSE:
24
+ 0.993/0.000
25
+ SSIM/MSE:
26
+ 0.953/0.003
27
+ SSIM/MSE:
28
+ 0.932/0.004
29
+ SSIM/MSE:
30
+ 0.837/0.003
31
+ SSIM/MSE:
32
+ 0.826/0.009
33
+ SSIM/MSE:
34
+ 0.875/0.010
35
36
37
+ Fig. 1. Our technique demonstrated on a modified Bistro Exterior scene containing 192 Γ— 64 Γ— 192 probes.
38
+ The third row shows the changes inside the tunnel as the gate opens over time. Our techniques responds faster
39
+ to a dynamic stimuli and offers 1.7-times higher performance compared to the Q-DDGI implementation even
40
+ with large probe grid containing excess of 2.3 million probes. Q-DDGI, detailed in section 5, is an extension of
41
+ vanilla DDGI making it more competitive and comparable against our approach.
42
+ We present an adaptive extension of probe based global illumination solution that enhances the response
43
+ to dynamic changes in the scene while while also enabling an order of magnitude increase in probe count.
44
+ Our adaptive sampling strategy carefully places samples in regions where we detect time varying changes in
45
+ radiosity either due to a change in lighting, geometry or both. Even with large number of probes, our technique
46
+ robustly updates the irradiance and visibility cache to reflect the most up to date changes without stalling
47
+ the overall algorithm. Our bandwidth aware approach is largely an improvement over the original Dynamic
48
+ Diffuse Global Illumination while also remaining orthogonal to the recent advancements in the technique. 1 2
49
+ CCS Concepts: β€’ Computing methodologies β†’ Ray tracing; Rasterization.
50
+ Additional Key Words and Phrases: Adaptive sampling, irradiance probes, global illumination, real-time
51
+ 1
52
+ INTRODUCTION
53
+ Global illumination (GI) strikingly improves the realism of a virtual scene, but its high computational
54
+ cost has been a long-standing challenge in its application to real-time rendering [22].
55
+ 1Project Page
56
+ 2Poster
57
+ 1
58
+ arXiv:2301.05125v1 [cs.GR] 12 Jan 2023
59
+
60
+ High Performance Graphics, Poster, July 11–14, 2022,
61
+ Datta et al.
62
+ Several real-time GI solutions have been proposed, such as screen space [43] techniques, which
63
+ support fully dynamic scenes but suffer from quality issues due to the limited availability of
64
+ information in screen space. On the other hand, baked texture light-maps only support static
65
+ geometry but remain popular due to their simplicity, low run-time cost, and quality. Precomputed
66
+ Radiance Transfer [51] combined with light probes [31] and light-maps [15] solved some of the issues
67
+ plaguing static light maps; in particular, these approaches support semi-dynamic geometry and self-
68
+ occlusion while adhering to a strict compute budget. The advent of real-time ray-tracing hardware
69
+ set the stage for modern fully dynamic GI. Dynamic real-time GI methods build upon the decades
70
+ of research in sampling, and amortization of shading and visibility across space (pixel/world), angle,
71
+ and time to improve convergence [40]. Adaptations of several offline techniques such as photon
72
+ mapping [17], many-light rendering [20, 62], and radiosity maps [54] have also been explored
73
+ in the context of modern [26, 27] ray-tracing capable hardware. However, presence of noise in
74
+ sampled algorithms require the use of strong denoisers. Machine learning denoisers [6, 66] have
75
+ demonstrable advantages in terms of quality compared to more traditional frequency [32] or
76
+ variance [46] based denoisers. However, the prospect of training a neural network, the added
77
+ complexity of integrating machine learning inference with traditional graphics pipeline, and the
78
+ proprietary nature of machine learning frameworks have stalled the industry-wide adoption of
79
+ these techniques. The recent probe-based algorithm, Dynamic Diffuse Global Illumination (DDGI)
80
+ [28], extending the classic irradiance probes, still remains an excellent choice due to its relative
81
+ simplicity, quality, and cloud streaming capabilities [14, 53]. However, scaling of DDGI in its original
82
+ formulation is limited, and approaches such as multi-grid hierarchy and probe rolling [29] are
83
+ necessary to scale it across large environments. Our adaptive approach focuses on dynamic contents
84
+ in environments containing millions of probes in a single hierarchy.
85
+ We propose Adaptive Dynamic GI (ADGI) algorithm where we trace a few pilot rays per frame to
86
+ scan the environment and build a coarse representative model of the dynamic events. Using Markov-
87
+ Chain sampling, we dynamically allocate resources to the critical areas, improving convergence in
88
+ those regions. While DDGI allocates a fixed number of samples per probe and uniformly distributes
89
+ samples across directions, ADGI non-uniformly samples the joint spatio-angular domain of the
90
+ discretized 5D light-field represented by the probes. Our approach essentially decouples resource
91
+ allocation from the number of probes resulting in a user-controlled performance target (FPS)
92
+ and improved scaling even with millions of probes. Additionally, our approach results in faster
93
+ convergence in static and dynamic environments given equal render time. Our approach is drop-in
94
+ compatible with the original implementation and its several other extensions such as probe rolling
95
+ and probe volume hierarchies [29].
96
+ We achieve these objectives by formulating a guided function approximation technique, which is
97
+ purposefully accurate in specific regions highlighted by our guiding function and thus eliminates
98
+ the need for uniform resource allocation. Furthermore, we develop a sampling methodology based
99
+ on temporal Markov-chain, which adapts naturally to a dynamic environment while also enabling
100
+ scaling across large number of probes. Finally, we discuss memory and bandwidth preserving color
101
+ compression schemes tailored specifically for our purpose.
102
+ 2
103
+ RELATED WORK
104
+ Probe-base approaches: Modern games rely extensively on light probes for static and dynamic
105
+ global illumination due to their ease of integration into the game engine pipeline at low run-time
106
+ cost. Some advocate a uniform grid probe placement due to their simplicity while others have
107
+ proposed non-unform probes due to their efficiency. Probe based techniques are usually prone to
108
+ light leakage. As such, uniform grid approaches [28, 31] use additional information, stored in the
109
+ probes to determine whether a probe is visible from a shade point. Non-uniform approaches may
110
+ 2
111
+
112
+ Adaptive Dynamic Global Illumination
113
+ High Performance Graphics, Poster, July 11–14, 2022,
114
+ Guiding function
115
+ β„Ž(π‘₯) = 𝑝 (π‘₯) Γ— 𝑙 (π‘₯)
116
+ 𝑝 (π‘₯)
117
+ 𝑙 (π‘₯)
118
+ π‘₯
119
+ π‘₯
120
+ State of
121
+ Environment
122
+ Sample
123
+ Feedback
124
+ (a) Construct the guide β„Ž(π‘₯).
125
+ π‘₯
126
+ Metropolis sampling
127
+ π‘₯𝑖
128
+ β„Ž(π‘₯)
129
+ (b) Sample π‘₯𝑖 ∼ β„Ž(π‘₯).
130
+ π‘₯
131
+ π‘₯𝑖
132
+ 𝑔(π‘₯)
133
+ (c) Evaluate objective 𝑔(π‘₯𝑖).
134
+ π‘₯
135
+ π‘₯𝑖
136
+ πΈπ‘Ÿ
137
+ 𝑔
138
+ ^𝑔
139
+ ^𝑔(π‘₯)
140
+ (d) ^𝑔(π‘₯) - Reconstruct 𝑔(π‘₯)
141
+ from 𝑔(π‘₯𝑖). πΈπ‘Ÿ indicates the
142
+ reconstruction error.
143
+ Fig. 2. Figure showing the steps in our adaptive-sampling strategy. We define a guiding function β„Ž(π‘₯) that
144
+ highlights (in yellow) the interesting regions of the domain. The samples π‘₯𝑖 obtained from β„Ž(π‘₯) are used to
145
+ evaluate the objective 𝑔(π‘₯𝑖). Our goal is to obtain an approximate representation of 𝑔(π‘₯), denoted as ^𝑔(π‘₯),
146
+ from the (π‘₯𝑖,𝑔(π‘₯𝑖)) pairs. As more samples are obtained from the highlighted region, the reconstruction error
147
+ is lower in the yellow area, as shown in sub-figure (d).
148
+ use carefully curated probe placement [63] combined with spatial data-structures like octrees to
149
+ determine the visibility of a probe from a surfel. McGuire et al. [28, 31] stores the depth values
150
+ of the surrounding geometry from a probe and use a similar idea as Variance-Shadow-Mapping
151
+ [9] to approximate visibility. However, non-uniform approaches has been mostly limited to static
152
+ geometry due to their high initial construction cost. Some approaches use rasterization [31, 63]
153
+ while other may use ray-tracing [28] to compute the probe content. Probe based techniques also
154
+ differ on how they store the information in the probes. Some use discrete textures [28, 31] while
155
+ other may use a compressed basis representation such as Spherical Harmonics [14, 55]. Spherical
156
+ harmonics implicitly pre-filters the content before storage but may cause light and dark ringing
157
+ issues. Memory bandwidth required for reading and writing from the probes is also a major concern.
158
+ Texture compression [31, 53] is usually the preferred choice to minimize memory bandwidth.
159
+ Bandwidth is also crucial for cloud streaming of probe data. In such scenarios, Spherical Harmonics
160
+ [14] representation may be preferable as they provide excellent compression for low frequency
161
+ data. At run-time, dynamic probe based [28] GI solutions uniformly distributes rays across probes
162
+ to update their content; this quickly becomes a bottleneck as the number of probes increases. Our
163
+ approach on the other hand, focuses on the optimal distribution of resources to maximize visual
164
+ fidelity. Various extensions have also been proposed to increase scalablity [29] of uniform grid
165
+ approaches such as multiple-volume hierarchies and probe rolling. Our approach remains largely
166
+ orthogonal and fully compatible with these extensions.
167
+ Adaptive sampling: Adaptive sampling has been used in the context of screen-space ray-traced
168
+ global illumination where more samples are accumulated in regions with high noise and high
169
+ frequency [12]. Adaptive sampling is also useful for filtering soft shadows [32], where pilot-rays
170
+ model the spatial frequency of shadow-penumbra and provide the number of additional samples
171
+ required at each pixel to improved convergence. Neural versions [13] of adaptive sampling has
172
+ also been proposed where a neural network generates a sampling-map that is tightly coupled to a
173
+ post-process neural-denoiser. Conceptually our approach is similar, but our execution is tailored
174
+ for the problem of temporally coherent sampling of probes. We refer readers to section 8 for an
175
+ extend related work in irradiance-caching, screen-space GI and MCMC techniques
176
+ 3
177
+
178
+ High Performance Graphics, Poster, July 11–14, 2022,
179
+ Datta et al.
180
+ 3
181
+ OVERVIEW
182
+ We focus on two primary issues with DDGI in its original formulation. First, the technique does
183
+ not allow for the non-uniform allocation of resources, resulting in unnecessary probe updates in
184
+ regions that are not crucial for visual fidelity. Seconds, it does not update the probes quick enough
185
+ to reflect transient changes in the scene environment. Our adaptive strategy involves detecting
186
+ the changes in the environment and allocating resources driven by the detected changes. While
187
+ the detection phase requires allocating additional resources, our empirical evaluations suggest our
188
+ non-uniform adaptive sampling compensates for the lost efficiency in the detection phase. Our
189
+ detection phase also enables fast probe updates for capturing transient changes in the scene. We
190
+ model our technique as guided function approximation where we approximate a continuous function
191
+ (e.g. 5D light-field) using a discrete (e.g. probes) representation driven by a guiding function.
192
+ A naive approach to approximate a continuous function is to discretize the domain and reserve a
193
+ representative sample for each discretization. The strategy is useful when the domain is relatively
194
+ small; however, as the domain gets larger or the number of discretizations increases, it is prohibi-
195
+ tively expensive to update all discretizations in real-time. This is one of the issues plaguing the
196
+ original DDGI technique. In many applications, it is not necessary to update the entire domain
197
+ uniformly; instead, we can tolerate more approximation errors in some regions than others. A
198
+ simple example is foveated rendering, where errors in the periphery are less intrusive than those
199
+ near the gaze center. In our case, we need the most accuracy in probes contributing to final shading.
200
+ We introduce the notion of guiding function, which highlights the regions where a higher
201
+ reconstruction accuracy is desired. We define the guide using a product of terms - the first term
202
+ represents the current state of the environment while the second term is a feedback from the
203
+ sampled cache. We sample the guide using a temporally coherent Markov-chain and use the
204
+ samples to update our approximate representation using a parallel thread-safe approach. Thus our
205
+ approach is summarized in three steps - defining a guiding function, sampling the guide, and using
206
+ the samples to update the approximate representation. We describe these steps in sections 3.2, 3.3
207
+ and 3.4 while we discuss various implementation specific details in section 4. See figure 2.
208
+ Our approach provides two distinct advantages compared to the original DDGI - approximation
209
+ quality and scalability. At any time, we concentrate our resources on a potentially challenging
210
+ area as opposed to the entire domain. Provided our guide correctly identifies the challenging
211
+ regions, the quality is improved due to a higher concentration of resources in the appropriate
212
+ region. Since we sample the guide independent of the number of discretizations, the decoupling
213
+ allows for a high number of statically allocated probes without affecting run-time performance.
214
+ Increased discretizations improve approximation quality while the independence of sampling from
215
+ the number of discretizations improves scalability. More specifically, we transparently increase
216
+ the number of discrete probes without affecting performance. The run-time performance depends
217
+ on the number of samples we generate; the samples are channeled to the appropriate areas by the
218
+ guiding function. Our Markov-chain sampling is highly parallel, temporally coherent, and scalable,
219
+ making it suitable for real-time temporally distributed reconstruction of large probe grids.
220
+ 3.1
221
+ Background
222
+ Here we briefly describe the original DDGI algorithm. DDGI consists of a 3D grid of directionally
223
+ resolved irradiance probes that are updated in real-time through hardware ray-tracing. The probes
224
+ also contains visibility information to prevent light leakage. The probe representation has many
225
+ benefits, it performs optimally for diffuse indirect transport and is relatively inexpensive to encode
226
+ and decode information to and from the probes. The algorithm evenly distributes ray-samples
227
+ outwards from the probe center at each active probe in a stochastic rotated spiral pattern. DDGI is
228
+ 4
229
+
230
+ Adaptive Dynamic Global Illumination
231
+ High Performance Graphics, Poster, July 11–14, 2022,
232
+ Uniform
233
+ probe grid
234
+ placement
235
+ Generating &
236
+ tracing rays
237
+ Evenly distribute
238
+ Probe state update
239
+ Update irr &
240
+ vis 2𝐷 atlas
241
+ Shade each
242
+ point
243
+ 8 cage probe
244
+ Fig. 3. The figure illustrates the main steps of DDGI algorithm. Algorithm defines a uniform grid of probes
245
+ and trace uniform-random rays in all direction from each probe. Based on the hit information, we compute
246
+ the visibility (vis) and irradiance (irr) and update the 2𝐷 atlas. We also update the probe states based on
247
+ visibility information (back-face hit ratio). Finally, for each shade-point, we query the eight bounding probes
248
+ surrounding it and interpolate them to compute incoming indirect illumination.
249
+ a two step algorithm. First, it updates the shading on the probe texels. Next a screen-space pass
250
+ where the up-to-date probe content is used for shading the camera-pixels. The probe texel values
251
+ are encoded into a spherical-mapped diffuse irradiance-texture with 8 Γ— 8 resolution. Probes also
252
+ captures the average ray-hit distance, and squared distances to the nearest geometry at 16 Γ— 16
253
+ resolution. DDGI temporally filters the probe texels by blending in the new values using a fixed
254
+ hysteresis. The visibility data is used to decide whether a probe is visible at a shade-point and also
255
+ used to infer whether a probe is inside a geometry and deactivated. The probe’s state is not limited
256
+ to on or off and can vary with scenarios [29]. The world-position of the screen-space pixel is used
257
+ as a key to the probe-texture lookup. The lookup interpolates the corresponding eight probes of
258
+ the grid voxel containing the shade-point. The algorithm is illustrated in figure 3. DDGI algorithm
259
+ is suitable for diffuse and slow changing phenomena in time. Therefore DDGI, combined with our
260
+ adaptive-sampling strategy is a reasonable real-time GI approximation for dynamic scenes.
261
+ 3.2
262
+ Guiding function
263
+ As summarised in section 3 and figure 2, a guiding function highlights the important areas in
264
+ the domain, i.e., challenging regions where more resources are required. These highlighted areas
265
+ receive more adaptive samples, reducing the approximation error in those regions. Mathematically,
266
+ the domain of the guiding function β„Ž : 𝑅𝑑 β†’ 𝑅 is the continuous 5D light field. Upon query, the
267
+ guide function returns a scalar value indicating the importance of a sampled point. In our case,
268
+ 𝑑 = 5 as the domain is a 5-dimensional space of world-space positions and directions, and the guide
269
+ encodes the importance of sampling a direction on a probe (texel’s importance).
270
+ We model the guiding function (β„Ž) as a product of two terms. The first term, we call 𝑓 : 𝑅𝑑 β†’ 𝑅,
271
+ represents the value in sampling a texel based on our understanding (limited) of whether such a
272
+ texel would contribute towards the final screen-space shading. The second term is the observed
273
+ sampled evidence (a.k.a irradiance cache) as they become available. Initially, the irradiance cache is
274
+ empty but filled progressively through sampling. We define the first term based on some heuristics
275
+ that describes our understanding of the probe-environment:
276
+ β€’ Probes closer to the camera,
277
+ β€’ Probes closer to geometric surfaces,
278
+ β€’ Directions on the probes facing away from geometric surfaces,
279
+ β€’ Directions on the probe with higher incoming irradiance,
280
+ β€’ Directions with temporal change in irradiance and visibility
281
+ We trace pilot rays from the probes to generate the information necessary to quantify the above
282
+ heuristics. We also call it the detection phase where we pre-scan the scene environment for changes.
283
+ We denote the individual heuristics as 𝑓𝑖 : 𝑅𝑑 β†’ 𝑅, and compose them into its final form 𝑓 as
284
+ shown in equation 1, where πœ™ represents a composition function. The composition function is
285
+ 5
286
+
287
+ High Performance Graphics, Poster, July 11–14, 2022,
288
+ Datta et al.
289
+ Algorithm 1: Metropolis algorithm
290
+ Input: β„Ž: Guide distribution, 𝑀 : No. of iterations
291
+ Input: 𝐾: No. of initial samples to reject
292
+ Output: π‘₯ : Sample
293
+ Ensure: 𝑀 β‰₯ 2, and 𝐾 < 𝑀
294
+ 1 𝑗 ← ShaderInvocationIndex()
295
+ 2 π‘₯0 ← 𝑆[𝑗]
296
+ // Initialize Markov-chain from memory
297
+ 3 while 𝑖 ← 0 to 𝑀 βˆ’ 1 do
298
+ 4
299
+ π‘₯𝑖+1 ← RandomWalk (π‘₯𝑖,β„Ž(π‘₯𝑖))
300
+ // Random walk step, algorithm 5
301
+ 5
302
+ if 𝑖 > 𝐾 then
303
+ /* Use sample π‘₯𝑖+1 for probe updates, see algorithm 2
304
+ */
305
+ 6 𝑆[𝑗] ← π‘₯𝑖 + 1
306
+ // Save Markov-chain state
307
+ simply a recipe to appropriately combine the individual heuristics. We quantify the individual
308
+ heuristics (𝑓𝑖) in section 4.1 and the composition (πœ™) in section 4.2.
309
+ 𝑓 = πœ™(𝑓0, 𝑓1, ..., 𝑓𝑖).
310
+ (1)
311
+ The second term uses the stored irradiance in the probes, denoted by ^𝑔, to modulate the first
312
+ term. We model the second term as - 𝑒π‘₯𝑝 (𝛼 Β· ^𝑔(π‘₯)/𝑓 (π‘₯)), where the scalar 𝛼 ∈ [0, ∞) indicates our
313
+ confidence in the irradiance probe content; a higher value indicating greater confidence. Note that a
314
+ stored texel with high irradiance value may or may not have a high contribution to the final shading.
315
+ Example - in a dynamic environment the probe content from the last frame is quickly outdated
316
+ and thus less useful. The parameter 𝛼 models this uncertainty. The term 𝑓 (π‘₯) in the denominator
317
+ ensures that we only trust ^𝑔(π‘₯) when 𝑓 (π‘₯) is low. Finally, we define the guiding function as:
318
+ β„Ž(π‘₯) = 𝑒π‘₯𝑝
319
+ οΏ½
320
+ 𝛼 Β· ^𝑔(π‘₯)
321
+ 𝑓 (π‘₯)
322
+ οΏ½
323
+ Β· 𝑓 (π‘₯).
324
+ (2)
325
+ 3.3
326
+ Sampling the guide
327
+ Next we sample the guiding function (equation 2). Mathematically, given an unnormalized distribu-
328
+ tion β„Ž : 𝑅𝑑 β†’ 𝑅, our goal is to obtain samples π‘₯𝑖 from β„Ž(π‘₯), where π‘₯𝑖 ∈ 𝑅𝑑.
329
+ Our sampling algorithm is straightforward. We use the Metropolis sampling, as shown in al-
330
+ gorithm 1 to sample β„Ž. The algorithm randomly initializes a state (π‘₯0 ∈ 𝑅𝑑) and moves the state
331
+ forward based on the acceptance of a newly proposed state. We generate the proposed states by
332
+ perturbing the current state with a zero-mean Gaussian noise, also known as Random-walk [5].
333
+ Parallelism: Note that algorithm 1 runs as a shader invocation, meaning several instances of
334
+ the chain run in parallel. Each instance is independent with its own memory to load and store the
335
+ chain state (denoted by S[] in algorithm 1). The instances generate thousands of samples per frame.
336
+ As an input to our algorithm, we explicitly specify the number of chains that run in parallel, thus
337
+ controlling the number of adaptive samples and performance. Contrasting with the original DDGI,
338
+ the number of samples in the original implementation is proportional to the number of probes
339
+ which increases cubically with scene dimensions. As such, it is difficult to scale up when the scene
340
+ gets larger or when using a denser probe grid. Our approach is independent of the discretization
341
+ resolution and scales better to higher probe counts without compromising approximation quality.
342
+ Mixing-time: Initially, a Markov chain requires many iterations for the chain to generate
343
+ samples from the target distribution (here β„Ž(π‘₯)), a phenomenon known as mixing time. We avoid
344
+ 6
345
+
346
+ Adaptive Dynamic Global Illumination
347
+ High Performance Graphics, Poster, July 11–14, 2022,
348
+ Table 1. List of symbols
349
+ Symbol
350
+ Description
351
+ Remarks
352
+ 𝑓
353
+ Heuristics model
354
+ Section 3.2
355
+ β„Ž
356
+ Guiding function/Target distribution
357
+ Section 3.2, 3.3
358
+ 𝑔
359
+ Objective function
360
+ Symbolic proxy for π‘”π‘Ÿ, 𝑔𝑐.
361
+ Section 3.4
362
+ ^𝑔
363
+ Approximation of objective function
364
+ Symbolic proxy for ^π‘”π‘Ÿ, ^𝑔𝑐.
365
+ Section 3.4
366
+ π‘”π‘Ÿ
367
+ 5D Light field
368
+ Section 3.4
369
+ 𝑔𝑐
370
+ Chebychev visibility
371
+ Section3.4
372
+ ^π‘”π‘Ÿ
373
+ Approximation of 5D light field
374
+ (Irradiance cache)
375
+ Section 3.4, 4.5
376
+ ^𝑔𝑐
377
+ Approximation of Chebychev visibility
378
+ (Visibility cache)
379
+ Section 3.4, 4.6
380
+ π‘₯ or π‘₯𝑖
381
+ Markov-chain samples
382
+ Symbolic proxy for 𝑝𝑖, πœ”π‘–.
383
+ Section 3.3
384
+ 𝑝𝑖
385
+ Positional (∈ 𝑅3) component of π‘₯𝑖
386
+ –
387
+ πœ”π‘–
388
+ Directional (∈ 𝑅2) component of π‘₯𝑖
389
+ –
390
+ this problem by bootstrapping the initial chain state from the last frame. As such, we keep the
391
+ number of iterations per frame small, but over frames, the chain effectively accrues many iterations.
392
+ Distribution stationarity: Markov chain sampling requires the target distribution β„Ž(π‘₯) remain
393
+ stationary. Due to a dynamic scene environment, the stationarity condition is seemingly violated.
394
+ This may affect the approximation quality of our technique if the distribution changes rapidly
395
+ between frames. However, we have several contingencies to deal with the issue. First, we target high
396
+ frame-rates, which minimizes the change in the target distribution between consecutive frames.
397
+ As an additional margin of safety, we reject initial 𝐾 samples per frame as shown in algorithm 1,
398
+ line 5. This ensures our usable samples are obtained closer to the target distribution. Note that
399
+ the evaluation time for β„Ž(π‘₯) negligible and thus rejecting few initial samples per frame does not
400
+ significantly impact performance. We also smooth out the target distribution (see section 4.1.4)
401
+ using spatio-temporal convolution to minimize abrupt changes in the target across frames.
402
+ Temporal tracking: Since our target distribution may vary with time, we require the samples
403
+ generated from the Markov-chain to closely follow the distribution to capture the transient changes
404
+ in the environment. We make some crucial modifications to our sampling algorithm to allow for
405
+ fast tracking of the target distribution, which we discuss in detail in section 4.9.
406
+ 3.4
407
+ Approximation
408
+ With samples obtained from the highlighted (figure 2(b)) parts of the domain, we focus on using
409
+ the samples to evaluate (figure 2(c)) and reconstruct (figure 2(d)) our objective function. The term
410
+ objective function refers to the quantity we aim to approximate. Mathematically, we denote our
411
+ objective function as 𝑔 : 𝑅𝑑 β†’ 𝑅𝑐, and its approximate reconstruction as ^𝑔. For ADGI, we have
412
+ two objective functions - the light field π‘”π‘Ÿ : 𝑅5 β†’ 𝑅3, and Chebychev-visibility 𝑔𝑐 : 𝑅5 β†’ 𝑅2
413
+ surrounding the probes. We denote their approximate reconstructions as the irradiance cache ^π‘”π‘Ÿ,
414
+ and the visibility cache - ^𝑔𝑐 respectively. See section 4.5 and 4.6 for more details.
415
+ Updating ^g: We evaluate the continuous objective function 𝑔 at collected sample points π‘₯𝑖 and
416
+ store the evaluations - 𝑔(π‘₯𝑖) into ^𝑔, as shown in algorithm 2. For ADGI, the evaluation step involves
417
+ 7
418
+
419
+ High Performance Graphics, Poster, July 11–14, 2022,
420
+ Datta et al.
421
+ Algorithm 2: Approximation algorithm
422
+ Input: π‘₯: Markov-chain samples
423
+ 1 function UpdateRepresentation(π‘₯):
424
+ 2
425
+ 𝑣 ← 𝑔(π‘₯)
426
+ // Evaluate sample, ray-trace
427
+ 3
428
+ AtomicMovingAvg(π‘₯, 𝑣)
429
+ // Populate ^𝑔, see algorithm 4
430
+ tracing a ray to query the local light field and visibility. At each Metropolis iteration, the evaluated
431
+ samples update the closest entry in the probes (^𝑔) within a critical section construct.
432
+ Representing ^g: Prior work represent ^𝑔 as either as discrete LUTs [28], continuous Spherical
433
+ Harmonics [14], Neural Networks [36], or any combination. In our case, the choice to use a discrete
434
+ representation is based on several factors. First, multiple parallel streams of Markov-chain samples
435
+ may update the same memory location in ^𝑔. As such, provisions are necessary to prevent race
436
+ conditions. We also need a representation that handles temporal accumulation and quickly update
437
+ itself to reflect any transient changes in the scene. Finally, the representation must be bandwidth
438
+ efficient to improve the read and write performance. We refer to section 4.5 and 4.9 for details.
439
+ 3.5
440
+ MCMC analysis
441
+ In this section, we analyze our adaptive sampling algorithm in the context of MCMC (Markov
442
+ Chain Monte Carlo). Note that our goal is not variance reduction through importance sampling;
443
+ rather the focus is guided approximation of the objective function via sampling the target function.
444
+ As such, unlike importance sampling, the sampling function is not necessarily correlated to the
445
+ integrand. With this distinction in mind, we first look at the equation driving importance sampling
446
+ using MCMC and then repurpose it for guided function approximation.
447
+ The following equation shows a typical case of importance sampling where the objective is to
448
+ compute the integral
449
+ ∫
450
+ β„Ž(π‘₯)𝑔(π‘₯)𝑑π‘₯ and there exists a strategy to sample from h(x). In many typical
451
+ scenarios (e.g. full Bayesian inference), the distribution β„Ž(π‘₯) is a proper distribution (
452
+ ∫
453
+ β„Ž(π‘₯)𝑑π‘₯ = 1)
454
+ but does not have an efficient sampling mechanism. This where Markov Chain MC is useful.
455
+ ∫
456
+ β„Ž(π‘₯)𝑔(π‘₯)𝑑π‘₯ β‰ˆ
457
+ οΏ½
458
+ 1
459
+ 𝑀
460
+ π‘€βˆ’1
461
+ βˆ‘οΈ
462
+ 𝑖=0
463
+ 𝑔(π‘₯𝑖)
464
+ � ∫
465
+ β„Ž(π‘₯)𝑑π‘₯, π‘₯𝑖 ∼ β„Ž(π‘₯).
466
+ (3)
467
+ In contrast, our choice of Markov Chain (Metropolis) is primarily technical - simplicity, GPU
468
+ parallelism and temporal sample tracking. Nevertheless, the same equations provide meaningful
469
+ insight - albeit in a different context of adaptive sampling. In our algorithm, we simply sum the
470
+ samples obtained from the target distribution without taking into account the sample density. This
471
+ is equivalent to computing the following:
472
+ 𝐼 = 1
473
+ 𝑀
474
+ π‘€βˆ’1
475
+ βˆ‘οΈ
476
+ 𝑖=0
477
+ 𝑔(π‘₯𝑖), π‘₯𝑖 ∼ β„Ž(π‘₯).
478
+ (4)
479
+ While our goal is to estimate
480
+ ∫
481
+ Ξ© 𝑔(π‘₯)𝑑π‘₯, the expectation of 𝐼 (rearranging equation 3) is:
482
+ E [𝐼] =
483
+ ∫
484
+ Ξ© β„Ž(π‘₯)𝑔(π‘₯)𝑑π‘₯
485
+ ∫
486
+ Ξ© β„Ž(π‘₯)𝑑π‘₯
487
+ ,
488
+ (5)
489
+ where Ω is the domain of integration. Clearly, the expected value of 𝐼 does not converge to the
490
+ correct estimate -
491
+ ∫
492
+ Ξ© 𝑔(π‘₯)𝑑π‘₯. However, there are two factors to consider - size of the domain Ξ© and
493
+ 8
494
+
495
+ Adaptive Dynamic Global Illumination
496
+ High Performance Graphics, Poster, July 11–14, 2022,
497
+ shape of β„Ž(π‘₯) in the domain. First consider the limit case where Ξ© β†’ 0. In this case, the integrals
498
+ collapses to a point evaluation and indeed the expected value of 𝐼 equals the unbiased estimate as
499
+ shown below.
500
+ 𝐿.𝐻.𝑆. = lim
501
+ Ξ©β†’0
502
+ ∫
503
+ Ξ© β„Ž(π‘₯)𝑔(π‘₯)𝑑π‘₯
504
+ ∫
505
+ Ξ© β„Ž(π‘₯)𝑑π‘₯
506
+ =
507
+ ∫
508
+ Ξ© β„Ž(π‘₯)𝑔(π‘₯)𝛿(π‘₯ βˆ’ π‘₯0)𝑑π‘₯
509
+ ∫
510
+ Ξ© β„Ž(π‘₯)𝛿(π‘₯ βˆ’ π‘₯0)𝑑π‘₯
511
+ = 𝑔(π‘₯0).
512
+ (6)
513
+ 𝑅.𝐻.𝑆. = lim
514
+ Ξ©β†’0
515
+ ∫
516
+ Ξ©
517
+ 𝑔(π‘₯)𝑑π‘₯ =
518
+ ∫
519
+ Ξ©
520
+ 𝑔(π‘₯)𝛿(π‘₯ βˆ’ π‘₯0)𝑑π‘₯ = 𝑔(π‘₯0).
521
+ (7)
522
+ In the above equation, 𝛿 is the Kronecker delta. The result is important as it shows with increasing
523
+ probe resolution, bias is reduced. However, reducing texel size is not always practical as more rays
524
+ and memory are required to populate and store a high resolution probe. Notice how the term β„Ž(π‘₯)
525
+ is cancelled in equation 6. When the domain of integration is sufficiently small, β„Ž(π‘₯) is practically
526
+ constant and the term cancels out in the denominator and numerator.
527
+ We now consider the shape of β„Ž(π‘₯). While the target h(x) varies globally, it is piece-wise constant
528
+ at a local scale due to its tabular nature. More crucially, the target β„Ž(π‘₯) is stored at a much lower
529
+ resolution compared to the irradiance probe ^𝑔(π‘₯). This implies β„Ž(π‘₯) is practically constant across a
530
+ texel of the irradiance probe. The expected value of 𝐼 for the π‘˜π‘‘β„Ž texel is thus given by:
531
+ E [πΌπ‘˜] =
532
+ ∫
533
+ π‘‡π‘˜ β„Ž(π‘₯)𝑔(π‘₯)𝑑π‘₯
534
+ ∫
535
+ π‘‡π‘˜ β„Ž(οΏ½οΏ½)𝑑π‘₯
536
+ =
537
+ ∫
538
+ π‘‡π‘˜ π‘π‘˜π‘”(π‘₯)𝑑π‘₯
539
+ ∫
540
+ π‘‡π‘˜ π‘π‘˜π‘‘π‘₯
541
+ =
542
+ ∫
543
+ π‘‡π‘˜ 𝑔(π‘₯)𝑑π‘₯
544
+ ∫
545
+ π‘‡π‘˜ 𝑑π‘₯
546
+ ,
547
+ (8)
548
+ where π‘‡π‘˜ represents the domain of π‘˜π‘‘β„Ž texel and π‘π‘˜ represents the piece-wise constant value of
549
+ β„Ž(π‘₯) when π‘₯ ∈ π‘‡π‘˜. The area estimate
550
+ ∫
551
+ π‘‡π‘˜ 𝑑π‘₯ is fixed for all texels and equivalent to 4πœ‹/#π‘Ÿπ‘’π‘ π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘›.
552
+ Thus, due to the tabular nature of our target function, the estimates of irradiance texels remain
553
+ un-biased. While performing texture filtering over irradiance texels, it is possible to compute an
554
+ unbiased estimate by weighing the texel values with π‘π‘˜ as follows:
555
+ 𝐼 𝑓 π‘–π‘™π‘‘π‘’π‘Ÿ
556
+ π‘˜
557
+ =
558
+ βˆ‘οΈ
559
+ 𝑗 ∈Nπ‘˜
560
+ π‘€π‘˜βˆ’π‘—πΌπ‘˜βˆ’π‘—, 𝑀𝑖 = 𝑐𝑖/
561
+ βˆ‘οΈ
562
+ 𝑗 ∈Nπ‘˜
563
+ 𝑐 𝑗,
564
+ (9)
565
+ where Nπ‘˜ represents the texels in the neighbourhood of texel π‘˜. The values 𝑐𝑖 are obtained by
566
+ querying the probes storing β„Ž(π‘₯). Note that bias is unavoidable as we blend samples temporally
567
+ in a dynamic environment. In a dynamic environment, the objective is evolving and the bias
568
+ manifests itself as temporal lag. Practically however, within a small time window, both β„Ž(𝑑) and
569
+ 𝑔(𝑑) are assumed constant and the samples can be blended using a windowed moving average.
570
+ Note that windowed moving average requires storing historical information. A cheaper but biased
571
+ approximation to windowed moving average is exponential moving.
572
+ 4
573
+ IMPLEMENTATION DETAILS
574
+ This section provides the several implementation details with a brief summary in figure 4.
575
+ 4.1
576
+ Heuristics construction
577
+ The section describes the construction of 𝑓 using the heuristics discussed in section 3.2. Our goal is
578
+ to measure and quantify the heuristics that highlight the probes which actively contribute to the
579
+ final shading and require additional resources for faster convergence. We represent the heuristics
580
+ either parametrically (equation 10) or using an explicit LUT representation as shown in figure 5(a).
581
+ The LUT is constructed such that each probe has eight texels corresponding to an octant. We trace
582
+ a ray for each octant; the rays return the hit distance and incoming irradiance at the hit-point.
583
+ From this information, we compute several quantities (equation 11 - 18) and store them in the
584
+ 9
585
+
586
+ High Performance Graphics, Poster, July 11–14, 2022,
587
+ Datta et al.
588
+ World space
589
+ Trace 8
590
+ pilot rays
591
+ Octahedral
592
+ Quantify heuristics
593
+ πœ™ (𝑓0, 𝑓1, ..., 𝑓𝑖)
594
+ Model feedback
595
+ 𝑒π‘₯𝑝 (𝛼 Β· ^𝑔/𝑓 )
596
+ Guide function
597
+ 𝑝 (π‘₯) Γ— 𝑙 (π‘₯)
598
+ Metropolis sampling
599
+ β„Ž(𝑝,πœ”)
600
+ (𝑝𝑖,πœ”π‘–)
601
+ Irradiance
602
+ Visibility
603
+ πœ”π‘–
604
+ βˆ’πœ”π‘–
605
+ 𝑝𝑖
606
+ 𝑝𝑖
607
+ Trace ray
608
+ Update irradiance, visibility
609
+ cache
610
+ Irradiance ( ^π‘”π‘Ÿ )
611
+ 8 Γ— 8
612
+ Visibility ( ^𝑔𝑐)
613
+ 16 Γ— 16
614
+ Deferred shading
615
+ Fig. 4. This figure illustrates our overall algorithm. We trace 8 pilot rays, one from each octant on the
616
+ probe and approximate the heuristic model 𝑓 (𝑝,πœ”). Using the heuristic and feedback, we define the guide
617
+ β„Ž(𝑝,πœ”) and sample it using Metropolis sampling. The sampled (𝑝𝑖,πœ”π‘–) are used to trace more adaptive ray
618
+ samples, gathering hit-distance and irradiance at the sample points. We update the probe-cache (^𝑔) with
619
+ adaptive-samples. The cache is used in the next shader and also looped back as feedback to model the target.
620
+ LUT/texture mapped to the probe octants. We define and evaluate the following heuristics for a
621
+ probe at position 𝑝 and a direction πœ”.
622
+ 4.1.1
623
+ Distance from camera. A probe far away from the camera is less likely to contribute to the
624
+ final shading. We represent this parametrically as described in equation 10, where 𝑝 represents
625
+ probe position, 𝑐 camera position and π‘˜ is a threshold set by the user.
626
+ 𝑓𝑐 (𝑝,πœ”) =
627
+ οΏ½
628
+ 1
629
+ if ||𝑝 βˆ’ 𝑐|| < π‘˜ ,
630
+ π‘’βˆ’( ||π‘βˆ’π‘ ||βˆ’π‘˜)
631
+ otherwise.
632
+ (10)
633
+ 4.1.2
634
+ Probe visibility. Only the probes encompassing a geometry participates in the deferred
635
+ shading. Thus, probes closer to a geometric surface are more important. Similarly, texels facing
636
+ away from the surface are queried more often for shading. We express both quantities together in
637
+ equation 11, where 𝑝 represents probe location and 𝑑 = π‘‘π‘Ÿπ‘Žπ‘π‘’(𝑝, βˆ’πœ”). The function trace returns
638
+ the distance of the nearest surface hit, and the scalar 𝑠 is the diagonal distance of a grid voxel.
639
+ 𝑓𝑣(𝑝,πœ”) = π‘’βˆ’2𝑑/𝑠
640
+ (11)
641
+ 4.1.3
642
+ Incoming radiance. We consider directions with high incoming radiance as more impor-
643
+ tant. To identify those directions, we query the radiance along each probe octant and use it as a
644
+ representative for incoming radiance.
645
+ π‘“π‘Ÿ (𝑝,πœ”) = π‘šπ‘–π‘›(π‘Ÿ, 𝛽)
646
+ 𝛽
647
+ ,
648
+ (12)
649
+ where π‘Ÿ = π‘™π‘’π‘š(𝑝,πœ”). The function lum returns the incoming luminance using direct illumination
650
+ at the surface hit point. The parameter 𝛽 controls the dynamic range and we set 𝛽 = 5.
651
+ 4.1.4
652
+ Probe visibility change . Detection of dynamic geometry is crucial for increased resource
653
+ allocation in regions affected by these changes. We detect dynamic geometry by computing a
654
+ temporal gradient of probe visibility followed by a spatio-temporal smoothing operation.
655
+ 𝑓0(𝑝,πœ”) = 𝑓 𝑑
656
+ 𝑣 (𝑝,πœ”) βˆ’ 𝑓 π‘‘βˆ’1
657
+ 𝑣
658
+ (𝑝,πœ”),
659
+ (13)
660
+ where 𝑓 𝑑
661
+ 𝑣 , 𝑓 π‘‘βˆ’1
662
+ 𝑣
663
+ represent visibility in the current and last time step respectively. Equation 13
664
+ implicitly states we keep the position and the direction fixed when measuring the time difference
665
+ across frames to avoid noisy gradients. The gradient is passed through a temporal trigger (π‘‡π‘Ÿ) as:
666
+ 10
667
+
668
+ Adaptive Dynamic Global Illumination
669
+ High Performance Graphics, Poster, July 11–14, 2022,
670
+ a. Probes storing prior-information (𝑓 ).
671
+ octant-mapped
672
+ pilot-rays
673
+ b. Irradiance cache ( ^π‘”π‘Ÿ ).
674
+ c. Visibility cache ( ^𝑔𝑐).
675
+ Fig. 5. Figure showing various probe-mapped textures and LUT in our technique.
676
+ 𝑓1(𝑝,πœ”) = π‘‡π‘Ÿ (𝑓0(𝑝,πœ”),πœƒ) ,
677
+ (14)
678
+ where π‘‡π‘Ÿ converts a pulse in time to a decaying signal controlled by the parameter πœƒ as shown
679
+ in figure 6(a). For simplicity, we drop the time axis from the function π‘‡π‘Ÿ. The function minimizes
680
+ temporal discontinuities, thus helping the Markov-chain to closely follow the target distribution
681
+ (β„Ž) across frames. Finally, we perform a spatial convolution as follows:
682
+ 𝑓Δ𝑣(𝑝,πœ”) =
683
+ βˆ‘οΈ
684
+ 𝑖,𝑗
685
+ 𝑓1(𝑝 βˆ’ 𝑝𝑖,πœ” βˆ’ πœ”π‘—).
686
+ (15)
687
+ The convolution step smooths out uncertainties in a single texel and also serves as a weak
688
+ predictor of possible locations of the dynamic geometry in the next frame. We use a 5 Γ— 5 Γ— 5 and
689
+ 3 Γ— 3 convolution in space and direction, respectively.
690
+ 4.1.5
691
+ Probe radiance change. Similar to the previous section, we detect a change in radiosity
692
+ using a temporal gradient of the probe radiance. We apply the same temporal trigger and spatial
693
+ convolution operator as in the previous section. The corresponding equations are as follows:
694
+ 𝑓2(𝑝,πœ”) = 𝑓 𝑑
695
+ π‘Ÿ (𝑝,πœ”) βˆ’ 𝑓 π‘‘βˆ’1
696
+ π‘Ÿ
697
+ (𝑝,πœ”),
698
+ (16)
699
+ 𝑓3(𝑝,πœ”) = π‘‡π‘Ÿ (𝑓2(𝑝,πœ”),πœƒ) ,
700
+ (17)
701
+ π‘“Ξ”π‘Ÿ (𝑝,πœ”) =
702
+ βˆ‘οΈ
703
+ 𝑖,𝑗
704
+ 𝑓3(𝑝 βˆ’ 𝑝𝑖,πœ” βˆ’ πœ”π‘—).
705
+ (18)
706
+ 4.2
707
+ Heuristics composition
708
+ Now that the individual heuristics are defined, as described in equation 1, we compose them for
709
+ the static and dynamic cases as follows:
710
+ 𝑓𝑠 (𝑝,πœ”) =
711
+ π‘ π‘‘π‘Žπ‘‘π‘–π‘
712
+ οΏ½οΏ½οΏ½οΏ½
713
+ 𝑓𝑐 𝑓𝑣 ,
714
+ (19)
715
+ 𝑓𝑑 (𝑝,πœ”) =
716
+ π‘‘π‘¦π‘›π‘Žπ‘šπ‘–π‘
717
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
718
+ 𝑓𝑐 𝑓𝑣(𝑓Δ𝑣 + πœ‡π‘“Ξ”π‘Ÿ) .
719
+ (20)
720
+ When the environment is static, we sample according to the camera and probe-to-surface distance
721
+ heuristics denoted by 𝑓𝑐 and 𝑓𝑣 in equation 19. In the dynamic case represented by equation 20, we
722
+ modulate the changes in the environment by the static term 𝑓𝑐 𝑓𝑣. The modulation indicates we are
723
+ more interested in changes close to the camera and geometric surfaces. The factor πœ‡ weighs the
724
+ strength of change in geometry versus change in lighting. We use πœ‡ = 2 in all our experiments.
725
+ 11
726
+
727
+ High Performance Graphics, Poster, July 11–14, 2022,
728
+ Datta et al.
729
+ π‘‡π‘Ÿ (𝑑; 𝑣,πœƒ)
730
+ 𝑣
731
+ Temporal-pulse
732
+ πœƒ
733
+ Linear decay
734
+ start
735
+ 𝑑
736
+ a. Transform temporal-pulse to a decaying signal.
737
+ 𝑓𝑐 β‰₯ 0.75
738
+ 𝑓𝑐 β‰₯ 0.5
739
+ |𝑐𝑙𝑖𝑝π‘₯𝑦 | ≀ 1.2
740
+ |𝑐𝑙𝑖𝑝π‘₯𝑦 | ≀ 1.4
741
+ b. Defining clip volumes for probes.
742
+ Fig. 6. Figure (a) shows the construction of temporal-triggerπ‘‡π‘Ÿ (𝑣,πœƒ). In figure (b), we call the volume bounded
743
+ by the blue frustum and black boundary as inner volume 𝑉𝑖𝑛. Similarly, outer volume π‘‰π‘œπ‘’π‘‘ is the volume
744
+ bounded by green frustum and outer grey boundary. All probes in π‘‰π‘œπ‘’π‘‘ participate in the heuristic modelling,
745
+ as described in section 4.4. Probes inside the blue frustum participate in adaptive sampling as described in
746
+ section 4.8, 4.9. We set the probe state 𝑁 = 16 for all probes outside 𝑉𝑖𝑛 but inside π‘‰π‘œπ‘’π‘‘, refer section 4.8.
747
+ 4.3
748
+ Heuristics storage
749
+ We store the quantities 𝑓𝑣, 𝑓𝑠, 𝑓𝑑 as a 6-10-10 bit encoded 32 bit integer at each octant of the probes.
750
+ The remaining 6 bits are used for other flags. When querying the LUT/texture, we use a mapping
751
+ function that maps the continuous position 𝑝 and direction πœ” to the corresponding texel in the
752
+ LUT. We note that 𝑓𝑐 is implicitly defined, hence do not require additional storage.
753
+ 4.4
754
+ Improving construction efficiency
755
+ The heuristics construction step is a potential bottleneck if we trace 8 rays per probe for all probes
756
+ in the scene. As such, we restrict the pilot-rays to the probes that are contained within an extended
757
+ camera frustum as shown in figure 6(b). To maximize the efficiency of our algorithm, we further
758
+ reuse the samples collected from the 8 pilot-rays to populate the irradiance ( ^π‘”π‘Ÿ) and visibility ( ^𝑔𝑐)
759
+ caches. We change the ray-directions at alternate frames in an AABBCCDD... pattern, improving
760
+ the detection of temporally varying light-field surrounding the probes. We measure the time-delta
761
+ (equation 13 and 16) between two frames with identical set of ray-queries, avoiding noisy gradients.
762
+ However, this effectively halves the detection frequency (frame-rate / 2) but improves the spatial
763
+ awareness. We use a stratified-random ray-direction such that there is always one ray per octant.
764
+ We update the irradiance and visibility cache at each alternate frame.
765
+ 4.5
766
+ Probe irradiance cache
767
+ As shown in figure 5(b), the irradiance cache ( ^π‘”π‘Ÿ) is represented as a uniform probe grid in space
768
+ where each probe stores the surrounding diffuse irradiance at a 8 Γ— 8 texel resolution using a
769
+ spherical mapping. At each texel, we store the irradiance in a custom RGB encoding with 9-9-8 bits
770
+ for the three channels. The remaining 6 bits (out of 32bit) store the sample accumulation count
771
+ (N), used for computing the moving average (see algorithm 3) of a sample stream in time. We
772
+ take several considerations into account for the choice of our encoding. Our encoding should be
773
+ bandwidth efficient and must support atomic updates on a commodity GPU. We found both DX12
774
+ and GLSL supports atomic operations on 32 bit integers. Finally, our encoding must faithfully
775
+ 12
776
+
777
+ Adaptive Dynamic Global Illumination
778
+ High Performance Graphics, Poster, July 11–14, 2022,
779
+ Algorithm 3: Moving Average algorithm
780
+ Input: π‘₯: Update location, 𝑣 : New sample, π‘π‘šπ‘Žπ‘₯ : Max sample count
781
+ Output: 𝑉 : Updated value, 𝑁: Sample count
782
+ 1 function MovingAvgUpdate(π‘₯, 𝑣, π‘π‘šπ‘Žπ‘₯):
783
+ 2
784
+ 𝑛 ← ^𝑔[π‘₯].𝑁
785
+ // Cumulative sample count
786
+ 3
787
+ π‘œ ← ^𝑔[π‘₯].𝑉
788
+ // Cumulative value
789
+ 4
790
+ 𝑉 ←
791
+ 𝑣
792
+ 𝑛+1 + π‘›Β·π‘œ
793
+ 𝑛+1
794
+ // Update cumulative value
795
+ 5
796
+ 𝑁 ← π‘šπ‘–π‘›(𝑛 + 1, π‘π‘šπ‘Žπ‘₯)
797
+ // Increment sample count
798
+ 6
799
+ return 𝑉, 𝑁
800
+ encode intensities beyond the standard definition. We apply a non-linear color compression across
801
+ the three color channels, 𝑖 ∈ [0..2] as shown in the equation below.
802
+ 𝑒𝑖 = π‘šπ‘–π‘› (𝑙𝑛(𝛾 Β· 𝑣𝑖 + 1), 𝛽)
803
+ 𝛽
804
+ .
805
+ (21)
806
+ We apply an inverse transform (𝑒π‘₯𝑝(𝛽 Β· 𝑒𝑖) βˆ’ 1) /𝛾 while decoding where 𝛽 = 5 and 𝛾 = 15.
807
+ More details regarding our choice of compression scheme is provided in appendix B and figure 11.
808
+ 4.6
809
+ Probe visibility cache
810
+ As shown in figure 5(c), texels in the visibility probes store the mean distances and mean squared
811
+ distances to the nearest geometry at 16x16 texel resolution. We call this ^𝑔𝑐 - our visibility cache.
812
+ Each texel stores the two channels with 13 bits of precision each while the rest 6 bits are used for
813
+ sample accumulation count. We normalize the distances with probe cage diagonal length. Similar to
814
+ irradiance cache, we apply a logarithmic encoding as per equation 21 for efficient use of available
815
+ precision. We use (𝛽,𝛾) values of (5, 15) and (8, 20) for the linear and squared channels respectively.
816
+ 4.7
817
+ Temporal sample accumulation mecahnism
818
+ We use a moving-average accumulation to store the samples in the irradiance and visibility caches. In
819
+ the algorithm 3, we have two parameters 𝑁 and π‘π‘šπ‘Žπ‘₯ to control the moving-average accumulation.
820
+ As we start accumulating samples, 𝑁 is incremented and the algorithm performs like a true moving
821
+ average. However, as 𝑁 approaches π‘π‘šπ‘Žπ‘₯ βˆ’ 1, the algorithm switches to an exponential moving
822
+ average form with hysteresis (π‘π‘šπ‘Žπ‘₯ βˆ’ 1)/π‘π‘šπ‘Žπ‘₯. Also, note that when the value of 𝑁 is low, the
823
+ cache updates itself quickly, but the stored values may be noisy. As 𝑁 increases, the new samples
824
+ are weighed less in their contribution to the cache. We exploit these parameters to control the
825
+ learning rate and noise in the static and dynamic cases as discussed in the following sections.
826
+ 4.8
827
+ Adaptive sampling - static
828
+ We split our adaptive sampling strategy into two stages - static and dynamic. We have two separate
829
+ Markov-chain sets, each focusing on different aspects of capturing the surrounding light-field.
830
+ While the static chain focuses more on the accuracy, the dynamic chain is tuned for capturing the
831
+ transient responses. We discuss the dynamic chain in detail in the next section.
832
+ We set up equation 2 as - β„Ž = 𝑒π‘₯𝑝(π‘šπ‘–π‘›(^π‘”π‘Ÿ/𝑓𝑠, 1)) Β· 𝑓𝑠. The feedback from irradiance cache ^π‘”π‘Ÿ is
833
+ obtained from the previous frame and from a higher mip-level (also used in deferred shader). The
834
+ lowest mip-level ^π‘”π‘Ÿ is continuously updated and thus avoided as feedback due to possible violation
835
+ of stationarity condition within a frame. We use the Metropolis sampling, algorithm 1, to generate
836
+ the samples π‘₯𝑖 ≑ (𝑝𝑖,πœ”π‘–). As summarized in the algorithm, 2, we use the samples to evaluate the
837
+ 13
838
+
839
+ High Performance Graphics, Poster, July 11–14, 2022,
840
+ Datta et al.
841
+ Algorithm 4: Atomic moving average algorithm
842
+ Input: π‘₯: Update location, 𝑣 : New update value
843
+ Output: Update ^𝑔[π‘₯]
844
+ 1 function AtomicMovingAvg(π‘₯, 𝑣):
845
+ 2
846
+ current ← ^𝑔[π‘₯]
847
+ /* Repeat until destination value stops changing
848
+ */
849
+ 3
850
+ do
851
+ 4
852
+ expected ← current
853
+ 5
854
+ next ← MovingAvgUpdate(π‘₯, 𝑣, 64)
855
+ 6
856
+ InterlockedCompareExchange(^𝑔[π‘₯], expected, next, current)
857
+ // Refer HLSL
858
+ 7
859
+ while current β‰  expected
860
+ continuous light field π‘”π‘Ÿ, which involves tracing a ray originating at 𝑝𝑖 along the direction πœ”π‘–.
861
+ We trace an additional shadow-ray per sample to compute the visibility in the opposite direction
862
+ (βˆ’πœ”π‘–) as the probe queries in the deferred shader for visibility is exactly 180β—¦ out of phase w.r.t
863
+ irradiance. Next we store the irradiance and visibility values in the irradiance ( ^π‘”π‘Ÿ) and visibility ( ^𝑔𝑐)
864
+ caches using an atomic update rule as presented in the algorithm 4. Atomic updates are required
865
+ as multiple invocations of the chain may update the same location in the irradiance and visibility
866
+ caches. Figure 4 summarizes the overall idea.
867
+ We set the random walk step size, denoted by 𝜎 ∈ 𝑅5 in algorithm 5, proportional to the size of
868
+ discretization in the irradiance and visibility cache. Thus positional step size is proportional to the
869
+ size of a voxel in the probe grid, while angular step size is roughly
870
+ √︁
871
+ πœ‹/256. Due to the small step
872
+ size, texels in the cache may accumulate more than one sample per texel, thereby accumulating
873
+ a large sample count over time. We also note that our cache behaves like a true moving average
874
+ between sample count 𝑁 = 0 to 64, which also contributes to better accuracy.
875
+ The static adaptive samples are useful for improving convergence in a static scene and for slow
876
+ changes that are undetected during prior construction. For example, slow changes in lighting such
877
+ as day-night cycles in games. We lower the hysteresis by setting 𝑁 = 16 for all probes in the region
878
+ {π‘‰π‘œπ‘’π‘‘} βˆ’ {𝑉𝑖𝑛} in figure 6(b). This enables the probe to quickly catch-up to the most recent values.
879
+ 4.9
880
+ Adaptive sampling - dynamic
881
+ We run a second set of Markov-chain when dynamic content is detected in the scene. When there
882
+ are dynamic elements, especially moving geometry, we run into two main issues. The generated
883
+ samples are not well distributed in the region of interest i.e. the areas where time varying changes
884
+ are present. When the step size is small, the chain cannot track the target distribution fast enough
885
+ to generate samples from the target, causing the samples to lag the moving target distribution.
886
+ The second problem is noise due to multi-sampling of the irradiance texel. Potentially, this can be
887
+ solved by increasing the hysteresis to improve temporal sample reuse. However, the reduced noise
888
+ comes at the cost of introducing objectionable temporal blur.
889
+ We solve the first issue by increasing the chain step size and by coarsening the target function
890
+ (𝑓𝑑). Practically, this amounts to grouping the heuristics-probes into virtual proxies. In our case, a
891
+ virtual proxy represents a group the 3 Γ— 3 Γ— 3 probes. This virtual probe has 8 directions and each
892
+ direction represents an axis-aligned octant. The value of a texel of the virtual probe is the max of
893
+ all 27 probes it represents along the corresponding direction. We also drop the sampled evidence by
894
+ setting 𝛼 = 0 in equation 2, as the stale irradiance cache ( ^π‘”π‘Ÿ) provide little useful information for
895
+ 14
896
+
897
+ Adaptive Dynamic Global Illumination
898
+ High Performance Graphics, Poster, July 11–14, 2022,
899
+ Table 2. Table showing probe grid details for various scenes used in our technique.
900
+ Scene
901
+ Probe Grid
902
+ Probe spacing
903
+ (in meters)
904
+ Irradiance ( ^π‘”π‘Ÿ)
905
+ Cache Resolution
906
+ Visibility ( ^𝑔𝑐)
907
+ Cache Resolution
908
+ Bistro - Exterior
909
+ 192 Γ— 64 Γ— 192
910
+ 0.5 Γ— 0.5 Γ— 0.5
911
+ 8 Γ— 8
912
+ 16 Γ— 16
913
+ Sponza - Diffuse
914
+ 192 Γ— 64 Γ— 192
915
+ 0.5 Γ— 0.5 Γ— 0.5
916
+ 8 Γ— 8
917
+ 16 Γ— 16
918
+ Sponza - Glossy
919
+ 192 Γ— 64 Γ— 192
920
+ 0.1 Γ— 0.1 Γ— 0.1
921
+ 16 Γ— 16
922
+ 16 Γ— 16
923
+ Table 3. Table showing probe encoding details for the various techniques we use in our comparison.
924
+ Technique
925
+ Irradiance
926
+ Cache Encoding
927
+ Visibility
928
+ Cache Encoding
929
+ Temporal
930
+ Hysteresis
931
+ Ours
932
+ ⌊R9βŒ‹βŒŠG9βŒ‹βŒŠB8βŒ‹ βˆ’ N
933
+ [R13][G13] βˆ’ N
934
+ Static: 0.98 (π‘π‘šπ‘Žπ‘₯ = 63)
935
+ Dyna: 0.91 (π‘π‘šπ‘Žπ‘₯ = 10)
936
+ Q-DDGI
937
+ ⌊R11βŒ‹βŒŠG11βŒ‹βŒŠB10βŒ‹ βˆ’ N
938
+ [R16][G16] βˆ’ N
939
+ 0.94
940
+ Reference
941
+ RGB32f
942
+ RG32f
943
+ N/A
944
+ sampling a time varying region. The chain step size is 3x, and 6x larger for position and directions,
945
+ respectively w.r.t the static case.
946
+ Since each sample from the coarse chain represents an entire octant, we trace 64 rays for the
947
+ octant for all underlying 3x3x3 probes in the group. We make the tracing step more efficient by
948
+ culling probes that are not used in deferred shading. The scheduling of ray-direction is deterministic,
949
+ passing through the center of a texel in the irradiance cache ( ^π‘”π‘Ÿ). This solves the problem of sampling
950
+ noise and also affords the opportunity to simplify the atomic updates. Since the rays are not random,
951
+ we do not benefit from multiple shader invocations updating the same octant. As such, the first
952
+ invocation to update the octant marks (atomically) it updated such that other invocations do not
953
+ repeat the same work move to the next.
954
+ We run the dynamic sampling after the static sampling step. During static sampling, if a probe
955
+ has non-zero dynamic component(𝑓𝑑 > 0), we quantize the ray directions to go through the
956
+ irradiance/visibility cache texel center to avoid injecting sampling noise in the texels.
957
+ 5
958
+ RESULTS AND COMPARISONS
959
+ We compare our results with Q-DDGI and a reference probe-based implementation in different
960
+ scenarios - static scene (fig. 7), dynamic geometry (fig. 1, 8, 10), and dynamic lighting (fig. 9).
961
+ Q-DDGI: Quantized-DDGI or Q-DDGI is a performance enhanced extension of original DDGI
962
+ [28], achieved without major modifications to the base algorithm. Q-DDGI is equipped with a more
963
+ compact irradiance and visibility cache representation that closely resembles ours. See table 3. We
964
+ also enable camera-frustum culling of probes in Q-DDGI as described in section 4.4 and figure 6.
965
+ These modifications allow Q-DDGI to have similar performance (table 4) at same probe count (table
966
+ 2) as ours across different scenes. We believe these modifications make our comparisons more fair.
967
+ We use 32 rays per probe for a total ray budget of 800-1600k (depending on scene) rays per frame.
968
+ Reference: Reference implementation uses a standard FP32 representation for irradiance and
969
+ visibility caches as shown in table 3. We also use a higher resolution 32Γ—32 irradiance and visibility
970
+ cache. Due to memory constraints, we are limited to a smaller probe-grid of size 32 Γ— 32 Γ— 32 using
971
+ same probe spacing (table 2) as other techniques. For each frame, we discard any previous values
972
+ in the probes and accumulate samples using a true-average with 64 rays per texel.
973
+ Ours: We use 4096 instances of static chain invocations and 1024 instances of dynamic chain
974
+ invocations. Overall, we use use between 500-900k (depending on scene) rays per frame.
975
+ 15
976
+
977
+ High Performance Graphics, Poster, July 11–14, 2022,
978
+ Datta et al.
979
+ Table 4. Performance breakdown of our technique and Q-DDGI. Our probe sampling stage is divided into
980
+ three sub-stages - heuristic construction (P), static adaptive sampling (S), and dynamic adaptive sampling (D).
981
+ Scene
982
+ Ours (in milliseconds)
983
+ Q-DDGI (in milliseconds)
984
+ Probe Sampling
985
+ (P + S + D)
986
+ Deferred
987
+ Total
988
+ Probe sampling
989
+ Deferred
990
+ Total
991
+ Bistro - Exterior
992
+ 4.01 + 2.23 + 4.73
993
+ = 11.0
994
+ 4.63
995
+ 15.6
996
+ 22.3
997
+ 4.47
998
+ 26.8
999
+ Sponza - Diffuse
1000
+ 1.21 + 1.85 + 3.18
1001
+ =6.24
1002
+ 3.62
1003
+ 9.86
1004
+ 9.69
1005
+ 3.51
1006
+ 13.2
1007
+ Sponza - Glossy
1008
+ 4.83 + 2.11 + 4.33
1009
+ =11.27
1010
+ 6.44
1011
+ 19.7
1012
+ 24.9
1013
+ 6.71
1014
+ 31.6
1015
+ Figure 1 and 8 shows a large scene (Bistro Exterior), with the tunnel’s entry and exit modified
1016
+ with dynamic gates. The tunnel interior walls are illuminated by indirect illumination alone,
1017
+ controlled by the direct light bouncing off the floor. The direct illumination on the floor is controlled
1018
+ by the dynamic entry gate. The scene tests the tracking capabilities of our algorithm; the dynamic
1019
+ Markov-chain should sample the probes close to the moving door. The scene also tests our color
1020
+ compression scheme under low-light and moving-average accumulation.
1021
+ Figure 9 shows the Sponza scene under dynamic lighting, testing the detection capabilities of
1022
+ ADGI in the absence of dynamic geometry. Figure 7 shows a static scene without dynamic geometry
1023
+ or lighting, testing the convergence of our static adaptive sampling when no dynamism is detected
1024
+ or the dynamic changes are too slow to detect, such as day-night cycles in games.
1025
+ Figure 10 shows a dynamic geometry (Stanford Buddha) under glossy indirect illumination
1026
+ with ambient lighting as direct component. The scene is stressful as the camera frustum contains
1027
+ many times more probes compared to other scenes due to the increased probe density required
1028
+ for glossy illumination. This scene tests the transient response of a dynamic geometry on a glossy
1029
+ floor. Thus the scene is less forgiving of spatio-temporal blurring.
1030
+ We measured the results on a desktop with Nvidia 2080Ti GPU and AMD 5600X CPU at
1031
+ 1920 Γ— 1080 resolution. The performance numbers cited in table 4 are only for ADGI and Q-
1032
+ DDGI algorithms. The GBuffer and direct-illumination passes require an additional 2ms and 3ms,
1033
+ respectively.
1034
+ 6
1035
+ LIMITATIONS
1036
+ We inherit similar limitations as the vanilla DDGI algorithm. The probe visibility from a shade-point
1037
+ is only approximate and requires modifications such as probe movement to minimize light leakage.
1038
+ The probe representation is not efficient in capturing glossy light-transport and requires a dense
1039
+ spatio-angular discretization of irradiance cache to capture glossy reflections.
1040
+ Accurate detection of transient spatio-temporal changes in a scene are difficult. The accuracy of
1041
+ detecting dynamic geometry reduces with the distance of the dynamic object from a probe. The
1042
+ same is true for dynamic lighting; especially high frequency localized lighting that is far from a
1043
+ probe is difficult to detect. Also, for the Markov chain to track the target distribution, the speed
1044
+ of motion should be capped comparable to the product of Markov-chain step size and average
1045
+ frame-rate. While many game engines keep track of the dynamic objects, facilitating the detection
1046
+ of changing in visibility, we still need ray-tracing to detect dynamic radiosity.
1047
+ 16
1048
+
1049
+ Adaptive Dynamic Global Illumination
1050
+ High Performance Graphics, Poster, July 11–14, 2022,
1051
+ 7
1052
+ CONCLUSION
1053
+ Our adaptive sampling approach improves upon the efficiency of the original DDGI algorithm. Our
1054
+ approach non-uniformly allocates resources in regions with time varying phenomena and captures
1055
+ transient localized changes in an environment containing millions of probes. By contrast, DDGI’s
1056
+ uniform allocation policy dilutes resource concentration in critical regions, especially when a large
1057
+ number of probes are present. These improvements reduce temporal lag and minimizes reliance on
1058
+ temporal blur to reduce noise. Our probe encoding scheme minimizes memory requirements by 4x
1059
+ (and by extension memory bandwidth) with minimal impact on quality while also enabling millions
1060
+ of probes in a scene. Our adaptive sampling stages have a fixed upper bound on the compute
1061
+ requirement and also decouples sampling from the number of probes, further reducing memory
1062
+ bandwidth requirement. These changes enable improved probe-based rendering while also enabling
1063
+ 1.5-2x performance improvements.
1064
+ 8
1065
+ RELATED WORK EXTENSION
1066
+ Irradiance caching Irradiance caching is another line of techniques attempting to overcome
1067
+ the high computation cost of GI. The irradiance caching method assumes that irradiance vary
1068
+ smoothly across the scene, and texture detail can be recovered using albedo modulation [64]. The
1069
+ interpolation and location of the various cache records is a critical, especially when the assumptions
1070
+ on smoothness do not hold. While robust, principled offline solutions exist [16, 24], real-time
1071
+ applications often resort to complex heuristics and impose harsh constraints to achieve online
1072
+ GI. Compression [56], sparse interpolation [49], pre-convolved environment maps [42, 45], spatial
1073
+ hashing [3] and using neural network [37] are instances of advancements in real-time irradiance
1074
+ caching. Although these approaches aim for real-time performance, their complexity and constraints
1075
+ make them challenging to implement and deploy.
1076
+ 𝑑 = 32 ms
1077
+ 64 ms
1078
+ 96 ms
1079
+ 128 ms
1080
+ 32 ms
1081
+ 64 ms
1082
+ 96 ms
1083
+ 128 ms
1084
+ Ours
1085
+ Q-DDGI
1086
+ MSE: 0.072
1087
+ 0.031
1088
+ 0.022
1089
+ 0.014
1090
+ SSIM: 0.750
1091
+ 0.894
1092
+ 0.910
1093
+ 0.940
1094
+ MSE:0.085
1095
+ 0.064
1096
+ 0.049
1097
+ 0.031
1098
+ SSIM:
1099
+ 0.561
1100
+ 0.758
1101
+ 0.828
1102
+ 0.888
1103
+ SSIM: 0.732
1104
+ 0.895
1105
+ 0.908
1106
+ 0.914
1107
+ SSIM: 0.524
1108
+ 0.745
1109
+ 0.838
1110
+ 0.874
1111
+ MSE : 0.076
1112
+ 0.038
1113
+ 0.025
1114
+ 0.020
1115
+ MSE : 0.087
1116
+ 0.067
1117
+ 0.052
1118
+ 0.041
1119
+ Green: Diminished luminance
1120
+ Red: Excess luminance
1121
+ Fig. 7. Comparing the convergence of our technique over time on a static Bistro Exterior scene. The figure
1122
+ demonstrates the effectiveness of our static adaptive sampling step. The two rows measure the difference in
1123
+ luminance w.r.t reference and highlight the error in red and green color.
1124
+ 17
1125
+
1126
+ High Performance Graphics, Poster, July 11–14, 2022,
1127
+ Datta et al.
1128
+ 𝑑 = start
1129
+ 𝑑 = start + 4𝑠
1130
+ 𝑑 = start + 8𝑠
1131
+ Direct + Indirect
1132
+ Tunnel interior (D + I)
1133
+ SSIM/MSE:
1134
+ 0.971/0.008
1135
+ SSIM/MSE:
1136
+ 0.974/0.008
1137
+ SSIM/MSE:
1138
+ 0.982/0.007
1139
+ SSIM/MSE:
1140
+ 0.951/0.010
1141
+ SSIM/MSE:
1142
+ 0.951/0.009
1143
+ SSIM/MSE:
1144
+ 0.967/0.009
1145
+ SSIM/MSE:
1146
+ 0.931/0.004
1147
+ SSIM/MSE:
1148
+ 0.948/0.004
1149
+ SSIM/MSE:
1150
+ 0.994/0.000
1151
+ SSIM/MSE:
1152
+ 0.913/0.009
1153
+ SSIM/MSE:
1154
+ 0.871/0.005
1155
+ SSIM/MSE:
1156
+ 0.804/0.004
1157
1158
1159
+ Fig. 8. Our technique compared with Q-DDGI on a modified Bistro Exterior scene augmented with a
1160
+ moving door. The scene has 192 Γ— 64 Γ— 192 probes and shows the convergence of the two techniques near
1161
+ a dynamic area in the scene. The second row shows the changes inside the tunnel as the door closes over
1162
+ time. Our technique is better able to allocate the resources closer to the dynamic areas resulting in faster
1163
+ convergence and higher performance.
1164
+ Path tracing The flexibility and generality offered by path tracing [18] is highly desirable for
1165
+ real-time rendering. However, path tracing has been out of reach for real-time applications due to its
1166
+ substantial computational requirements. Even with the advent of hardware-accelerated ray tracing
1167
+ [23], it is only possible to trace a few tens of rays at each pixel in real-time. Therefore, effective
1168
+ sampling strategies and high-quality denoising algorithms [38, 46, 47] are essential. Many sampling
1169
+ methods try to learn the representation of incident illumination during rendering [1, 8, 34, 44, 60].
1170
+ While these approaches can provide substantial error reduction, constructing these structures in
1171
+ parallel on a GPU incurs a significant overhead that seem unsuitable for real-time applications.
1172
+ Recently proposed ReSTIR GI [41] provides an efficient real-time sampling strategy by reusing the
1173
+ paths spatially and temporally but the algorithm becomes complicated after second bounce and still
1174
+ requires denoising for the final stage. Deep learning has also been applied to path guiding, including
1175
+ work by [35, 36]. These approaches demonstrated a substantial reduction in error due to more
1176
+ effective path sampling, though their performance remain insufficient for real-time applications.
1177
+ Screen space approaches: Approximating physically plausible illumination at real-time frame
1178
+ rates with screen space methods is popular in games. Screen space methods are fast, GPU-friendly,
1179
+ and simple to implement. Screen space ambient occlusion (SSAO) [2, 33] is part of many real-time
1180
+ rendering engines. Following SSAO, screen Space Directional Occlusion (SSDO) [43] is used for
1181
+ near-field direct and indirect diffuse lighting. Sousa et al. [52] proposed Screen Space Reflections
1182
+ (SSR) using a 2D ray-tracing approach directly in screen space to obtain the indirect specular
1183
+ component. Recently Screen-Space Global Illumination (SSGI) [43, 50, 52] methods offer a viable
1184
+ solution to real-time GI. However, these methods are limited by the information visible from the
1185
+ observer’s position, thus making it difficult to engineer a robust solution.
1186
+ Importance sampling and Bayesian modeling: Importance sampling provides a tool to
1187
+ reduce the cost of brute force integration by selectively evaluating elements of the integrand based
1188
+ on prior knowledge, i.e. an educated guess. Previous works in importance sampling proposed
1189
+ different methods to apply importance sampling to various Monte-Carlo integration existing in
1190
+ rendering equations [21, 48, 57]. Although Markov Chain Monte Carlo(MCMC) methods have been
1191
+ 18
1192
+
1193
+ Adaptive Dynamic Global Illumination
1194
+ High Performance Graphics, Poster, July 11–14, 2022,
1195
+ 𝑑 = start
1196
+ 𝑑 = start + 2𝑠
1197
+ 𝑑 = start + 3𝑠
1198
+ Direct only
1199
+ Indirect - Ours
1200
+ Indirect - Q-DDGI
1201
+ SSIM/MSE: 0.920/0.005
1202
+ SSIM/MSE: 0.966/0.006
1203
+ SSIM/MSE:0.957/0.007
1204
+ SSIM/MSE: 0.868/0.012
1205
+ SSIM/MSE: 0.763/0.023
1206
+ SSIM/MSE: 0.794/0.021
1207
+ Difference in luminance w.r.t reference
1208
+ Error - Ours
1209
+ Error - Q-DDGI
1210
+ Green: Diminished luminance
1211
+ Red: Excess luminance
1212
1213
1214
+ Fig. 9. Figure comparing the convergence of our technique under dynamic lighting controlled by the direct
1215
+ component shown in the first row. The last two rows measure the difference in luminance w.r.t reference and
1216
+ highlight the error in red and green color.
1217
+ used in Bayesian learning from the early days of neural networks [39], and Stochastic-Gradient
1218
+ MCMC has been proposed [65] with various applications [25], our approach is neither Monte
1219
+ Carlo-based nor Neural-network learning. We exploit Bayesian inference and Markov Chains as our
1220
+ mathematical means to sample the important texels on the probe, by defining our guide function
1221
+ (prior), likelihood, and posterior.
1222
+ Markov Chain: Markov Chains are used broadly in Monte Carlo path-tracing. For example,
1223
+ Veach and Guibas [58] used Metropolis Sampling to explore the space of all possible paths. Kelemen
1224
+ et al. [19] later applied the exact sampling in the space of random numbers, i.e., in Primary Sample
1225
+ 19
1226
+
1227
+ High Performance Graphics, Poster, July 11–14, 2022,
1228
+ Datta et al.
1229
+ 𝑑 = start
1230
+ 𝑑 = start + 10𝑠
1231
+ 𝑑 = start + 15𝑠
1232
+ Glossy indirect - Ours
1233
+ Glossy indirect without texture
1234
+ Ours
1235
+ Q-DDGI
1236
+ SSIM/MSE:0.996/0.017
1237
+ SSIM/MSE:0.995/0.019
1238
+ SSIM/MSE:0.992/0.018
1239
+ SSIM/MSE:0.990/0.019
1240
+ SSIM/MSE:0.989/0.020
1241
+ SSIM/MSE:0.987/0.028
1242
+ Ours
1243
+ Reference
1244
+ Q-DDGI
1245
+ Ours
1246
+ Reference
1247
+ Q-DDGI
1248
+ Ours
1249
+ Reference
1250
+ Q-DDGI
1251
+ SSIM: 0.979
1252
+ 0.981
1253
+ SSIM: 0.995
1254
+ 0.993
1255
+ SSIM: 0.998
1256
+ 0.992
1257
+ MSE : 0.013
1258
+ 0.028
1259
+ MSE : 0.016
1260
+ 0.034
1261
+ MSE : 0.014
1262
+ 0.058
1263
1264
1265
+ Fig. 10. Figure comparing glossy indirect reflection on a scene lit by ambient lighting. The scene tests transient
1266
+ response due to the moving Buddha geometry over a glossy floor.
1267
+ Space. The most recent work by Bitterli et al. [4] combines a simple path tracing integrator with
1268
+ MCMC by using the random seeds of high variance paths as starting points for the Markov Chains.
1269
+ Although Markov Chains are encountered extensively beneficial in solving Monte Carlo sampling,
1270
+ our point of view on sampling and employing the Markov Chain to draw samples from the guide
1271
+ function is distinct.
1272
+ Bayesian inference: Bayesian modeling is a widespread methodology in computer vision and
1273
+ graphics. Brouillat et al. [5] and Marques et al. [30] pioneered the use of Bayesian Monte Carlo
1274
+ (BMC) [11] in light transport simulation. In contrast, [59] keep the efficient classic, frequentist
1275
+ MC approach and apply Bayesian modeling to optimize their sampling distributions for direct
1276
+ illumination estimates across the scene. Similar approach is used by Vorba et al. [61], who employ
1277
+ a maximum a posteriori (MAP) formulation to regularize training of parametric mixture models for
1278
+ optimized indirect illumination sampling. Our approach uses Bayesian modeling in the context of
1279
+ light-probes to detect important probes and directions based on sampled evidence.
1280
+ REFERENCES
1281
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+ Mixture Models for Light Transport Simulation. ACM Trans. Graph. 33, 4, Article 101 (jul 2014), 11 pages.
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+ //doi.org/10.1145/2601097.2601203
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+ //doi.org/10.1145/2601097.2601203
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+ and Tracing. In Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games (Montreal,
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1458
+ https:
1459
+ //doi.org/10.1145/3306131.3317024
1460
+ [64] Gregory J. Ward, Francis M. Rubinstein, and Robert D. Clear. 1988. A Ray Tracing Solution for Diffuse Interreflection.
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+ [65] Max Welling and Yee Whye Teh. 2011. Bayesian Learning via Stochastic Gradient Langevin Dynamics. In Proceedings
1463
+ of the 28th International Conference on International Conference on Machine Learning (Bellevue, Washington, USA)
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+ (ICML’11). Omnipress, Madison, WI, USA, 681–688.
1465
+ [66] Lei Xiao, Salah Nouri, Matt Chapman, Alexander Fix, Douglas Lanman, and Anton Kaplanyan. 2020.
1466
+ Neural
1467
+ Supersampling for Real-Time Rendering.
1468
+ ACM Trans. Graph. 39, 4, Article 142 (July 2020), 12 pages.
1469
+ https:
1470
+ //doi.org/10.1145/3386569.3392376
1471
+ 23
1472
+
1473
+ High Performance Graphics, Poster, July 11–14, 2022,
1474
+ Datta et al.
1475
+ A
1476
+ METROPOLIS-HASTINGS
1477
+ Markov Chain Monte Carlo (MCMC) allows sampling from the posterior without computing
1478
+ the marginal. [10]. Metropolis-Hastings (Metropolis), which we exploit in this work, is a specific
1479
+ implementation of MCMC [7]. The Metropolis–Hastings algorithm can draw samples from any
1480
+ probability distribution with probability density 𝑃(π‘₯), provided a function β„Ž(π‘₯) proportional to
1481
+ the density 𝑃(π‘₯). The Metropolis algorithm works by generating a sequence of sample values so
1482
+ that, as more samples are produced, the distribution of samples more closely approximates the
1483
+ desired distribution. These sample values are produced iteratively, meaning the next sample being
1484
+ dependent on the current sample (thus making the sequence of samples into a chain). Let β„Ž(π‘₯)
1485
+ be a function that is proportional to the desired probability density function 𝑃(π‘₯) (a.k.a. a target
1486
+ distribution). The Metropolis Markov Chain algorithm with random walk is defined as follows:
1487
+ Algorithm 5: Random-walk algorithm
1488
+ Input: π‘₯𝑖: Current state, 𝑦𝑖 : Probability of current state
1489
+ Input: 𝜎 : Step size or std-dev of Gaussian noise
1490
+ Output: π‘₯𝑖+1: Next state, 𝑦𝑖+1 : Probability of next state
1491
+ 1 function RandomWalk(π‘₯𝑖, 𝑦𝑖):
1492
+ 2
1493
+ π‘₯𝑖+1 ← π‘₯𝑖 + N (𝜎)
1494
+ // Propose a new state
1495
+ 3
1496
+ 𝑦𝑖+1 ← β„Ž(π‘₯𝑖+1)
1497
+ 4
1498
+ πœ‡ ← min
1499
+ οΏ½
1500
+ 𝑦𝑖+1
1501
+ 𝑦𝑖 , 1
1502
+ οΏ½
1503
+ // Compute acceptance ratio
1504
+ 5
1505
+ πœ– ∼ π‘ˆ (0, 1)
1506
+ // Sample uniform distribution
1507
+ 6
1508
+ if πœ– > πœ‡ then
1509
+ /* Reject proposed state
1510
+ */
1511
+ 7
1512
+ π‘₯𝑖+1 ← π‘₯𝑖
1513
+ 8
1514
+ 𝑦𝑖+1 ← 𝑦𝑖
1515
+ 9
1516
+ return π‘₯𝑖+1,𝑦𝑖+1
1517
+ Initialization: Choose an arbitrary point π‘₯π‘–βˆ’1 as the initial observation in the sample-space and
1518
+ choose an arbitrary probability density N (π‘₯𝑖 | π‘₯π‘–βˆ’1) that suggests the next sample candidate π‘₯𝑖,
1519
+ given the previous sample value π‘₯π‘–βˆ’1. In our work, N is assumed to be symmetric. A usual choice
1520
+ is to let N (π‘₯𝑖 | π‘₯π‘–βˆ’1) be a Gaussian distribution centered at π‘₯π‘–βˆ’1, so that points closer to π‘₯π‘–βˆ’1 are
1521
+ more likely to be visited next, making the sequence of samples resemble a random walk [7]. The
1522
+ random walk algorithm is described in algorithm 5.
1523
+ B
1524
+ PROBE COMPRESSION
1525
+ We tested several 26-bit encoding and settled on a non-linear RGB encoding represented by
1526
+ ⌊R9βŒ‹βŒŠG9βŒ‹βŒŠB8βŒ‹ βˆ’N in figure 11. In this encoding, the RGB color is first passed through a logarithmic
1527
+ non-linearity as per equation 21 such that the quantization errors are distributed evenly across
1528
+ intensities. We perform a round-to-lowest-integer (βŒŠβŒ‹) quantization for all channels, although round-
1529
+ to-nearest-integer ([ ] ) is more accurate. Our quantization scheme ensures the moving-average
1530
+ updates produce dark colors when the intensity of new samples are low. In a round-to-nearest set-
1531
+ ting, due to a round-up error, the colors may never go to zero. Interestingly, YCbCr encoding allows
1532
+ round-to-lowest for the Y channel and round round-to-nearest for Cb and Cr channels, however,
1533
+ they perform poorly in both luminance and color preservation metrics as shown in figure 11.
1534
+ 24
1535
+
1536
+ Adaptive Dynamic Global Illumination
1537
+ High Performance Graphics, Poster, July 11–14, 2022,
1538
+ Intensity:
1539
+ [0 - 0.25]*
1540
+ [0.25 - 1.0]
1541
+ [1.0 - 5.0]*
1542
+ Decoded RGB
1543
+ Error
1544
+ Decoded RGB
1545
+ Error
1546
+ Decoded RGB
1547
+ Error
1548
+ ⌊Y8βŒ‹[ Cb9] [ Cr9] βˆ’ N
1549
+ ⌊Y8βŒ‹[ Cb9] [ Cr9]
1550
+ ⌊R9βŒ‹ ⌊G9βŒ‹ ⌊B8βŒ‹ βˆ’ N
1551
+ ⌊R9βŒ‹ ⌊G9βŒ‹ ⌊B8βŒ‹
1552
+ LUM: 0.0024
1553
+ LUM: 0.0025
1554
+ LUM: 0.0025
1555
+ LUM: 0.0004
1556
+ LUM: 0.0011
1557
+ LUM: 0.0043
1558
+ LUM: 0.0045
1559
+ LUM: 0.0049
1560
+ LUM: 0.0050
1561
+ LUM: 0.0007
1562
+ LUM: 0.0021
1563
+ LUM: 0.0086
1564
+ COR: 0.9972
1565
+ COR: 0.9999
1566
+ COR: 1.0000
1567
+ COR: 1.0000
1568
+ COR: 1.0000
1569
+ COR: 1.0000
1570
+ COR: 0.9944
1571
+ COR: 0.9997
1572
+ COR: 1.0000
1573
+ COR: 1.0000
1574
+ COR: 1.0000
1575
+ COR: 1.0000
1576
+ Fig. 11. Figure comparing 26-bit color encodings on slices of the 3D color-space with dynamic range. We
1577
+ compare the reconstruction error measured in Luminance and Color Correlation with RGB32f reference. The
1578
+ log-non-linear encodings marked with - N suffix shifts the bit error from lower to higher intensities - which
1579
+ are less frequent in indirect illumination. βŒŠβŒ‹ and [ ] denotes round-low and round-nearest quantizations
1580
+ respectively. * Color map visualizations are normalized.
1581
+ The parameters in equation 21 are obtained by performing a grid search minimizing the recon-
1582
+ struction error w.r.t RGB32f reference across various color and intensity combinations as shown in
1583
+ figure 11. Luminance error is the r.m.s. value of the difference between the two color-maps. Color
1584
+ accuracy is measured using a normalized dot product between the two flattened color-maps.
1585
+ 25
1586
+
JNE4T4oBgHgl3EQfhg0d/content/tmp_files/load_file.txt ADDED
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