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+1
+A Direct Construction of Near-Optimal Multiple
+ZCZ Sequence Sets
+Nishant Kumar, Sudhan Majhi, Senior Member, IEEE, and Ashish K. Upadhyay
+Abstract—In this paper, for the first time, we present a direct
+construction of multiple zero-correlation zone (ZCZ) sequence
+sets with inter-set zero-cross correlation zone (ZCCZ) from
+generalised Boolean function. The presented ZCZ sequence sets
+are optimal and their union is near-optimal ZCZ sequence set.
+This work partially settles the open problem introduced by Tang
+et al. in their 2010 paper using direct construction. The proposed
+construction is presented by two layer graphical representation.
+Finally, the construction is compared with existing state-of-the-
+art.
+Index Terms—Generalised Boolean function (GBF), zero-cross
+correlation zone (ZCCZ), zero-correlation zone (ZCZ), multiple
+ZCZ sequence sets.
+I. INTRODUCTION
+Z
+-complementary pairs (ZCPs) were introduced by Fan et
+al. [1] to overcome the limitation on the lengths of Golay
+complementary pairs (GCPs) [2]–[5]. The idea of ZCPs was
+generalized to Z-complementary code set (ZCCS) by Feng
+et al. in [6]. A ZCCS refers to a set of K codes, each of
+which consists of M constituent sequences of identical length
+L, having ideal aperiodic auto- and cross-correlation properties
+inside the ZCZ width (Z) [7], [8]. When Z = L and K = M,
+the set is called complete complementary code CCC [9]–
+[11]. To reduce the “near-far effect” and ensure interference-
+free communication in asynchronous CDMA systems, ZCZ
+sequences were introduced in the late 1990s [12]. When the
+received signal delays within ZCZ, ZCZ sequences can be
+employed to remove or reduce MAI and multipath interference
+(MPI) in quasi synchronous CDMA (QS-CDMA) systems
+[13], [14]. Although the ZCZ spreading sequences prevent co-
+channel interference within each cell, inter-cell interference
+across neighbouring cells is unavoidable [15].
+To address the aforementioned shortcoming, the idea of
+multiple ZCZ sequence sets with inter set zero-cross corre-
+lation zone (ZCCZ) has recently been proposed [16]–[23].
+A multiple ZCZ sequence set comprises ZCZ sequence sets
+as its subsets and the cross-correlation function between two
+arbitrary sequences from different subsets has either ZCCZ
+or low cross-correlation zone (LCCZ). Authors in [24] and
+[25] used generalised bent function and perfectly non-linear
+functions respectively to construct multiple ZCZ sequence
+sets. But they tend to achieve only multiple ZCZ sequence
+set with interset LCCZ instead of ZCCZ. In [26], authors
+presented construction of multiple ZCZ sequence sets using
+discrete Fourier transform (DFT) matrices. Furthermore, an
+asymmetric ZCZ (A-ZCZ) sequence set is a multiple ZCZ
+sequence set and the ZCCZ between two arbitrary sequences
+from distinct subsets has a large ZCCZ [20]. To obtain A-ZCZ
+sequence sets, interleaving techniques on perfect sequences
+are presented in the literature [21]–[23]. Since, perfect se-
+quences are available only for very few lengths therefore these
+constructions also have very limited lengths. Additionally,
+the DFT matrices [18]–[20] and Hadamard matrices [16]
+are also used to construct A-ZCZ sequences. But, all these
+constructions are indirect. The limitation of A-ZCZ sequence
+set is that the large ZCCZ is obtained at the cost of optimality
+of ZCZ sequence sets.
+In [17], Tang et al. proposed a method for constructing
+multiple binary ZCZ sequence sets from mutually orthog-
+onal Golay complementary set (MOGCS) with good inter-
+set cross-correlation property and provided an open problem
+as* “we propose the following open problem: Construct N
+ZCZ sequence sets Zi, 0 ≤ i < N, satisfy: 1. Each Zi is
+an (K, Z, L)-ZCZ sequence set with KZ/L = 1/2; 2. The
+sets have a common zero correlation zone of length Zc with
+Zc = Z/N”.
+Motivated by the above open problem given in [17], in this
+letter, we propose a direct construction of near-optimal mul-
+tiple ZCZ sequence sets using generalised Boolean function
+(GBF). Since, proposed construction is based on GBFs, there-
+fore it is suitable for rapid hardware generation. A graphical
+analysis of our proposed construction has also been provided.
+Also, it is the first time that a direct construction of multiple
+ZCZ sequence sets with ZCCZ is presented. The proposed
+construction generalizes construction given in [13] and it is
+optimal over several constructions of A-ZCZ sequence sets
+presented in [16], [18]–[23], [27].
+II. NOTATIONS AND DEFINITIONS
+A. Definition and Correlation Functions
+Let a1 = (a10, a11, . . . , a1(L−1)) and a2 = (a20, a21, . . . ,
+a2(L−1)) be two sequences of equal length L, having entries
+from complex numbers. For an integer u, we define aperiodic
+cross-correlation function (ACCF) of sequences a1 and a2 as
+γ(a1, a2)(u) =
+��L−1−u
+i=0
+a1ia∗
+2(i+u),
+0 ≤ u < L,
+�L+u−1
+i=0
+a1(i−u)a∗
+2i,
+−L < u < 0.
+(1)
+Moreover, ACCF is termed as aperiodic auto-correlation func-
+tion (AACF) if a1 = a2 and denoted as γ(a1)(u). Next, we
+define periodic cross-correlation function (PCCF) in terms of
+ACCF as
+φ(a1, a2)(u) = γ(a1, a2)(u) + γ∗(a2, a1)(L − u).
+(2)
+*The notations has been changed as per this work.
+arXiv:2301.02144v1  [cs.IT]  5 Jan 2023
+
+2
+Definition 1: Let C = {C0, C1, . . . , CP −1} be a collection
+of P codes (matrices) having M rows and L columns. Define
+Cη = [aη
+0 aη
+1 . . . aη
+M−1]T
+M×L,
+(3)
+where aη
+ν (0 ≤ ν ≤ M − 1, 0 ≤ η ≤ P − 1) is the νth row
+sequence or νth constituent sequence and [·]T represents trans-
+pose of matrix [·]. Then the ACCF of two codes Cη1, Cη2 ∈ C
+is defined as
+γ(Cη1, Cη2)(u) =
+M−1
+�
+ν=0
+γ(aη1
+ν , aη2
+ν )(u).
+(4)
+Definition 2: Let C be a code set as defined in (3) which
+satisfies following correlation properties
+γ(Cη1, Cη2)(u) =
+�
+�
+�
+�
+�
+LM,
+η1 = η2 and u = 0,
+0,
+η1 = η2 and 0 < |u| < L,
+0,
+η1 ̸= η2 and |u| < L.
+(5)
+Then C is known as (P, M, L)-MOGCS and each code in C
+is called GCS. Moreover, if P = M then C is known as CCC
+set and denoted by (P, P, L)-CCC.
+Definition 3: Let Zl = {zl
+0, zl
+1, . . . , zl
+K−1} be a collection
+of K L-length sequences, i.e.,
+zl
+i = (zl
+i0, zl
+i1, . . . , zl
+iL−1),
+0 ≤ i ≤ K − 1.
+Then, Z is called (K, Z, L)-ZCZ sequence set if it satisfies
+following,
+φ(zl
+i, zl
+j)(u) =
+�
+�
+�
+�
+�
+0,
+i = j and 1 ≤ |u|≤Z,
+0,
+i ̸= j and 0 ≤ |u|≤Z,
+L,
+i = j and u = 0,
+(6)
+where 0 ≤ i, j ≤ K − 1 and Z is termed as ZCZ width.
+Definition 4: Let Z be a collection of N, (K, Z, L)-ZCZ
+sequence sets then Z = {Z1, Z2, . . . , ZN} is known as a
+multiple ZCZ sequence set with ZCCZ equal to Zc, if for
+0 ≤ |u| < Zc, φ(zl
+i, zl′
+j )(u) = 0, ∀1 ≤ l ̸= l′ ≤ N and
+0 ≤ i, j ≤ K − 1.
+Definition 5 (Tang-Fan-Matsufuji Bound [28]): Let Z be
+any (K, Z, L)-ZCZ sequence set. Then, KZ ≤ L. If for any
+Z, KZ = L (or K(Z + 1) = L) then Z is called optimal (or
+near-optimal) ZCZ sequence set. However, in case of binary
+ZCZ sequence set the bound is reduced to 2KZ ≤ L.
+B. Generalised Boolean Function (GBF) [29]
+We define a complex valued sequence corresponding to a
+GBF, f : {0, 1}m −→ Zq of m variables as
+Ψ(f) =
+�
+ωf0, ωf1, . . . , ωf2m−1�
+,
+(7)
+where fj = f(j0, j1, . . . , jm−1), ω = exp
+�
+2π√−1/q
+�
+, and
+(j0, j1, . . . , jm−1) is the binary vector representation of j,
+where as in the remainder of this letter, q is an even integer
+not less than 2. Corresponding to a GBF f with m variables
+the sequence Ψ(f) is of length 2m.
+Definition 6: Let J = {j0, j1, . . . , jk−1} ⊂ {0, 1, . . . , n−1}
+and xJ = [xj0, xj1, . . . , xjk−1]. For a constant e ∈ {0, 1}k,
+f|xJ=e is known as restriction of f over e and is obtained by
+substituting xjβ = eβ (β = 0, 1, ..., k − 1) in the function f.
+Moreover, the sequence Ψ(f|xJ=e) is the same as sequence
+Ψ(f) of length 2m except for the positions ijβ ̸= eβ for each
+0 ≤ β < k, at these positions Ψ(f|xJ=e) has the zero entries.
+C. Quadratic Forms and Graphs [30]
+Let f be GBF of order r over m variables. If f|xJ=e
+is a quadratic GBF, then graph of f|xJ=e, i.e., G(f|xJ=e)
+has vertex set V , where V = {x0, x1, . . . , xm−1}\{xj0, xj1,
+. . . , xjk−1}. If there is a term qβ1β2xβ1xβ2 (0 ≤ β1 < β2 <
+m, xβ1, xβ2 ∈ V ) in the GBF f|xJ=e with qβ1β2 ̸= 0 (qβ1β2 ∈
+Zq) then by connecting the vertices xβ1 and xβ2 by an edge,
+the graph G(f|xJ=e) can be obtained. For k = 0, the graph
+of f|xJ=e is the same as that of f.
+D. Generalized Reed-Muller Codes
+Definition 7: Let q ≥ 2 and 0 ≤ r ≤ m,, then a linear code
+over Zq generated by the Zq-valued sequences corresponding
+to the monomials of degree at most r in x0, x1, . . . , xm−1 is
+said to be rth order generalised Reed-Muller (RM) code and
+denoted as RMq(r; m).
+E. The Existing Construction of Multiple CCCs
+Lemma 1 ( [31]): Let m, k, and s are integers with
+0 ≤ s ≤ k ≤ m−2. Define Js = {jk−s, jk−s+1, . . . , jk−1} =
+{m − s, m − s + 1, . . . , m − 1}, J = {j0, j1, . . . , jk−1−s} ⊂
+Zm−s, I
+=
+{i0, i1, . . . , im−k−1}
+=
+Zm−s\J, x
+=
+�
+xj0, xj1, . . . , xjk−s−1
+�
+, xs =
+�
+xjk−s, xjk−s+1, . . . , xjk−1
+�
+. Let
+π be a permutation on symbols {0, 1, . . . , m − k − 1}. Let f
+be a quadratic GBF over the m variables x0, x1, . . . , xm−1,
+such that for e ∈ {0, 1}k−s,
+f|x=e = Q +
+m−k−1
+�
+β=0
+uβxiβ +
+s−1
+�
+β=0
+vβxjk−s+β + v,
+(8)
+where
+Q = q
+2
+m−k−2
+�
+β=0
+xiπ(β)xiπ(β+1),
+(9)
+uβ ∈ Zq ∀ 0 ≤ β ≤ m−k −1, vβ ∈ Zq ∀ 0 ≤ β ≤ s−1, and
+v ∈ Zq Let γ1 and γ2 be two end vertices of the path G(Q),
+t1 = �s−1
+β=0 bk+1+β2β, t2 = �k
+β=0 bβ2β, where bβ ∈ {0, 1}
+for 0 ≤ β ≤ k +s. For the natural order generated by (t1, t2),
+Define the set S(t1,t2) by
+�
+�
+�f + q
+2
+�
+�
+k−1
+�
+β=0
+dβxjβ + dxγ1 +
+k−1
+�
+β=0
+bβxjβ
++bkxγ2 +
+k−1
+�
+β=k−s
+dβbs+1+β
+�
+� : dβ, d ∈ {0, 1}
+�
+�
+� .
+(10)
+Let St1 =
+�
+S(t1,t2) : 0 ≤ t2 ≤ 2k+1 − 1
+�
+, 0 ≤ t1 ≤ 2s − 1.
+Then {St1 : 0 ≤ t1 ≤ 2s − 1} is a collection of 2s CCCs, and
+any two GCSs from different CCCs St1 and St′
+1 with 0 ≤
+t1 ̸= t′
+1 ≤ 2s − 1 have a ZCCZ of width 2m−s.
+For the fixed values of t1 and t2, S(t1,t2) is a GCS. Let us
+denote,
+S(t1,t2) =
+�
+s(t1,t2)
+0
+s(t1,t2)
+1
+. . . s(t1,t2)
+2k+1−1
+�T
+,
+(11)
+
+3
+where s(t1,t2)
+ν
+(0 ≤ ν ≤ 2k+1 − 1) is νth row sequence of
+S(t1,t2).
+Lemma 2 ( [13]): Let q = 2 and x0, x1, . . . , xk, xk+1 be k+
+2 binary variables. Also, let h be a Boolean function defined
+on x0, x1, . . . , xk, xk+1 as follow
+h =
+k+1
+�
+β=1
+cβxβx0 +
+�
+1≤µ<ν≤k
+dµνxµxν +
+k+1
+�
+α=0
+eαxα + e′, (14)
+where ck+1 = 1, cβ ∈ Z2 for 1 ≤ β ≤ k, dµν, eα, e′ ∈ Z2. Let
+h denotes the binary vector corresponding to function h, i.e.,
+h = [h0, h1, . . . , h2k+2−1] .
+For 0 ≤ τ ≤ 2k+1 − 1, we have
+(−1)hτ +hτ+1 + (−1)hτ+2k+1+hτ+1+2k+1 = 0,
+(15)
+where the operation in the subscripts is done in modulo 2k+2.
+III. PROPOSED CONSTRUCTION
+In this section, we provide a GBF which generates the
+required multiple ZCZ sequence sets.
+Theorem 1: Let x0, x1, . . . , xm+k+1 are m + k + 2 binary
+variables. Define a GBF f(x0, x1, . . . , xm−1) on m variables
+same as in Lemma 1, i.e., removing J = {j0, j1, . . . , jk−1−s}
+having k − s vertices from the graph of f results in s isolated
+vertices in Js and a path on m−k vertices in I. Define another
+GBF h(xm, xm+1, . . . , xm+k+1) on k + 2 variables as
+h =
+k+1
+�
+r=1
+crxm+rxm +
+�
+2≤µ<ν≤t
+dµνxm+µxm+ν
++
+k+1
+�
+β=1
+eβxm+β + e′,
+(16)
+where ck+1 ̸= 0, cr ∈ Z2 for 1 ≤ r ≤ k, dµν, eβ ∈ Z2. For
+a fixed value of t1, define the set Zt1 = {Ψ(zt1
+t2) : 0 ≤ t2 ≤
+2k+1 − 1} by
+�
+�
+�f + h + q
+2
+�
+�
+k−1
+�
+β=0
+xm+βxjβ + xm+kxγ1 +
+k−1
+�
+β=0
+bβxjβ
++bkxγ2 +
+k−1
+�
+β=k−s
+xm+βbs+1+β
+�
+�
+�
+�
+� .
+(17)
+Then Z
+=
+�
+Zt1
+: 0 ≤ t1 ≤ 2s − 1} is a collection of
+2s (2k+1, 2m ,2m+k+2)-ZCZ sequence sets having ZCCZ
+equals to 2m−s − 1.
+Proof: Using (10), (11), (17) and taking natural order
+generated by t2, we get Zt1 =
+�
+Zt1
+0 , Zt1
+1
+�
+, where
+�
+Zt1
+0 , Zt1
+1
+�
+is horizontal concatenation of matrices Zt1
+0 and Zt1
+1 and these
+matrices are defined as,
+Zt1
+0 =
+�
+������
+s(t1,0)
+0
+ωk0
+s(t1,0)
+1
+ωk1
+. . .
+s(t1,0)
+l−1 ωkl−1
+s(t1,1)
+0
+ωk0
+s(t1,1)
+1
+ωk1
+. . .
+s(t1,1)
+l−1 ωkl−1
+...
+...
+...
+...
+s(t1,l−1)
+0
+ωk0
+s(t1,l−1)
+1
+ωk1
+. . .
+s(t1,l−1)
+l−1
+ωkl−1
+�
+������
+,
+Zt1
+1 =
+�
+������
+s(t1,0)
+0
+ωkl
+s(t1,0)
+1
+ωkl+1
+. . .
+s(t1,0)
+l−1 ωk2l−1
+s(t1,1)
+0
+ωkl
+s(t1,1)
+1
+ωkl+1
+. . .
+s(t1,1)
+l−1 ωk2l−1
+...
+...
+...
+...
+s(t1,l−1)
+0
+ωkl
+s(t1,l−1)
+1
+ωkl+1
+. . .
+s(t1,l−1)
+l−1
+ωk2l−1
+�
+������
+,
+where l = 2k. Now, we need to prove that Zt1 is a (2k+1, 2m
+, 2m+k+2)-ZCZ sequence set. For 0 ≤ i, j ≤ 2k+1−1, periodic
+correlation of Ψ(zt1
+i ) and Ψ(zt1
+j ) at any time shift 0 ≤ τ ≤
+2m is given by (12). Next, by (15), (12) and aperiodic sum
+property of CCCs, we get,
+φ(Ψ(zt1
+i ), Ψ(zt1
+j ))(τ) = 2 ·
+2k+1−1
+�
+m=0
+γ
+�
+ci
+m, cj
+m
+�
+(τ)
+(18)
+=
+�
+2k+m+2,
+if τ = 0 and i = j,
+0,
+otherwise.
+Which proves that Zt1 is a (2k+1, 2m, 2m+k+2)-ZCZ sequence
+sets ∀ 0 ≤ t1 ≤ 2s − 1. Now, let 0 ≤ t1 ̸= t′
+1 < 2s and
+0 ≤ i, j ≤ 2k+1 − 1 then for 0 ≤ τ ≤ 2m−s − 1, the value
+of φ(Ψ(zt1
+i ), Ψ(zt′
+1
+j ))(τ) is given by (13). Now, by (15), (13)
+and ZCCZ property of CCCs in Lemma 1, we get
+φ(Ψ(zt1
+i ), Ψ(zt′
+1
+j ))(τ) = 0,
+∀ 0 ≤ τ ≤ 2m−s − 1.
+Remark 1: Theorem 1 constructed 2s ZCZ sequence sets
+with parameter (2k+1, 2m, 2m+k+2) having common ZCZ
+equals to 2m−s − 1. Since 2k+1 · 2m/2m+k+2 = 1/2 and
+Zc = 2m−s − 1 = (Z + 1)/N.
+φ(Ψ(zt1
+i ), Ψ(zt1
+j ))(τ) =2 ·
+2k+1−1
+�
+m=0
+γ
+�
+s(t1,i)
+m
+, s(t1,j)
+m
+�
+(τ) + [(−1)hl−1+hl + (−1)h2l−1+h0]γ∗ �
+s(t1,j)
+0
+, s(t1,i)
+2l−1
+�
+(L − τ)
++
+2l−2
+�
+m=0
+[(−1)hm+hm+1 + (−1)hm+2l+hm+1+2l]γ∗ �
+s(t1,j)
+m+1 , s(t1,i)
+m
+�
+(L − τ).
+(12)
+φ(Ψ(zt1
+i ), Ψ(zt′
+1
+j ))(τ) =2 ·
+l−1
+�
+m=0
+γ
+�
+s(t1,i)
+m
+, s(t′
+1,j)
+m
+�
+(τ) + [(−1)hl−1+hl + (−1)h2l−1+h0]γ∗ �
+s(t′
+1,j)
+0
+, s(t1,i)
+2l−1
+�
+(L − τ)
++
+l−2
+�
+m=0
+[(−1)hm+hm+1 + (−1)hm+l+hm+1+l]γ∗ �
+s(t′
+1,j)
+m+1 , s(t1,i)
+m
+�
+(L − τ).
+(13)
+
+4
+Remark 2: Since the set of isolated vertices in Theorem
+1 contribute to multipleness of constructed multiple ZCZ
+sequence set. Hence, if we put s = 0, i.e., Js = φ in Theorem
+1 then our construction reduces to construction presented in
+[13]. Therefore, construction provided in [13] is a special case
+of the proposed construction.
+Corollary 1: Collection of all the ZCZ sequences in Theo-
+rem 1, i.e., {Ψ(zt1
+t2) : 0 ≤ t2 ≤ 2k+1 − 1, 0 ≤ t1 ≤ 2s − 1}
+is a near-optimal (2k+s+1, 2m−s − 1, 2m+k+2)-ZCZ sequence
+set.
+Proof: Directly follows from Theorem 1.
+Remark 3: It is the first time in the literature that the
+direct construction of optimal multiple ZCZ sequence sets is
+provided such that their union is a near-optimal ZCZ sequence
+set. Which makes our construction advantageous over several
+constructions of A-ZCZ sequence sets which are presented in
+the literature [16], [18]–[23], [27]. The detailed comparison of
+the proposed work is provided in Table I.
+Remark 4: From equation (17), it can be seen that the
+proposed multiple ZCZ sequence sets are obtained from sec-
+ond order cosets of generalised RM code. Since, RM codes
+have efficient encoding, good error correction properties and
+important practical advantage of being easy to decode [32].
+Hence, our proposed construction has advantage over any other
+non-GBF based construction.
+IV. GRAPHICAL INTERPRETATION OF THE PROPOSED
+CONSTRUCTION
+This section interprets the proposed construction with
+graphical point of view.
+Fig. 1 depicts a graphical repre-
+    
+. 
+. 
+. 
+. 
+. 
+. 
+. . . 
+𝑖𝜋(0) 
+𝑖𝜋(𝑚−𝑘−1) 
+𝑖𝜋(1) 
+𝑖𝜋(2) 
+𝑗𝑘−1−𝑠 
+𝑗0 
+m-s 
+m−1 
+ 
+m 
+m+k-1-s 
+I 
+J 
+J s 
+Lower Layer 
+Upper Layer 
+. 
+. 
+. 
+. 
+. 
+. 
+m+k 
+m+k-1 
+m+k-s 
+m+k+1 
+Fig. 1: Graphical representation of (17).
+sentation of (17). The graph has a two-layered structure with
+a horizontal straight line which is separating the upper and
+bottom layers. The upper layer and lower layer correspond to
+graphs of Boolean functions f and h respectively. These layers
+are interconnected through the set of edges
+{xj0xm, xj1xm+1, . . . , xjk−1−sxm+k−1−s, xm−sxm+k−s,
+xm−s+1xm+k−s+1, . . . , xm−1xm+k−1},
+and the vertex xm+k is connected to any of the end vertices
+of the path in I. Interestingly, the ZCZ of each ZCZ sequence
+set is equals to the power of number of vertices in the upper
+layer of the graph and ZCCZ of ZCZ sequence sets equals to
+one less than the power of number of vertices in the upper
+layer of graph except isolated vertices.
+Example 1: Let m = 4, q = 2, s = 1, and k = 2. Assume
+J = {0}, Js = {3}, I = {1, 2} and GBFs
+f = x0x1 + x0x2 + x0x3 + x1x2 + x1 + x2,
+h = x4x5 + x4x6 + x4x7 + x4.
+(19)
+Generate two sequence sets Z0 and Z1 as
+Z0 = {Ψ(f+h+x0x4+x2x6+x3x5+b0·x0+b1·x3+b2·x1
++ 0 · x5) : b0, b1, b2 ∈ Z2},
+Z1 = {Ψ(f +h+x0x4+x2x6+x3x5+b0·x0+b1·x3+b2·x1
++ 1 · x5) : b0, b1, b2 ∈ Z2}.
+(20)
+Then Z0 and Z1 are two optimal (8, 16, 256)-ZCZ sequence
+sets having inter-set ZCCZ equals to 8. Moreover, Z = Z0∪Z1
+is also an optimal (16, 7, 256)-ZCZ sequence set. In Fig. 2, a
+graph corresponding to quadratic form, i.e., f + h + x0x4 +
+x2x6 + x3x5 of Example 1 is presented.
+5 
+6 
+4 
+7 
+2 
+1 
+0 
+3 
+I 
+J 
+J 
+s 
+Upper Layer 
+Lower Layer 
+Fig. 2: Graphical representation of f +h+x0x4+x2x6+x3x5.
+V. CONCLUSION
+In this paper, we partially answered the open problem
+provided by Tang et al. [17]. For the first time in the
+literature, we proposed a direct construction of multiple
+(2k+1, 2m, 2m+k+2)-ZCZ sequence sets having ZCCZ equals
+to (Z + 1)/N = 2m−s using GBF.
+TABLE I: Comparison of the proposed construction with [19], [20], [22], [23], [26].
+Ref.
+Method
+Parameter1
+Optimality2
+ZCCZ
+No. of sets
+Constraints
+[20, Th. 1]
+Indirect
+(L, M − 1, LP)
+No
+2M − 1
+N
+N = ⌊ T
+M ⌋ > 1, L = KM, M > 1, K > 1
+[20, Th. 2]
+Indirect
+(T, M, TL)
+No
+TL
+N
+N = ⌊ T
+M ⌋ > 1, L = KM, M > 1, K > 1
+[19]
+Indirect
+(M, M − 1, PM)
+Yes
+PM − 1
+N
+N = ⌊ T
+M ⌋, N > 1, M > 1
+[22]
+Indirect
+(L, P, TLP)
+No
+TLP
+T
+gcd(T, P) = 1, gcd(L, P) = 1(orL|PorP|L)
+[23]
+Indirect
+(2M, Z, 2TP)
+No
+2TP
+T
+⌊ P −2
+Z ⌋ = M or ⌊ P −1
+Z ⌋ = M, Z ≤ 2
+[26]
+Indirect
+(N 2, N, N)
+Yes
+Z + 1
+M
+N is order of DFT matrix, N = M(Z + 1)
+This paper
+Direct
+(2k+1, 2m, 2m+k+2)
+Yes
+2m−s − 1
+2s
+0 ≤ s ≤ k ≤ m − 2
+1 Parameter of each ZCZ sequence set.
+2 Optimality of each ZCZ sequence set.
+
+5
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+[29] P. Kumar, P. Sarkar, S. Majhi, and S. Paul, “A direct construction of even
+length ZCPs with large ZCZ ratio,” Cryptography and Communications,
+pp. 1–10, 2022.
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+for OFDM system,” IEEE Trans. on Commun., vol. 68, no. 10, pp.
+6245–6262, 2020.
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+Golay complementary sequences, and Reed-Muller codes,” IEEE Trans.
+on inf. theory, vol. 45, no. 7, pp. 2397–2417, 1999.
+

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+Article
+Experimental and Theoretical Study of Solitary-like Wave
+Dynamics of Liquid Film Flows over a Vibrated Inclined Plane
+Ivan S. Maksymov 1*
+and Andrey Pototsky 2
+Maksymov, I.S.; Pototsky, A.
+Experimental and theoretical study of
+solitary-like wave dynamics of liquid
+film flows over a vibrated inclined
+plane. Preprints 2022, 1, 0.
+https://doi.org/
+Received:
+Accepted:
+Published:
+1
+Optical Sciences Centre, Swinburne University of Technology, Hawthorn, VIC 3122, Australia;
+[email protected]; Tel.: +61-3-3921-4805
+2
+Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC 3122, Australia;
+[email protected]; Tel.: +61-3-9214-4653
+Abstract: Solitary-like surface waves that originate from the spatio-temporal evolution of falling liquid
+films have been the subject of theoretical and experimental research due to their unique properties that
+are not readily observed in the physical system of other nature. Here we investigate, experimentally and
+theoretically, the dynamics of solitary-like surface waves in a liquid layer on an inclined plane that is
+subjected to a harmonic low-frequency vibration in the range from 30 to 50 Hz. We demonstrate that the
+vibration results in a decrease in the average and peak amplitude of the long solitary-like surface waves.
+However, the speed of these waves remains largely unaffected by the vibration, implying that they may
+propagate over large distances almost without changing their amplitude, thus rendering them suitable
+for a number of practical applications, where the immunity of pulses that carry information to external
+vibrations is required.
+Keywords: falling liquid films; solitary waves; surface waves; vibrations
+1. Introduction
+Solitary waves—physical waves that maintain their shape and move with a constant
+velocity due to a cancellation of nonlinear effects and dispersive processes in the medium
+[1]—have been a long-term subject of fundamental and applied research studies in the fields
+of optics [2], fluid dynamics [3], magnetism [4], acoustics [5], electronics [6] and biology
+[7,8]. However, despite a good understanding of the physical properties of solitary waves of
+different kinds, their experimental studies often involve expensive and difficult to operate
+equipment such as intense laser beams and nonlinear-optical materials in the field of optics
+[2] and sources of high-power microwave radiation in the field of magnetism [4], respectively.
+Yet, in some systems such as biological nerve fibres [7,8] the observation of solitary-like waves
+requires significant preparatory works and is possible mostly when a number of specific
+experimental conditions are satisfied. Such technical challenges complicate both fundamental
+studies and verification of numerous theoretical works predicting that solitary waves could
+be used in communication [9,10], sensing [11] and data processing [12] devices and systems.
+There also exists a class of material solitary-like surface waves that originate from spatio-
+temporal evolution of falling liquid films [13,14]. Since the equipment needed to create falling
+liquid films is, in general, simpler than that used in experiments in the fields of optics and
+magnetism, the waves of this kind have attracted attention of many scientists [15–28] following
+the pioneering experiments conducted by the Kapitzas [29]. In fact, while such solitary-like
+surface waves share many physical features with the other known types of solitary waves,
+they can exhibit unique physical properties not observed in other systems [24,30,31]. For
+instance, they can merge instead of passing through each other without significant change,
+with the latter being the case of two solitary waves governed by the well-known KdV equation
+[3,32]. The analysis of solitary-like surface waves in flowing liquid films is also important
+because liquid films, as well as similar physical systems [33–35], are often encountered in
+arXiv:2301.03300v1  [physics.flu-dyn]  9 Jan 2023
+
+2 of 15
+the fields of earth and planetary sciences [36,37] and in technological processes [38], where
+the liquids of interest can also experience temperature gradients [14,28] and vibrations [39–
+41]. Given this, the effect of vibrations on the wave dynamics of film flows has become
+an independent subject of fundamental and applied research [42–46]. In particular, it has
+been shown that vibrations can suppress certain waves on the surface of flowing liquid films
+[42] but in a relevant experiment [47] it has been demonstrated that vibrations can promote
+unusual regimes of spontaneous drop movement. Speaking broadly, the study of the effect of
+vibrations should also help develop communication, sensing and data processing systems
+that are immune to undesirable mechanical impacts on devices that use liquids as a medium
+that provides the critical functionality (see, e.g., [48–50]).
+Although, traditionally, greater attention has been paid to the wave dynamics on free-
+falling vertical liquid ���lms [13,14], studies of surface waves on liquid films flowing over
+slightly inclined planes have also been conducted given an essentially the same physics as
+in the case of vertical systems [24,42]. However, reports on experimental results involving
+the effect of vibrations are rather scarce and scattered in the literature sources [39,40,45]. In
+particular, in [39] it has been shown that the vibration of a horizontal tube with a liquid thin
+film flowing over it results in the appearance of ripple waves at the vibration frequency. The
+amplitude of the so-created waves depends on the vibration amplitude and can reach the
+amplitude of periodic waves existing on the film surface without vibration. Subsequently,
+high-amplitude vibrations result in an increase in the film thickness and a concomitant increase
+in the speed of the waves. However, the opposite conclusions were drawn in [40], which
+is, most likely, a result of the differences in the system (a liquid film under two-phase flow
+conditions) investigated in that paper. It is also well-known that in horizontal liquid layers a
+harmonic vibration excites two different types of standing surface waves: harmonic waves
+that oscillate at the vibration frequency and subharmonic waves that oscillate at the half of the
+vibration frequency [51]. However, the presence of a mean flow across the layer changes the
+response frequency of the excited waves [42–44]. Surface waves excited by harmonic vibration
+in a liquid film flowing over a vertical plane were investigated experimentally in [45] and the
+results obtained in that work validated the linear theory developed in [42–44].
+Thus, mostly the experimental work [45] represents an attempt to systematically study
+the physics of wave motion on a vibrated plane. However, in general, building a setup
+involving liquids flowing down a vibrated vertical surface requires non-standard equipment
+built according to demanding technical specifications. In particular, the liquid should be
+supplied to the inlet located at the upper part of the plane so that the flow rate is not affected
+by the vibration. This is because the thickness of the liquid film is known to be very sensitive
+to external disturbances, including vibrations caused by the pump used to deliver the liquid
+from a reservoir to the inlet [22]. Moreover, the shaker producing the vibration should be
+connected to the vertically positioned surface via a vibration transmission structure. Some of
+the engineering challenges of creating such a structure are the need to move a considerable
+total mass of the supporting structure and liquid with high precision, and to ensure that the
+amplitude of the vibration across the plane area is uniform [45]. To resolve the problem of
+non-uniform vibration amplitude, in Ref. [45] it is was suggested that qualitatively similar
+results could be obtained vibrating just one side of the plane, i.e. vibrating just a portion of the
+liquid, thus also significantly reducing the total mass that needs to be moved by the shaker.
+In this paper, we present and discuss a technically simple and compact experimental
+setup for the investigation of solitary-like surface waves on a slightly inclined plane positioned
+on top of a vibrating table and equipped with an auxiliary channel that recycles the liquid
+used in experiment, thus decreasing the chance of spills of the liquid and its unwanted contact
+with the measurement and imaging equipment, and also decreasing the total mass that needs
+to be moved by the shaker. We employ this setup to demonstrate that the instabilities of
+
+3 of 15
+the thin liquid film caused by the vibrations result in a decrease in the peak amplitude of
+the solitary-like surface waves. We conclude that, despite these changes, the speed of the
+solitary-like waves does not appreciably change due to vibration. As a result, these waves can
+propagate for long distances without changing their shape and, therefore, can be used in the
+practical applications discussed in this work. We also demonstrate the advantage of using
+frequency-wavevector dispersion maps for the analysis of the properties of rolling waves, thus
+extending the toolbox of experimentalists working on this class of wave motion phenomena.
+Our experimental results are validated using the Shkadov model [52,53]—a boundary-layer
+hydrodynamic model derived from the Navier-Stokes equation under the assumption of
+self-similar parabolic longitudinal velocity flow field across the layer.
+Figure 1. Sketch of the experimental setup used to observe the solitary-like surface waves. For the sake
+of clarity, only the main constructive features are shown, including the inclined plate, where the waves
+are observed, the pathway for recycling of the used liquid and the vibrating table. The dimensions and
+relative positions of the components in this sketch are not to scale.
+2. Background and Experimental Methods
+When a single-layer liquid film flows down an inclined plane with a no-slip boundary,
+the resulting Nusselt flat film flow profile assumes a parabolic longitudinal velocity shape
+having the largest velocity at the free surface [13,14,24]. In this flow regime, a long-wavelength
+surface instability develops when the average flow rate exceeds a certain critical value [15].
+When the disturbances are excited naturally, in general four regimes of different wave be-
+haviour can be observed in the downstream regions of the inclined plane [13]. The first
+regime is observed in a section of the plane that is adjacent to the inlet of the liquid, where
+small disturbances caused by the inlet structure are amplified while moving downstream
+and forming predominantly monochromatic waves. The second regime is observed in the
+following downstream region, where the monochromatic waves grow in amplitude and then
+develop higher-order frequency harmonics due to nonlinear effects. Then, as a result of com-
+plex nonlinear interactions, two-dimensional solitary-like waves are formed, and then they
+propagate further downstream exhibiting unique properties that, in part, coincide with those
+of other known solitary waves but, in general, are unique [24]. Finally, three-dimensional
+waves start to form due to transverse variations [13,14].
+It is noteworthy that not all aforementioned regimes can necessarily be observed in
+practice [13]. Yet, it is well-known that when the initial natural disturbance at the inlet is
+nearly monochromatic, the waves emerging in the region located immediately after the inlet
+can first inherit the frequency of the disturbance and then evolve into a solitary-like wave far
+
+UV light
+digital camera
+inlet
+rolling waves
+dwnd
+inclined plane
+H
+recycled liquid
+vibrating table
+shaker
+accelerometer
+fluorescent
+liquid4 of 15
+downstream [13,14,24]. However, when either the thickness of the liquid film or the fluid flow
+is periodically modulated at the inlet, solitary-like surface waves develop almost immediately
+after leaving the inlet area [14,24], which indicates that the nonlinear evolution of the flow
+over an inclined plane is dominated by solitary-like waves independently of whether their
+formation was deliberately forced or resulted naturally.
+Figure 1 shows a sketch of the setup that enables observing the formation of both forced
+and natural (unforced) solitary-like surface waves. The setup is assembled on a vibrating
+table that is driven by a shaker (35 W, 20–80 Hz response, Dayton Audio, USA) and where
+the vibration amplitude is controlled by an analog accelerometer (ADXL326, Analog Devices,
+USA). The inclined plane, where the waves are observed, is a 5-cm-wide and 50-cm-long rigid
+aluminium plate. The surface of the plate was chemically treated to improve the formation
+of the liquid film. The inclination angle is θ = 3 o. The inclined plate was mounted on top
+of wider open channel used to recycle the liquid by redirecting it to the main reservoir. A
+low-vibration DC voltage pump driven via a customised electronic circuit was used to supply
+water from the reservoir to the inlet. The electronic modulation of the pump flow rate enabled
+controlling the thickness of the liquid film and creating solitary-like waves. At the stage of
+preparation to the experiments, an organic fluorescent dye (Tintex, Australia) was added to
+tap water in the concentration of 1 g per litre, thus leading to the emission of bright green
+fluorescence light when the surface of the inclined plate was illuminated with UV-A light. All
+experiments were conducted in a darkened room using an overhead digital camera capable
+of recording videos in a slow motion regime. The resulting videos were post-processed in
+Octave software using customised computational procedures enabling the extraction of the
+wave amplitude from the intensity profile of the fluorescence images. All experiments were
+conducted in an acoustically isolated room with environmental humidity and temperature
+levels.
+Figure 2. Instantaneous profiles the surface waves in a liquid film flowing over the inclined plate. The
+profiles were obtained from the fluorescence images. The frequency of the flow forcing resulting in the
+formation of solitary-like waves is 2 Hz. (a) No vibration. (b) The inclined plate is vibrated with the
+frequency of 48 Hz and the peak vibration amplitude of 1g.
+3. Experimental Results
+Figure 2 shows the representative images of the solitatry-like surface waves propagating
+over a downstream section of the inclined plate, when the vibration is turned off (Fig. 2a) and
+when the plate is vibrated with the frequency of 48 Hz and the peak amplitude of 1g (Fig.
+2b), being g the gravitational acceleration. The frequency of the forcing of the solitary-like
+
+Amplitude (arb. units)
+(a)
+Amplitude (arb. units)
+(b)
+1
+0
+0.05
+0.05
+0.1
+0.1
+0.15
+0.15
+0.2
+X
+0.2
+Downstream distance (m)
+0.25
+0.25
+X
+0.3
+0.3
+0.06
+0.04
+0.04
+0.02
+X
+0.02
+Plate width (m)
+0.35
+0
+Plate width (m5 of 15
+waves is 2 Hz in both panels of Fig. 2. The images were obtained from the selected individual
+fluorescence frames of the recorded videos of the propagating waves. Without vibration (Fig.
+2a), we can observe a train of downstream-propagating solitary pulses. A closer inspection
+also reveals the existence of periodic waves with an amplitude that is much smaller than
+that of solitary-like waves. When the plate is subjected to vibration (Fig. 2b), we continue
+observing a train of solitary pulses with an approximately the same pulse periodicity as in Fig.
+2a. However, the peak amplitude of the pulses is lower than in the case without vibration. Yet,
+in agreement with the relevant theory [42,44] and experiment on the vertical plane [45], we
+also observe the short wavelength ripples arising due to the onset of the Faraday instability.
+Using our fluorescence intensity analysis software, we register the profiles of the waves at
+the points located on the centreline of the inclined plate along the downstream direction, and
+we plot the so-obtained data as a function of time. The resulting spatiotemporal false-colour
+maps are plotted in Fig. 3 with the observation period of 2 s for the scenario of no vibration
+(Fig. 3a) and with the 48 Hz vibration (Fig. 3b). In those figures, we can see the traces of
+several solitary-like waves that propagate in the downstream direction. The traces are more
+distinguishable and have a higher false-colour amplitudes in the case of no vibration than
+with the vibration, which confirms our observation of a decrease in the peak amplitude of the
+solitary pulses due to the vibration in Fig. 2. The ripple waves caused by the vibration-induced
+Faraday instabilities can also be seen in Fig. 3b. It is noteworthy that the separation between
+the traces and the relative position of the traces in the time-downstream coordinate space
+are very similar without and with the vibration. This indicates that, even though the peak
+amplitude of the solitary-like waves is affected by the vibration, in general the vibration does
+not change the shape of the soliton pulses.
+Figure 3. False-colour maps showing the spatiotemporal traces of solitary-like waves forced at the
+frequency of 2 Hz. (a) No vibration. (b) The inclined plate is vibrated with the frequency of 48 Hz and
+the peak vibration amplitude of 1g.
+Then, we apply a two-dimensional Fourier transformation to the spatiotemporal data to
+obtain the dispersion maps as a function of frequency f and wavevector k. Since the speed of
+a wave is given by u = ω/k = 2π f /k, using the resulting dispersion maps we can identify
+the bands of constant f /k ratio that correspond to waves travelling along the inclines plate at
+
+(a) 2
+(b) 2
+1.5
+0.5
+1.5
+0.5
+S
+S
+0
+0
+0.5
+-0.5
+0.5
+-0.5
+0
+OF
+0
+0.1
+0.2
+0.3
+0.4
+0
+0.1
+0.2
+0.3
+0.4
+Downstream distance (m)
+Downstream distance (m)6 of 15
+a constant speed. Yet, we apply the standard f-k filtering procedures to remove noise from the
+dispersion characteristics [54]. Figure 4 shows the dispersion maps for the case of no vibration
+(Fig. 4a) and with the 48 Hz vibration (Fig. 4b). While the negative frequency regions of the
+dispersion maps originate from the mathematical properties of the Fourier transformation,
+the sign of the wavevector has the physical meaning as it determines the direction of the wave
+propagation.
+We first analyse the dispersion map in Fig. 4a and its zoomed image presented in Fig. 5a,
+where we can see a set of high-magnitude discrete bands that are superimposed on a broader
+continuum band of a lower magnitude. The frequencies of the discrete bands correspond to
+the forcing frequency of the solitary-like waves 2 Hz and its higher-order harmonics of 4, 6
+and 8 Hz and so forth. The origin of the harmonics is due to the nonlinear effects as discussed
+below. The spectrum of the discrete bands changes as the frequency of forcing of the solitary-
+like waves is changed. When the modulation of the pump flow was turned off, i.e. with no
+wave forcing, the discrete bands completely disappeared. However, a continuum band was
+always observed independently of whether the forcing was turned on or off. Subsequently, we
+associate the continuum band with natural periodic rolling waves propagating on the surface
+of the liquid film flowing over the inclined surface. According to the frequency-wavevector
+spectral analysis theory [54], a fit of the observed bands with a straight line produces the
+velocity of the solitary-like wave of 0.27 ± 0.02 m/s.
+When the plate is vibrated (Fig. 4b), in addition to the dispersion bands discussed in Fig.
+4a we observe two new isolated bands that can be associated with the Faraday instability.
+Moreover, the close-up of the dispersion map (Fig. 5b) shows that the magnitude of the
+discrete modes decreased due to the vibration, which is an observation that is consistent with
+our conclusions made earlier in the text. Yet, the bands in Fig. 5b can also be fitted with a
+straight line that corresponds to the wave velocity of 0.27 ± 0.02 m/s.
+Figure 4. Dispersion maps of the solitary-like waves forced at the frequency of 2 Hz. (a) No vibration.
+(b) The inclined plate is vibrated with the frequency of 48 Hz and the peak vibration amplitude of 1g.
+The plots are slightly oversaturated for the sake of a better visual presentation.
+Empirically, the presence of the discrete bands at the forcing frequency of 2 Hz and
+its higher-order harmonics can be explained using the well-established analogy between
+the rolling waves and acoustic waves [55]. Indeed, the solitary-like surface waves in Fig.
+2 can be regarded as large-amplitude shock-like disturbances (in the sub-field of physically
+
+(a)
+0.1
+(b)
+0.1
+40
+40
+0.08
+0.08
+20
+20
+Frequency (Hz)
+0.06
+Frequency (Hz)
+0.06
+0
+0
+0.04
+0.04
+20
+-20
+0.02
+0.02
+-40
+-40
+0
+-1.5
+-1
+-0.5
+0
+0.5
+1.5
+-1.5
+-1
+-0.5
+0
+0.5
+1.5
+Wavevector (mm-1)
+Wavevector (mm-1)7 of 15
+similar roll waves in open channel such a discontinuity is called the hydraulic jump [14,21,34]).
+Shock waves are also well-known in the field of nonlinear acoustic, where their formation
+is accompanied by strong nonlinear effects such as the generation of higher-order harmonic
+frequencies [50]. Considering longitudinal acoustic waves that can be described as alternating
+areas of compression and rarefaction in the medium, we can show that the points of the crests
+of an acoustic wave travel faster than the speed of sound in the medium, but the points of
+the wave troughs travel slower [50]. This physical process underpins the formation of an
+acoustic shock wave [55]. In turn, in the field of rolling waves, the crest of a large-amplitude
+solitary-like wave is connected to its trough by a discontinuity, where the flow regime abruptly
+changes from a supercritical condition and where the fluid moves faster than the wave, to a
+subcritical one, where the fluid moves slower [14,34]. As a result, the spectrum of the wave
+becomes enriched by higher-order harmonic of the frequency of forcing.
+Qualitatively similar results were obtained at the vibration frequencies in the range from
+30 Hz to 50 Hz, and they were validated by our theoretical analysis, the results of which are
+presented in the following section.
+Figure 5. Close-ups of the dispersion maps presented in Fig. 4. The linear fits of the dispersion bands
+and the corresponding wave velocities are shown.
+4. Theory
+The theoretical description of a non-steady flow in the presence of deformable interfaces,
+such as the flow in a thin liquid layer down a vibrated incline, is a notoriously difficult
+hydrodynamic problem. The exact analysis is only available in the linear case, i.e. when
+the deformation amplitude of the liquid-air interface is much smaller than the average film
+thickness [42]. In the general case, when a harmonic vibration is applied both in the perpen-
+dicular and parallel directions with respect to the the inclined plate, the unperturbed base
+flow is given by a superposition of a steady Nusselt flow and of an additional harmonically
+oscillating flow parallel to the incline with a flat free surface [42]. The Navier-Stokes equation
+for an incompressible fluid can be linearised about the base flow and the Floquet theory-based
+stability analysis determines if the flow is stable or unstable.
+The full nonlinear problem with large amplitude deformation of the film surface can
+only be studied approximately using simplified hydrodynamic models. Here we use the
+well-known Shkadov model [52,53], which can be derived from the Navier-Stokes equation
+
+0.27 m/s
+0.27 m/s
+(a)
+10
+0.1
+(b)
+10
+0.1
+0.08
+0.08
+5
+5
+Frequency (Hz)
+0.06
+Frequency (Hz)
+0.06
+0
+0.04
+0.04
+-5
+-5
+0.02
+0.02
+-10
+-10
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+Wavevector (mm-1)
+Wavevector (mm1)8 of 15
+by assuming a self-similar parabolic longitudinal velocity profile. The model developed by
+Shkadov has been used earlier to study nonlinear solitary waves in falling liquid films in
+the absence of vibration [22,23] and to investigate the onset of Faraday waves in vertically
+vibrated isolated liquid drops [56].
+Thus, we consider a liquid film with the local film thickness h(x, t) flowing down an
+inclined solid plate that makes an angle θ with the horizontal, as shown in Fig. 1. In our model,
+the x-axis is chosen to be parallel to the plate with the positive direction pointing down the
+incline. To capture rolling waves, we use a one-dimensional version of the Shkadov model,
+which is formulated as a set of two coupled nonlinear equations for h(x, t) and the local flux
+across the layer q(x, t) = � h(x,t)
+0
+u(x, z, t) dz, where u(x, z, t) is the longitudinal flow velocity
+and z-axis is perpendicular to the incline
+ρ
+�
+∂tq + 6
+5∂x
+�q2
+h
+��
+=
+−3µq
+h2 + σh∂3
+xh − ρg(t) cos(θ)h∂xh + ρg(t) sin(θ)h,
+∂th + ∂xq
+=
+0,
+(1)
+where µ is the dynamic viscosity, σ is the liquid-air surface tension and the time-dependent
+gravity acceleration due to vibration is g(t) = g(1 + a cos(ωt)). The inclination of the plate
+leads to a re-distribution of the vertical vibration into a longitudinal g(t) sin(θ) and an orthog-
+onal g(t) cos(θ) components, respectively.
+The base flow corresponds to a time-periodic spatially homogeneous flux q0(t) and a
+flat film surface h0 = const. From Eqs. (1) we obtain the following expression by setting
+∂xq0(t) = ∂xh0 = 0:
+q0(t) = g sin(θ)h3
+0
+3ν
+�
+1 +
+a cos(ωt)
+(2h2
+0/3l2ac)2 + 1 + 2h2
+0
+3l2ac
+a sin(ωt)
+(2h2
+0/3l2ac)2 + 1
+�
+,
+(2)
+where lac = √
+2ν/ω represents the length of the acoustic boundary layer.
+In the absence of vibration, i.e. when a = 0, the base flow is the time-independent Nusselt
+flow, where the linear stability is well-known in the case of a falling film, i.e. at θ = 90o
+[22,23]. For an arbitrary inclination angle θ, the instability sets in when Re > cot(θ), where
+Re = q0/ν = g sin(θ)h3
+0
+3ν2
+is the Reynolds number. To put this condition into perspective, for a
+water film on a θ = 3o incline, the flow is unstable when h0 > 0.48 mm. The corresponding
+instability is called gravitational instability and it leads to the onset of long surface waves
+propagating downstream. The wavelength of unstable waves is longer than λc = 2π/kc,
+where kc is the critical wave vector of the gravitational instability
+kc =
+�
+ρg sin(θ)
+σ
+�
+g sin(θ)h3
+0
+3ν2
+− cot(θ)
+��1/2
+.
+(3)
+Neutrally stable waves with the wavelength λc = 2π/kc propagate downstream with a speed
+c, which is twice as large as the surface speed in the Nusselt flow, i.e. c = g sin(θ)h2
+0/ν.
+When the vibration is switched on, the Faraday instability mode develops and it competes
+with the gravitational instability mode. The linear stability of a flat film flowing down an
+incline under the combined action of the longitudinal and orthogonal vibration has been
+investigated in Ref. [43] using a theoretical approach based on the exact linearisation of the
+Navier-Stokes equation [42]. In the relevant work Ref. [57], an integral boundary layer model
+has been formulated and applied to study nonlinear Faraday waves in liquid films on a
+horizontal plate subjected to horizontal and vertical vibrations. In Refs. [44,45], the nonlinear
+dynamics of a liquid film falling down a vertical vibrated plate is investigated theoretically
+
+9 of 15
+and experimentally. However, it should be emphasised that large-amplitude surface waves in
+a liquid film flowing down an incline in the presence of both the longitudinal and orthogonal
+vibrations have not been studied earlier.
+To study the stability of the base flow Eqs. (2) we use the standard plane-wave ansatz
+q(x, t) = q0(t) + ˜q(t)eikx and h(x, t) = h0 + ˜h(t)eikx, where k is the wavevector of the small-
+amplitude perturbation. By differentiating the second equation in Eqs. (1) with respect to time
+and the first equation with respect to x, the flux perturbation ˜q can be eliminated to yield a
+complex-valued Mathieu-like equation for the film thickness perturbation ˜h
+∂tt ˜h + A(t)∂t ˜h + B(t)˜h = 0,
+(4)
+with A(t) = 3ν
+h2
+0 + 12
+5 ik q0(t)
+h0
+and B(t) = ikg(t) sin(θ) + σ
+ρ h0k4 + g(t) cos(θ)h0k2 + ik6ν q0(t)
+h3
+0
+−
+6
+5k2 q0(t)2
+h2
+0 .
+According to the Floquet theory, the solution of Eq. (4) is given by ˜h(t) = H(t)eλt,
+where H(t) is some bounded periodic function with the period T = 2π/ω and λ is the
+Floquet exponent. The solution is stable when the real part of the largest Floquet exponent
+is negative, i.e. Re(λ) < 0 and it is unstable otherwise. The Floquet exponents are related to
+the monodromy matrix M via Re(λ) = 1
+T ln(|Λ|), where Λ is the eigenvalue of M. The 2 × 2
+complex-valued monodromy matrix M is given by the fundamental solution matrix that is
+obtained by writing Eq. (4) as a system of two first-order equations and integrating it over one
+period T with the unit 2 × 2 matrix as the initial condition.
+For the inclination angle θ = 30, we choose the thickness of the water film h0 = 0.6 mm,
+which is slightly above the critical value for the gravitational instability of h0 = 0.48 mm. The
+marginal stability curves that correspond to Re(λ) = 0 are shown in Fig. 6 for four different
+vibration frequencies f = 18, 20, 25, 48 Hz. The critical wavevector of the gravitational
+instability Eq. (3) is marked by kc in Fig. 6d and it remains unaffected by the vibration. The
+shaded regions in Fig. 6d indicate the unstable areas. The Faraday instability sets in at the
+vibration amplitude ac that corresponds to the tip of the lowest Faraday tongue. The value
+of ac, as extracted from Fig. 6, slightly increases with f, namely: ac = 0.33 for f = 18 Hz,
+ac = 0.35 for f = 20 Hz, ac = 0.38 for f = 25 Hz and ac = 0.48 for f = 48 Hz. This observation
+confirms the earlier statement that, for the range of frequencies between 30 Hz and 50 Hz, the
+surface waves are much more sensitive to the changes of the vibration amplitude a than to the
+changes of the vibration frequency f. Indeed, comparing Figs. 6c,d we see only a marginal
+difference in the critical amplitude ac when the frequency is doubled. On the other hand,
+increasing the value of a from a = 0.5 to a = 1 will significantly broaden the band of unstable
+wavevectors of the Faraday instability, thus significantly changing the dynamics of the surface
+waves.
+
+10 of 15
+0.2
+0.4
+0.6
+0.8
+1
+k (mm
+-1)
+0
+1
+2
+3
+4
+5
+a
+0.2
+0.4
+0.6
+0.8
+1
+k (mm
+-1)
+0
+1
+2
+3
+4
+5
+a
+0.2
+0.4
+0.6
+0.8
+1
+k (mm
+-1)
+0
+1
+2
+3
+4
+5
+a
+0
+0.5
+1
+1.5
+2
+k (mm
+-1)
+0
+1
+2
+3
+4
+5
+a
+(a)
+(b)
+(c)
+(d)
+f=18 Hz
+f=20 Hz
+f=25 Hz
+f=48 Hz
+kc
+ac
+Figure 6. Marginal stability curves of the base flow Eq. (2) for a 0.6 mm water film on a θ = 30 incline
+vibrated at (a) f = 18 Hz, (b) f = 20 Hz, (c) f = 25 Hz and (d) f = 48 Hz. The shaded regions in
+panel (d) highlights the unstable areas, kc indicates the critical wave vector of the gravitational long-
+wave instability Eq. (3) and ac marks the critical vibration amplitude when the Faraday instability sets in.
+To better understand the temporal signature of the surface waves in response to vibration,
+we compute the imaginary part of the Floquet exponent Im(λ) = ω
+2π (arg(Λ)) + ωn, where,
+as before, Λ is the eigenvalue of the monodromy matrix and n is an arbitrary integer. Any
+neutrally stable wave, i.e. Re(λ) = 0, can be represented in the form ˜h(x, t) = eikxH(t)eλt =
+H(t)eikx+iIm(λ)t, where H(t) is a bounded 2π/ω-periodic function. Therefore, the temporal
+spectrum of such a neutrally stable wave contains delta-peaks located at ω
+2π (arg(Λ)) + ωn.
+The temporal spectrum of a growing wave with Re(λ) > 0 contains the same delta peaks that
+will appear slightly smeared.
+At this stage, it is important to emphasise that the temporal response of the surface waves
+that develop on the surface of a liquid layer on a vibrated incline is not necessarily harmonic
+(frequencies ωn) or subharmonic (frequencies ω/2 + ωn). This feature is in stark contrast to
+the standard Faraday instability in horizontal liquid layers, when the neutrally stable waves
+are always harmonic or subharmonic standing waves [51]. In some special cases, however,
+such as discussed in Ref. [45] for transversally vibrated falling liquid films, the magnitude
+of ω
+2π (arg(Λ)) may be close to zero or ω/2, leading to an almost harmonic or subharmonic
+response. For the fluid parameters used in the present study, the frequency of the Faraday
+mode is significantly shifted from ω or ω/2, as shown in Fig. 4b.
+
+11 of 15
+Next, we simulate the experimental conditions, at which the results shown in Fig. 4b
+was obtained, to gain a better understanding of how the vibration changes the dynamics of
+the waves in the early stages of evolution. Thus, we numerically integrate Eqs. (1) over the
+time interval of 3 seconds in the system of length of 60 cm with periodic boundaries. The
+vibration amplitude is a = 0.8 and the other parameters are the same as in Fig. 6d. As the
+initial conditions, we use zero flux and random initial perturbation of the flat film surface
+with the amplitude of 10−3 mm. The dispersion map is obtained taking the two-dimensional
+Fourier transformation of the solution h(x, t). The contour lines of the dispersion map that
+correspond to the level of 3% of its maximum are shown by the thick lines in Fig. 7. The thin
+solid lines in Fig. 7 correspond to the dispersion curves Im(λ)(k), computed from Eq. (4) for
+a = 0.8 and f = 48 Hz. It can be seen that the results of the direct simulation of the full system
+Eq. (1) are in perfect agreement with the dispersion curves of the small-amplitude surface
+waves.
+-1
+-0.5
+0
+0.5
+1
+k (mm
+-1)
+-40
+-20
+0
+20
+40
+f  (Hz)
+Figure 7. Contour plot (thick lines) of the dispersion map of the solution of Eqs. (1) computed over the
+time interval of 3 seconds with a random initial perturbation of the flat film surface. The thickness of the
+water film is h = 0.6 mm and the vibration parameters are a = 0.8 and f = 48 Hz, similarly to Fig. 4b.
+The thin solid lines correspond to the imaginary part of the Floquet exponent Im(λ) of the most unstable
+mode.
+The dispersion map in Fig. 7 is dominated by the delta-peaks located at f = ±14 and f =
+±34 Hz, thus confirming that the primary response of the liquid film to a harmonic vibration
+is neither harmonic, nor subharmonic. Qualitatively, the shift of the response frequency away
+from the standard for the Faraday instability subharmonic mode can be explained as follows.
+In a horizontal layer vibrated at frequency ω, the Faraday instability sets in the form of a
+standing wave oscillating at the subharmonic frequency ω/2. Any standing wave can be
+represented as a superposition of two plane waves travelling at the phase speed of c = ω/(2k)
+in the opposite directions, i.e. h(x, t) = Aeiωt/2+ikx + Aeiωt/2−ikx + cc. When the layer is
+slightly inclined with the positive direction pointing downstream, it would be reasonable to
+
+12 of 15
+assume that the plane wave propagating downstream will increase its phase speed by some
+amount δc, but the wave propagating upstream will decrease its phase speed by the same
+amount δc. Assuming that the wavevector remains unaffected by a small inclination angle,
+the resulting solution is represented by h(x, t) = Aei(ω/2−kδc)t/2+ikx + Aei(ω/2+kδc)t/2−ikx + cc.
+Therefore, the temporal spectrum of h(x, t) will contain delta-peaks located at ±(ω/2 + kδc)
+and (±(ω/2 − kδc)), in agreement with Fig. 7.
+Alongside the delta-peaks, the dispersion map in Fig. 7 also contains a band of linearly
+unstable plane waves with the wavevectors k < kc. These long waves are amplified as the
+result of the gravitational instability mode. It can be seen that the gravitational band falls
+perfectly on the central dispersion curve that passes through the origin. The central dispersion
+branch in Fig. 7 is almost indistinguishable from the dispersion curve in the absence of
+vibration (not shown). This allows us to conclude that a relatively strong vibration (sufficiently
+strong to excite Faraday waves) has almost no effect on the phase speed c = Im(λ)/k of the
+long gravitational surface waves.
+To study the interaction between the Faraday waves and gravitational surface waves in
+the nonlinear regime, we solve Eqs. (1) over the time interval of 15 seconds with and without
+vibration and compare the respective dispersion maps in Fig. 8.
+Figure 8. (a) Dispersion map obtained from the solution of Eqs. (1) in the absence of vibration for a
+0.6 mm water film on a θ = 3o incline. (b) Dispersion map of the solution of Eqs. (1) when the inclined
+plane vibrated at f = 48 Hz with the amplitude a = 0.8. The scaling for the vertical axis is in arbitrary
+logarithmic units.
+It is evident from Fig. 8 that vibration leads to a suppression of the long surface waves.
+Indeed, the magnitude of the dispersion band that corresponds to the gravitational waves is
+significantly smaller when the film is vibrated. This result is in qualitative agreement with
+Fig. 2.
+5. Conclusions
+In conclusion, our experiments with a sub-millimetre thick water layer on a slightly
+inclined vertically vibrated plate demonstrate that low-frequency vibration in the range
+between 30 and 50 Hz suppresses the development of long rolling surface waves propagating
+downstream. These surface waves appear as the result of the long-scale gravitational instability
+of the base flow in the absence of vibration [15,16] and may also be excited by mechanically
+perturbing the flow at the inlet. A relatively small thickness of the water layer (under 1 mm)
+is required to suppress the three-dimensional instability of the rolling waves that is known
+to develop at large flow rates. Experimental findings are verified using a boundary-layer
+
+15
+(a)
+15
+(b)
+10
+10
+5
+5
+0
+0
+-5
+-5
+-10
+-10
+-15 
+-15
+50
+50
+40
+40
+30F
+30
+20
+20
+10
+10
+f (Hz)
+f (Hz)
+-10
+k (mm-1)
+-10
+k (mm-1)
+-20F
+-20
+-30
+0.5
+-30
+0.5
+0
+40
+0.5
+-40
+0
+-0.5
+-50
+50
+-113 of 15
+hydrodynamic model [52,53] obtained from the Navier-Stokes equation by assuming a self-
+similar parabolic longitudinal flow velocity. Linear stability and nonlinear dynamics of the
+surface waves obtained with the model qualitatively confirm the main experimental findings.
+Without vibration, the Fourier content of surface waves is represented by a broad band
+of unstable wave vectors k < kc, where kc is a critical cut-off wave vector of the gravitational
+instability (see Eq. 3). As the instability unfolds, the wavelength of the dominant wave quickly
+increases until it develops into a solitary-like wave [24]. For fluids with a relatively small
+viscosity, such as water, the characteristic time required for solitary rolling waves to develop
+on a 3o incline is of the order of several seconds. In the nonlinear regime, the Fourier content
+of the surface waves is dominated by solitary-like waves characterised by a small wavevector
+with a background of smaller amplitude shorter waves, which is shown in Fig. 8a and Fig. 4a.
+We observe that the properties of the surface waves change dramatically when the layer
+is vibrated. Thus, a relatively weak vibration (the vibration amplitude a < g) leads to the
+onset of the secondary Faraday instability in the form of short waves with a wavelength
+of λ ≈ 5 . . . 10 mm when vibrated at f = 48 Hz. In agreement with the earlier theoretical
+studies [42–44], the temporal frequency of the Faraday waves is shifted away from the
+harmonic (48 Hz) and subharmonic (24 Hz) response that is typical of Faraday instability
+in horizontal liquid layers. In fact, the inclination angle of the plate acts as a wave filter,
+splitting a standing Faraday wave into two plane waves: one propagating upstream and
+one propagating downstream. Similarly to the Doppler effect, the wave that propagates
+downstream increases its speed and, therefore, increases its temporal frequency, while the
+wave that propagates upstream decreases its speed and frequency. For water layers vibrated
+at 48 Hz, we observe the following shifts in frequency away from the subharmonic response:
+from 24 Hz to approximately 40 Hz for the downstream wave and from 24 Hz to approximately
+8 Hz for the upstream wave.
+In the nonlinear regime, the interaction between shorter Faraday waves and longer
+gravitational waves leads to the broadening of their respective bands in the f-k dispersion
+map. Most importantly, we find that the average and peak amplitudes of the long-scale
+gravitational waves are significantly reduced when vibration is applied. This result is rather
+intriguing since the total influx of energy is larger in the vibrated system when both gravity
+and vibration together drive the flow, unlike in the non-vibrated case, where the only source
+of energy is due to gravity. Yet, nonlinear wave interaction leads to an uneven re-distribution
+of energy amongst the Faraday and gravitational waves in favour of the former. The physical
+mechanism responsible for the suppression of gravitational waves remains an open question;
+however, it is plausible to assume that the fast-oscillating fluid flow in the form of circulation
+patterns [58] in pulsating Faraday waves may slow down the redistribution of fluid on the
+large scale, required for the growth and development of the gravitational waves. This result is
+even more surprising since we did not observe any noticeable change in the velocity of the
+gravitational waves induced by vibration.
+Apart from a contribution of the fundamental knowledge, the results presented in this
+work may be used to better understand and further improve certain technological processes
+that rely on falling liquid films. Yet, the demonstrated immunity of the solitary-like waves to
+external vibration and their intriguing nonlinear dynamical behaviour will be of interest to
+researchers working on emergent technologies, where both solitary waves and fluidic systems
+play an important role [7,8,12,48,59–61].
+Author Contributions: I.S.M. conducted the experiments. A.P. conducted the theoretical analysis. Both
+authors wrote the manuscript.
+Conflicts of Interest: The authors declare no conflict of interest.
+
+14 of 15
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+Reconfigurable magnetic-field-free superconducting
+diode effect in multi-terminal Josephson junctions
+Fan Zhang,1 Mostafa Tanhayi Ahari,2
+Asmaul Smitha Rashid,3 George J. de Coster,4
+Takashi Taniguchi,5 Kenji Watanabe,6 Matthew J. Gilbert,7,2
+Nitin Samarth,1∗ Morteza Kayyalha3∗
+1Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA
+2Materials Research Laboratory, The Grainger College of Engineering, University of Illinois,
+Urbana-Champaign, IL 61801, USA
+3Department of Electrical Engineering, The Pennsylvania State University, University Park,
+PA 16802, USA
+4DEVCOM Army Research Laboratory, 2800 Powder Mill Rd, Adelphi, MD, 20783, USA
+5International Center for Materials, Nanoarchitectonics
+National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
+6Research Center for Functional Materials
+National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
+7Department of Electrical Engineering, University of Illinois, Urbana-Champaign, IL 61801, USA
+∗Correspondding author: [email protected].
+∗Correspondding author: [email protected].
+The superconducting diode effect (SDE) has attracted growing interest in re-
+cent years as it potentially enables dissipationless and directional charge trans-
+port for applications in superconducting quantum circuits. Here, we demon-
+strate a materials-agnostic and magnetic-field-free approach based on four-
+terminal Josephson junctions (JJs) to engineer a superconducting diode with
+1
+arXiv:2301.05081v1  [cond-mat.supr-con]  12 Jan 2023
+
+a record-high efficiency (∼ 100%). We show that the SDE is reconfigurable
+by applying control currents to different terminals. We attribute the observed
+SDE to the asymmetry of the effective current-phase relation (CPR), which we
+derive from a circuit-network model. Our findings demonstrate the emergence
+of a new form of the CPR in multi-terminal JJs that can emulate macroscopic
+transport signatures of superconducting systems with broken inversion and
+time-reversal symmetries.
+Introduction
+In linear electrical networks, the concept of reciprocity implies a symmetric relationship be-
+tween the applied current and measured voltage. In other words, the voltage magnitude remains
+the same if the polarity of the current source is reversed from positive to negative (1). Vio-
+lating this fundamental symmetry in semiconductor technology has led to a plethora of new
+devices including diodes, transistors, rectifiers, and photodetectors (2,3,4,5). In superconduc-
+tors, engineering non-reciprocity requires simultaneous breaking of time-reversal and inversion
+symmetries, known collectively as chiral symmetry (6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16). The
+superconducting diode effect (SDE) is defined as an asymmetry in the critical current when the
+current sweep direction is reversed.
+Macroscopic transport signatures of the SDE are determined by the current-phase relation
+(CPR), which typically depends on the band structure and microscopic details of the underlying
+material system. Therefore, careful engineering of the interplay between spin-orbit interactions,
+topological phases, and magnetic fields can lead to the observation of the SDE. To this point, the
+SDE has been experimentally reported in a multitude of systems including but not limited to:
+non-centrosymmetric superconductors with the magneto-chiral anisotropy (17, 18, 19, 20, 21),
+Josephson junctions (JJs) based on Dirac semimetals with finite-momentum Cooper pairing
+2
+
+(22), two-dimensional (2D) van der Waals heterostructures (23, 21, 24, 25, 26), three-terminal
+JJs based on InAs in the presence of a magnetic field (27), and a network of graphene JJs (28).
+In this work, we develop a reconfigurable, materials-agnostic, and magnetic-field-free method
+to engineer a synthetic CPR that emulates macroscopic transport signatures of systems with bro-
+ken inversion and time-reversal symmetries. We consider a multi-terminal Josephson junction
+(MTJJ) fabricated in graphene, which has a symmetric Fermi surface with no spin-orbit cou-
+pling (both inversion and time-reversal symmetries are preserved). We show that the MTJJ
+can emulate the SDE, which is a macroscopic transport signature predicted to emerge in sys-
+tems with broken inversion and time-reversal symmetries. We further show that the SDE is
+reconfigurable: the MTJJ exhibits the typical Josephson effect with no SDE under a symmetric
+current-bias configuration. However, it manifests the SDE under an asymmetric current-bias
+configuration (see Fig. 1 and fig. S1 for details of asymmetric and symmetric bias configura-
+tions, respectively). The observed SDE is also magnetic-field-free and reversible with efficien-
+cies as large as ∼ 100%. We explain the experimental observations by modeling our system
+using a circuit network of coupled resistively-shunted junctions (RSJs). We find that the semi-
+classical RSJ model accurately captures the non-reciprocal transport effect in the system. We
+calculate an effective non-sinusoidal CPR arising from circuit network effects among the super-
+conducting terminals. We show that this effective CPR is not symmetric under sign-reversal of
+the superconducting phase, resulting in the observation of the SDE. Our combined experimental
+and theoretical findings establish a new materials-agnostic platform based on MTJJs to engineer
+novel forms of CPRs and non-reciprocal superconducting properties for potential applications
+in superconducting cryogenics and quantum technology.
+3
+
+Results and Discussion
+We fabricate four-terminal JJs on hBN/graphene/hBN van der Waals heterostructures which
+are edge-contacted by Ti(10 nm)/Al(100 nm) superconducting electrodes. Figure 1A shows an
+atomic force microscope (AFM) image of a representative four-terminal JJ. We characterize
+the junction in two asymmetric bias-current configurations (see our prior work (29) and fig.
+S1 in the Supplementary Materials for results obtained from symmetric configurations). In
+configuration 1 (2), we ground terminal 2 (3) and apply a constant control current to terminal 3
+(2). In both configurations, we measure the SDE by sweeping the current I1 of terminal 1 and
+applying a control current I4 to terminal 4. Figure 1B shows V12 vs I1 measured in Config. 1
+at I3 = 0 nA, I4 = 10 nA, Vg = 30 V, B = 0 G, and T = 12 mK. We observe that the critical
+current (I+
+c ) for positive sweep direction, marked by red arrows, is around 4 nA, whereas the
+critical current (I−
+c ) for the negative sweep direction, marked by black arrows, is around −24
+nA. We also observe different critical currents (I+
+c or I−
+c ) and return currents (I+
+r or I−
+r ) for
+each sweep direction, likely due to the Joule heating effect (30). We note that increasing the
+temperature reduces the impact of Joule heating, thereby resulting in similar critical and return
+currents (see fig. S4 in the Supplementary Materials).
+To elucidate the origin of the observed SDE, we consider a circuit-network model of coupled
+RSJs (Fig. 1C). The RSJ model represents individual junctions by a two-fluid system in which
+the total junction current is the sum of a pair current ip
+jk(t) and a dissipative quasiparticle current
+iq
+jk(t) (31,32). Here, we assume diffusive transport in which the pair current is given by the first
+Josephson relation ip
+jk(t) = Ijk
+c sin(φjk(t)), where Ijk
+c
+is the critical current between terminals
+j and k and φjk(t) ≡ φj(t) − φk(t) is the gauge-invariant phase difference satisfying the sec-
+ond Josephson relation dφjk(t)/dt = (2e/ℏ)Vjk(t). The quasiparticle current is due to a finite
+voltage Vjk(t) across the junction, iq
+jk(t) = GjkVjk(t), where Gjk is a constant phenomenolog-
+4
+
+ical conductance tensor (see the Supplementary Materials for the RSJ parameters). A circuit
+network model of JJs typically includes a parallel capacitance as well. In our graphene-based
+JJs, however, the junction capacitance is negligible and, hence, we only consider the resistance
+of the junction (33). Imposing current conservation (Kirchhoff’s current law) at terminal j, we
+obtain
+Ij =
+�
+k
+(ip
+jk + iq
+jk).
+(1)
+In this study, we are interested in the emergent non-reciprocal superconducting properties in
+the four-terminal JJ. Therefore, we only consider small bias currents such that no quasiparticle
+current flows between the terminals, i.e., iq
+jk = 0. In this case, starting from Eq. 1 and assuming
+I3 = 0 nA for Config. 1, we may analytically obtain expressions for the other terminal currents
+as
+I1 = I14
+c sin φ14 + I13
+c sin φ13 + I12
+c sin φ1,
+I4 = I41
+c sin φ41 + I43
+c sin φ43 + I42
+c sin φ4,
+0 = I32
+c sin φ3 + I31
+c sin φ31 + I34
+c sin φ34.
+(2)
+For fixed (φ1, I4) in Eq. 2, we find φ3, φ4, and, consequently, the following effective CPR:
+I1(φ1, I4) =
+�
+n
+an sin nφ1 + bn cos nφ1,
+(3)
+where n is an integer and (an, bn) are the amplitudes of the nth harmonic for the sine and cosine
+functions, respectively. Comparing our numerical simulation to Eq. 3, we find that b0, a1, and
+a2 are the only dominant factors in our device (Fig. 1D). This leads to
+I1(φ1, I4) ≈ b0 + a1 sin φ1 + a2 sin 2φ1,
+(4)
+where b0 ∝ I4 and (a1, a2) are independent of I4 (see Supplementary Materials for more de-
+tails).
+5
+
+The symmetry of this effective CPR depends on the current-bias configuration.
+For ex-
+ample, Eq. 4 is symmetric under simultaneous sign reversal of φ1 and I4, i.e., I1(φ1, I4) =
+−I1(−φ1, −I4). However, for a fixed I4 ̸= 0, Eq. 4 represents a CPR which is asymmetric
+under sign reversal of φ1 , i.e., I1(φ1, I4) ̸= −I1(−φ1, I4). This is because for fixed I4 ̸= 0, b0
+does not change sign when φ1 → −φ1. In general, the non-reciprocity arises from an asymme-
+try in the free energy of the system for opposite current directions ±I1 and fixed I4. To see this,
+we consider the expression for the free energy of the system
+F(I1, I4, φ1, φ3, φ4) = (ℏ/2e)
+�
+I1φ1 + I4φ4 +
+�
+j<k
+Ijk
+c cos φjk
+�
+.
+(5)
+We obtain the individual superconducting phases, φj, that minimize the free energy, δF/δφj =
+0 for all individual contacts, j. We note that this is the static solution of the RSJ model,
+which is equivalent to solving Eq. 2 for the phases. When I4 ̸= 0, the minimized free en-
+ergy Fmin ≡ F(I1, I4) is an asymmetric function of I1. In other words, when I4 ̸= 0, a fixed
+free energy F(I1, I4) corresponds to different current values for positive and negative current
+directions. Hence, the critical currents in opposite directions I±
+c are different (see fig. S6 in the
+Supplementary Materials for more details). We note that the RSJ model used in our theoretical
+analysis is materials agnostic. It applies to a broad range of junctions with a sinusoidal CPR,
+regardless of the Fermi surface or the band structure of the normal material.
+Figure 1D depicts the CPRs obtained from Eq. 4 (solid lines) and the numerical simulation
+(dotted lines) for I4 = −10 nA, 0 nA, and 10 nA. We observe that while I1(φ1) is symmetric for
+I4 = 0 nA, it exhibits an asymmetric behavior with non-zero current at zero phase when I4 =
+±10 nA, i.e., I1(φ1) ̸= −I1(−φ1) and I1(φ1 = 0) ̸= 0. We note that the numerically calculated
+CPR (dashed lines) from the RSJ model is limited to ∼ (−π/2, π/2) as the quasiparticle current
+is non-zero outside of this range and, hence, φ1 is no longer time-independent. This is because
+from the viwpoint of the classical washboard potential when I1 approaches the critical current,
+6
+
+the local minima of the tilted washboard potential become horizontal inflection points so that
+the the phase particle is not at a stable equilibrium, i.e., it is no longer time independent (34).
+Figures 1, E and F depict the numerically calculated pair currents flowing between different
+terminals for I1 = −|I−
+c | = −19 nA (E) and I1 = |I−
+c | = 19 nA (F), respectively. In both
+panels, I3 = 0 nA and I4 = 10 nA. We find that the junctions are superconducting for I1 =
+−|I−
+c | = −19 nA. However for I1 = |I−
+c | = 19 nA, we see that the pair current is severely
+decreased, which is signifying a transition to a finite resistance state. This observation points to
+the emergence of the SDE in terminal 1. While there is a small difference between the simulated
+and measured critical currents, we emphasize that the RSJ model successfully captures the
+qualitative behavior of the experimental data.
+To better understand the role of I4 in controlling the SDE, we measure differential resistance
+maps versus I1 and I4 at I3 = 0 nA (Config. 1). Figures 2, A and B plot dV13/dI1 and
+dV43/dI4 maps versus I1 and I4, respectively. For panel A (B), we sweep I1 (I4) for each fixed
+I4 (I1) value. The black arrows mark the sweep directions for I1 and I4 (see figs. S2, A and
+B for maps obtained by sweeping I1 and I4 in the opposite directions). We also numerically
+solve the RSJ model (Eq. 1) to obtain (φ1(t), φ4(t), φ3(t)), assuming terminal 2 is grounded
+(φ2(t) = 0). We then calculate the dc voltages, relative to the ground, by taking the time average
+as ⟨Vj2(t)⟩ = (ℏ/2e)⟨dφj(t)/dt⟩ ≡ Vj2. Figures 2, C and D plot the simulated differential
+resistance maps corresponding to the experimental results of Figures 2, A and B, respectively.
+Our experimental and theoretical findings demonstrate the emergence of the SDE for I4 ̸= 0.
+More specifically, we observe that the value of I+
+c (|I−
+c |) increases (decreases) as we decrease
+I4, e.g., I+
+c ∼ 2 nA (|I−
+c | ∼ 25 nA) at I4 = 12 nA and I+
+c ∼ +25 nA (|I−
+c | ∼ 2 nA) at I4 = −12
+nA.
+We use the difference in I+
+c and |I−
+c | to show a voltage rectification when I4 = −12 nA and
+7
+
+I1 is a pulsed current whose amplitude is 26 nA which is larger than |I−
+c | but smaller than I+
+c
+(Figs. 3, A and B). We calculate the diode efficiency as:
+Q ≡ I+
+c + I−
+c
+I+
+c − I−
+c
+.
+(6)
+We only consider the SDE and Q inside the critical current contour, where transport is non-
+dissipative (no voltage is developed across the terminals). Figure 3C depicts the critical currents
+I+
+c and I−
+c extracted from the differential resistance maps (blue solid lines), pulsed current
+measurements (red symbols), and the RSJ model (black dashed lines). Figure 3D plots the
+corresponding diode efficiency Q calculated from Eq. 6. We observe that I+
+c , I−
+c , and Q linearly
+depend on I4. More specifically, we can see from the effective CPR (Eq. 4) that I±
+c ≈ ±Ic + b0,
+where Ic depends on a1 and a2. As a result, Q ≈ b0/Ic ∝ −I4/Ic in our MTJJs.
+Within the shaded areas in Fig. 3D, the voltage obtained from the pulsed measurement occa-
+sionally fluctuates between zero and non-zero values at T = 12 mK. This is due to the hysteresis
+in switching currents (Ic and Ir) caused by the Joule heating effect (28). We find that the fluctu-
+ations disappear at higher temperatures, and we obtain a diode efficiency of ∼ 100% at T = 220
+mK (see fig. S5 for more details). We present the full dependency of Q on control currents I3
+and I4 and magnetic field in fig. S3 (see the Supplementary Materials for more details). We
+finally point out that in our setup, a non-linear voltage (second harmonic signal) develops across
+the terminals during temperature transition from the normal state to the superconducting state
+if a non-zero dc current is applied to any of the terminals. This non-linear behavior, however,
+emerges in two-terminal JJs as well (see the Supplementary Materials for more details).
+We now turn our attention to the influence of an external magnetic field on the SDE. Figures 4,
+A-C plot the differential resistance dV13/dI1 versus the normalized magnetic flux Φ/Φ0 (bottom
+axis) and perpendicular magnetic field B (top axis) for three different values of I4 = −12 nA
+8
+
+(A), 0 nA (B), and 12 nA (C), respectively. We obtain the magnetic flux from Φ = B × A,
+where A = 1.76 µm2 is the area of the MTJJ and Φ0 = h/2e is the superconducting magnetic
+flux quantum. We observe that the superconducting quantum interference pattern (Fraunhofer
+pattern) is symmetric for I4 = 0 nA, whereas it is vertically shifted up and down for I4 = −12
+nA and I4 = 12 nA, respectively. The vertical shift in the Fraunhofer pattern induces a disparity
+between I+
+c and I−
+c and, consequently, leads to the SDE. To better understand the quantum
+interference pattern, in Figs. 4, D-F, we plot the magnetic flux dependence of the critical currents
+I±
+c , obtained from the CPR (Eq. 4) with b0 = −0.7I4 and I4 = −12 nA (D), 0 nA (E), and 12
+nA (F). The effective CPR is symmetric under simultaneous sign reversals of b0 and B, i.e.,
+|I+
+c (b0, B)| = |I−
+c (−b0, −B)|. However, for fixed I4 ̸= 0 (fixed b0 ̸= 0), this symmetry is
+broken for the current flow, which directly follows from the corresponding broken symmetry in
+the free energy (see fig. S7 in the Supplementary Materials). More specifically, the b0 term in
+Eq. 4 is independent of φ1 and, accordingly, the external magnetic flux. Therefore, it creates a
+constant vertical shift in the Fraunhofer patterns of Figs. 4, D and F.
+Conclusion
+In conclusion, we have demonstrated that MTJJs display a reconfigurable SDE at zero magnetic
+field. We have found diode efficiencies (up to ∼ 100%) which depend linearly on a control bias
+current I4. To elucidate the origin of the SDE, we have modeled our junctions using cou-
+pled RSJs. The model predicts the emergence of an effective CPR in our junctions. We have
+shown experimentally and theoretically that the bias-current configuration plays a crucial role
+in breaking symmetries of this effective CPR. More specifically, we have demonstrated that for
+a constant I4 ̸= 0, the effective CPR is asymmetric under sign-reversal of the superconducting
+phase, resulting in the SDE. We have further shown that the exact dependence of the diode ef-
+ficiency on I4 is determined by the phase-independent term (b0) of the CPR. Finally, we have
+9
+
+demonstrated that no SDE emerges from the external perpendicular magnetic field at I4 = 0.
+Instead, we have shown that non-zero I4 values lead to a constant shift in the Fraunhofer pat-
+terns. The effective CPR derived from the RSJ model captures this shift and reproduces similar
+Fraunhofer patterns. Our joint experimental and theoretical study demonstrates that MTJJ can
+serve as a materials-agnostic platform to engineer the magnetic-field-free SDE with tunable
+and reversible diode efficiencies. This platform could potentially enable new low-temperature
+superconducting logics such as memories (35) and directional cryogenic electronics such as
+circulators (36).
+Materials and Methods
+We exfoliate graphene (Kish graphite) and hBN (National Institute for Materials Science, Japan)
+flakes from bulk crystals. We assemble hBN/graphene/hBN van der Waals heterostructures us-
+ing a standard dry transfer technique, followed by annealing in H2/Ar gas at 350 °C to remove
+polymer residues from the heterostructures. The heterostructures are then patterned with elec-
+tron beam lithography, followed by dry etching (O2/CHF3), to define the junction area. Another
+e-beam lithography step is performed to define the contact patterns. Finally, Ti(10 nm)/Al(100
+nm) is evaporated to create superconducting edge contacts
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+M. Kjaergaard, A. Gyenis, A. Blais, Nonreciprocal devices based on voltage-tunable junc-
+tions. arXiv preprint arXiv:2209.06194 (2022).
+Acknowledgements: We acknowledge funding from the National Science Foundation (NSF)
+Innovation and Technology Ecosystems (No. 2040667). F Z and N S acknowledge support from
+the University of Chicago. G J C acknowledges support from the ARAP program of the Office
+of the Secretary of Defense. M J G and M T A acknowledge funding from US ARO Grant
+W911NF-20-2-0151 and the NSF through the University of Illinois at Urbana-Champaign Ma-
+terials Research Science and Engineering Center DMR-1720633. K W and T T acknowledge
+support from the JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).
+Author Contributions M K and N S conceived the research. F Z and A S R fabricated and
+14
+
+characterized the devices. F Z and M K performed low temperature measurements. F Z ana-
+lyzed the transport data with inputs from M K and N S. M T A, M J G, and G J C performed the
+modeling and simulations. T T and K W grew the hBN crystals. F Z, M T A, and M K wrote
+the manuscript with comments from all authors.
+Competing Interests The authors declare that they have no competing financial interests.
+Data and materials availability: Additional data and materials are available online.
+15
+
+Fig. 1. Superconducting diode effect in a multi-terminal geometry. (A) An AFM image of a
+representative four-terminal JJ. The JJ is made of hBN-graphene-hBN (dashed rectangle) edge
+contacted with Ti/Al superconducting terminals. Arrows show bias current directions used in
+Config. 1. (B) Voltage V12 versus current I1 measured at I3 = 0 nA and I4 = 10 nA in Config.
+1. Red arrows mark the I1 sweep direction from −100 nA to 100 nA (positive direction) in
+which the positive critical current I+
+c and the positive return current I+
+r are extracted. Black
+arrows mark the I1 sweep direction from 100 nA to −100 nA (negative direction) in which
+the negative critical current I−
+c and the negative return current I−
+r are extracted. The blue dots
+mark the positions of I+
+c and I−
+c , and the orange dots mark the positions of I+
+r and I−
+r . (C)
+Schematic of the circuit-network of coupled RSJs utilized to simulate our four-terminal JJs. The
+links between each pair of terminals, e.g., j and k, are characterized by a sinusoidal CPR, i.e.,
+I(φjk) = Ijk
+c sin φjk(t), and a shunted conductance Gjk. (D) Synthetic CPR (I1 vs φ1) obtained
+from the numerical simulation (dotted lines) and equation Eq. 4 (solid lines) at I4 = −10, 0,
+and 10 nA. Here b0 = −0.7I4, (a1, a2) = (14, −1.8) for I4 = 0, and (a1, a2) = (12, −1) for
+|I4| = 10. The numerically calculated φ1 is limited to (−π/2, π/2) because the junctions are
+no longer superconducting beyond this range in the RSJ model. (E, F) A schematic of the pair
+16
+
+A
+c
+D
+Gjk
+20
+1μm
+k
+10
+Ic jk sinΦjk(t)
+(nA)
+0
+-10
+14 = -10 nA
+14 = 0 nA
+I4 = 10 nA
+-20h
+-1.0
+-0.5
+0
+0.5
+1.0
+ / 元
+B
+E
+F
+14 = 10 nA
+14 = 10 nA
+I4 = 10 nA
+10
+4
+4
+(Λn)
+Ic
+l3 = O nA
+l3 = O nA
+0
+1.+
+1
+3
+1
+3
+I1 = -I/el
+I1 = +[/c-l
+-10
+= -19 nA
+= 19 nA
+2
+2
+-40
+-20
+0
+20
+I1 (nA)current distribution obtained from the numerical simulation for I1 = −|I−
+c | (E) and I1 = +|I−
+c |
+(F). Both schematics are for I4 = 10 nA and I3 = 0 nA. The line thickness is proportional to
+the pair current amplitude and the arrows indicate the pair current direction.
+Fig. 2. Critical current contours in Config. 1. (A, B) Color map of the differential resistance
+dV12/dI1 (A) and dV42/dI4 (B) versus I1 and I4 in Config. 1 at I3 = 0 nA. The black arrows
+mark the sweep directions for I1 and I4. (C, D) Theoretical simulation of the differential resis-
+tance dV12/dI1 (C) and dV42/dI4 (D) versus I1 and I4 obtained from the coupled RSJ model.
+17
+
+A 50
+B
+50
+1.5
+1.5
+1.0
+1.0
+25
+25
+0.5
+0.5
+0
+4
+dV12
+4
+dV42
+-25
+-25
+dl1
+dl4
+(k2)
+(k2)
+-50
+-50
+-50
+-25
+0
+25
+50
+-50
+-25
+0
+25
+50
+I (nA)
+I1 (nA)
+c
+D
+50
+0.6
+50
+0.6
+0.4
+0.4
+25
+25
+0.2
+0.2
+(nA)
+-0
+0
+-0
+4
+dV12
+4
+dV42
+-25
+-25
+dl1
+dl4
+(k2)
+(k2)
+-50
+-50
+-50
+-25
+0
+25
+50
+-50
+-25
+0
+25
+50
+I (nA)
+I1 (nA)Fig. 3. Voltage rectification in multi-terminal JJs. (A) Pulsed current I1 versus time mea-
+sured in Config. 1. The amplitude and frequency of I1 are 26 nA and 0.05 Hz, respectively.
+I3 = 0 nA and I4 = −12 nA. (B) Voltage drop V on terminals 1 (V12), 3 (V32), and 4 (V42)
+versus time. Voltages are all measured with respect to ground (terminal 2). (C, D) Critical
+current I+,−
+c
+in positive and negative directions (C) and the extracted diode efficiency Q (D)
+versus I4. Red circles are obtained from pulse measurements, blue solid lines are extracted
+from differential resistance maps (Figs. 2, A and B), and black dashed lines are calculated from
+the RSJ model (Figs. 2, C and D).
+18
+
+A
+14 = -12 nA
+c
+30
+30
+Measured
+20
+Extracted
+ +
+Simulated
+20
+c
+10
+(nA)
+10
+(nA)
+0
+0
+-10
+-10
+0
+-20
+000
+-20
+-30
+1
+1
+1
+-5
+-15
+-10
+0
+5
+10
+15
+-30
+14 (nA)
+B
+D
+20
+100
+V12
+Measured
+V32
+Extracted
+10
+Simulated
+50
+42
+(Λn)
+0
+0
+Q
+a
+-10
+-50
+-100
+-20
+30
+60
+90
+120
+-15
+-10
+-5
+0
+5
+10
+15
+time (s)
+14 (nA)Fig. 4. Fraunhofer patterns under different control bias currents in Config. 2. (A-C) Color
+maps of the differential resistance dV13/dI1 versus I1 (left axis), magnetic flux Φ/Φ0 (bottom
+axis), and magnetic field B (top axis) at I4 = −12 nA (A), 0 nA (B), and 12 nA (C). The
+magnetic flux Φ is calculated as Φ = B × A, where A = 1.76 µm2 is the area of the MTJJ.
+Here, Φ0 = h/2e is the superconducting magnetic flux quantum. Data is measured in another
+device with the same geometry using Config. 2. All measurements are performed at Vg = 0
+V and T = 12 mK. (D-F) Theoretical simulation of the superconducting interference pattern:
+normalized critical current Ic/Ic0 versus Φ/Φ0 at I4 = −12 nA (D), 0 nA (E), and 12 nA (F).
+Here, Ic0 is the critical current at Φ = 0 and I4 = 0.
+19
+
+A
+B (G)
+B
+B (G)
+c
+B (G)
+-90
+-45
+0
+45
+90
+-90
+-45
+0
+45
+90
+06-
+-45
+0
+45
+90
+60
+1.0
+I4 = -12 nA
+14 = O nA
+I4 = 12 nA
+30
+-0.5
+(nA)
+0
+dV13
+-30
+dl1
+-60
+(kΩ2)
+-8
+-4
+4
+-8
+-4
+0
+4
+8
+-8
+-4
+0
+8
+0
+4
+8
+Φ Φo
+Φ /Φo
+Φ /Φo
+D
+E
+F
+2
+14 = -12 nA
+14 = O nA
+14 = 12 nA
+1
+0
+Nc
+.1
+-2
+-3
+-2
+-1
+0
+1
+2
+3 
+-3
+-2 
+-1
+1
+2
+3
+-3
+-2
+0
+-1
+0
+1
+2
+3
+Φ /Φo
+Φ /Φo
+Φ /Φo

89E1T4oBgHgl3EQfCALf/content/tmp_files/2301.02860v1.pdf.txt

@@ -0,0 +1,2141 @@
+THERMODYNAMICAL MODELING OF MULTIPHASE FLOW
+SYSTEM WITH SURFACE TENSION AND FLOW
+HAJIME KOBA
+Abstract. We consider the governing equations for the motion of the viscous
+fluids in two moving domains and an evolving surface from both energetic
+and thermodynamic points of view. We make mathematical models for multi-
+phase flow with surface flow by our energetic variational and thermodynamic
+approaches.
+More precisely, we apply our energy densities, the first law of
+thermodynamics, and the law of conservation of total energy to derive our
+multiphase flow system with surface tension and flow. We study the conserva-
+tive forms and conservation laws of our system by using the surface transport
+theorem and integration by parts. Moreover, we investigate the enthalpy, the
+entropy, the Helmholtz free energy, and the Gibbs free energy of our model by
+applying the thermodynamic identity. The key idea of deriving surface tension
+and viscosities is to make use of both the first law of thermodynamics and our
+energy densities.
+1. Introduction
+Figure 1. Moving Domains, Surfaces and Notations
+We are interested in a mathematical modeling of a soap bubble floating in the
+air. When we focus on a soap bubble, we can see the fluid flow in the bubble. We
+2020 Mathematics Subject Classification. 80M30, 35Q79, 76-10, 80-10, 35A15.
+Key words and phrases. Multiphase flow, Surface tension, Surface flow, Mathematical model-
+ing, First law of thermodynamics, Energetic variational approach.
+This work was partly supported by the Japan Society for the Promotion of Science (JSPS)
+KAKENHI Grant Number JP21K03326.
+1
+arXiv:2301.02860v1  [math-ph]  7 Jan 2023
+
+PA,PB, Ps: Density
+n2
+UA, UB, Us: Velocity
+μA, μB, μs, 入A,入B, 入s: Viscosity
+nr
+TA,TB, s: Pressure
+I(t)
+A,OB,Os: Temperature
+A(t)
+B(t)
+eA, eB, es: Internal energy
+KA, KB, Ks: Thermal conductivity
+hA,hB,hs: Enthalpy
+SA, SB, Ss: Entropy
+2
+AB
+= A(t) UI(t) U 2B(t2
+HAJIME KOBA
+call the fluid flow in the bubble a surface flow. We can consider a surface flow
+as a fluid-flow on an evolving surface. To make a mathematical model for a soap
+bubble floating in the air, we have to study the dependencies among fluid-flows in
+two moving domains and surface flow. We consider the governing equations for
+the motion of the viscous fluids in the two moving domains and surface from both
+energetic and thermodynamic points of view. More precisely, we apply the first law
+of thermodynamics and our energy densities to derive our multiphase flow system
+with surface tension and flow.
+Let us first introduce fundamental notations. Let t ≥ 0 be the time variable,
+and x(= t(x1, x2, x3)) ∈ R3 the spatial variable.
+Fix T > 0.
+Let Ω ⊂ R3 be
+a bounded domain with a smooth boundary ∂Ω.
+The symbol nΩ = nΩ(x) =
+t(nΩ
+1 , nΩ
+2 , nΩ
+3 ) denotes the unit outer normal vector at x ∈ ∂Ω.
+Let ΩA(t)(=
+{ΩA(t)}0≤t<T ) be a bounded domain in R3 with a moving boundary Γ(t). Assume
+that Γ(t)(= {Γ(t)}0≤t<T ) is a smoothly evolving surface and is a closed Riemannian
+2-dimensional manifold. The symbol nΓ = nΓ(x, t) = t(nΓ
+1, nΓ
+2, nΓ
+3) denotes the unit
+outer normal vector at x ∈ Γ(t). For each t ∈ [0, T), assume that ΩA(t) ⋐ Ω. Set
+ΩB(t) = Ω \ ΩA(t). It is clear that Ω = ΩA(t) ∪ Γ(t) ∪ ΩB(t) (see Figure 1). Set
+ΩA,T =
+�
+0<t<T
+{ΩA(t) × {t}}, ΩB,T =
+�
+0<t<T
+{ΩB(t) × {t}},
+ΓT =
+�
+0<t<T
+{Γ(t) × {t}}, ΩT = Ω × (0, T), ∂ΩT = ∂Ω × (0, T).
+In this paper we assume that the fluids in ΩA,T , ΩB,T , and ΓT are compressible
+ones.
+Let us state physical notations.
+For ♯ = A, B, S, let ρ♯ = ρ♯(x, t), v♯ =
+v♯(x, t) = t(v♯
+1, v♯
+2, v♯
+3), π♯ = π♯(x, t), θ♯ = θ♯(x, t), e♯ = e♯(x, t), κ♯ = κ♯(x, t)
+and µ♯ = µ♯(x, t), λ♯ = λ♯(x, t) be the density, the velocity, the pressure, the
+temperature, the internal energy, the thermal conductivity, and two viscosities of
+the fluid in Ω♯(t), where ΩS(t) := Γ(t). The symbols h♯ = h♯(x, t), ς♯ = ς♯(x, t),
+F H
+♯
+= F H
+♯ (x, t), and F G
+♯ = F G
+♯ (x, t) denote the enthalpy, the entropy, the Helmholtz
+free energy, and the Gibbs free energy of the fluid in Ω♯(t), respectively (see Figure
+1). We call µ♯ the share viscosity and µ♯+λ♯ the dilatational viscosity. In particular,
+we often call µS the surface share viscosity, µS+λS the surface dilatational viscosity.
+We assume that ρ♯, v♯, π♯, θ♯, e♯, κ♯, µ♯, λ♯, h♯, ς♯, F H
+♯ , and F G
+♯ are smooth functions
+in R4.
+Remark 1.1. We call vS a total velocity, and πS a total pressure. Total velocity
+means that vS can be divided into surface velocity uS and motion velocity wS, that
+is, vS = uS + wS. Total pressure means one that includes surface pressure and
+tension. In this paper, we focus on the total velocity and the total pressure.
+Let us introduce several operators and notations. For each f = f(x, t) ∈ C1(R4)
+and V = V (x, t) = t(V1, V2, V3) ∈ [C1(R4)]3, DA
+t f := ∂tf + (vA · ∇)f, DB
+t f :=
+∂tf+(vB·∇)f, DS
+t f := ∂tf+(vS·∇)f, gradf := ∇f, divV := ∇·V , gradΓf := ∇Γf,
+divΓV := ∇Γ·V , (V ·∇)f := V1∂1f +V2∂2f +V3∂3f, (V ·∇Γ)f := V1∂Γ
+1 f +V2∂Γ
+2 f +
+V3∂Γ
+3 f, where ∇ := t(∂1, ∂2, ∂3), ∂i := ∂/∂xi, ∂t := ∂/∂t, ∇Γ := t(∂Γ
+1 , ∂Γ
+2 , ∂Γ
+3 ), and
+∂Γ
+i f := �3
+j=1(δij −nΓ
+i nΓ
+j )∂jf = ∂if −nΓ
+i (nΓ ·∇)f. Define the orthogonal projection
+
+MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW
+3
+PΓ to a tangent space by
+PΓ = PΓ(x, t) = I3×3 − nΓ ⊗ nΓ =
+�
+�
+1 − nΓ
+1nΓ
+1
+−nΓ
+1nΓ
+2
+−nΓ
+1nΓ
+3
+−nΓ
+2nΓ
+1
+1 − nΓ
+2nΓ
+2
+−nΓ
+2nΓ
+3
+−nΓ
+3nΓ
+1
+−nΓ
+3nΓ
+2
+1 − nΓ
+3nΓ
+3
+�
+� ,
+and the mean curvature HΓ in the direction nΓ by HΓ = HΓ(x, t) = −divΓnΓ,
+where I3×3 is the 3×3 identity matrix, and ⊗ denotes the tensor product. It is easy
+to check that PΓnΓ = t(0, 0, 0) and PΓ∇f = ∇Γf.
+Let us explain the key restrictions on the boundaries ∂ΩT and ΓT . We assume
+that
+(1.1)
+�
+�
+�
+�
+�
+vB = t(0, 0, 0)
+on ∂ΩT ,
+vA · nΓ = vB · nΓ = vS · nΓ
+on ΓT ,
+PΓvA = PΓvB = rPΓvS
+on ΓT ,
+�
+(nΩ · ∇)θB = 0
+on ∂ΩT ,
+θA = θB = θS
+on ΓT ,
+where r ∈ {0, 1}. We call PΓvA = PΓvB = rPΓvS a slip boundary condition if r = 1
+and a no-slip boundary condition if r = 0. Note that we do not consider phase
+transition in this paper.
+This paper has three purposes.
+The first purpose is to derive the following
+multiphase flow system with surface tension and flow:
+(1.2)
+�
+�
+�
+�
+�
+DA
+t ρA + (divvA)ρA = 0
+in ΩA,T ,
+DB
+t ρB + (divvB)ρB = 0
+in ΩB,T ,
+DS
+t ρS + (divΓvS)ρS = 0
+on ΓT ,
+(1.3)
+�
+�
+�
+�
+�
+ρADA
+t eA + (divvA)πA = divqA + eDA
+in ΩA,T ,
+ρBDB
+t eB + (divvB)πB = divqB + eDB
+in ΩB,T ,
+ρSDS
+t eS + (divΓvS)πS = divΓqS + eDS + qB · nΓ − qA · nΓ
+on ΓT ,
+(1.4)
+�
+�
+�
+�
+�
+ρADA
+t vA = divTA
+in ΩA,T ,
+ρBDB
+t vB = divTB
+in ΩB,T ,
+ρSDS
+t vS = divΓTS + �TBnΓ − �TAnΓ
+on ΓT ,
+where
+(1.5)
+�
+�
+�
+�
+�
+qA = qA(θA) := κAgradθA,
+qB = qB(θB) := κBgradθB,
+qS = qS(θS) := κSgradΓθS,
+(1.6)
+�
+�
+�
+�
+�
+eDA = eDA(vA) := µA|D(vA)|2 + λA|divvA|2,
+eDB = eDB(vB) := µB|D(vB)|2 + λB|divvB|2,
+eDS = eDS(vS) := µS|DΓ(vS)|2 + λS|divΓvS|2,
+�
+�
+�
+�
+�
+D(vA) := {(∇vA) + t(∇vA)}/2,
+D(vB) := {(∇vB) + t(∇vB)}/2,
+DΓ(vS) := {(PΓ∇ΓvS) + t(PΓ∇ΓvS)}/2,
+
+4
+HAJIME KOBA
+(1.7)
+�
+�
+�
+�
+�
+TA = TA(vA, πA) := µAD(vA) + λA(divvA)I3×3 − πAI3×3,
+TB = TB(vB, πB) := µBD(vB) + λB(divvB)I3×3 − πBI3×3,
+TS = TS(vS, πS) := µSDΓ(vS) + λS(divΓvS)PΓ − πSPΓ,
+(1.8)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�TA = �TA(vA, πA) =
+�
+µAnΓ · (nΓ · ∇)vA + λA(divvA) − πA if r = 0,
+TA(vA, πA) if r = 1,
+�TB = �TB(vB, πB) =
+�
+µBnΓ · (nΓ · ∇)vB + λB(divvB) − πB if r = 0,
+TB(vB, πB) if r = 1.
+Here |D(vA)|2 = D(vA) : D(vA), |D(vB)|2 = D(vB) : D(vB), and |DΓ(vS)|2 =
+DΓ(vS) : DΓ(vS).
+The symbol : denotes the Frobenius inner product, that is,
+M : N = �3
+i,j=1[M]ij[N]ij, where M, N are two 3×3 matrices, and [M]ij denotes
+the (i, j)-component of the matrix M. We call qA, qB, qS the heat fluxes, eDA, eDB,
+eDS the energy densities for the energy dissipation due to the viscosities, D(vA),
+D(vB) strain rate tensors, DΓ(vS) a surface strain tensor, TA, TB stress tensors, and
+TS a surface stress tensor. We often call TS the surface stress tensor determined by
+the Boussinesq-Scriven law. More precisely, under the restrictions ( 1.1) we apply
+our energy densities and thermodynamic approaches to derive ( 1.2)-( 1.4). See
+Section 4 for details.
+Remark 1.2. (i) Using nΓ · nΓ = 1, PΓ∇f = ∇Γf, (nΓ · ∇Γ)f = 0, and HΓ =
+−divΓnΓ, we easily check that
+divΓ(πSPΓ) = gradΓπS + πSHΓnΓ,
+2DΓ(vS) = PΓ{(∇vS) + t(∇vS)}PΓ = PΓ{(∇ΓvS) + t(∇ΓvS)}PΓ.
+Note that 2[DΓ(vS)]ij = ∂Γ
+i vS
+j + ∂Γ
+j vS
+i − nΓ
+i (nΓ · ∂Γ
+j vS) − nΓ
+j (nΓ · ∂Γ
+i vS),
+∇vS =
+�
+�
+∂1vS
+1
+∂2vS
+1
+∂3vS
+1
+∂1vS
+2
+∂2vS
+2
+∂3vS
+2
+∂1vS
+3
+∂2vS
+3
+∂3vS
+3
+�
+� , ∇ΓvS =
+�
+�
+∂Γ
+1 vS
+1
+∂Γ
+2 vS
+1
+∂Γ
+3 vS
+1
+∂Γ
+1 vS
+2
+∂Γ
+2 vS
+2
+∂Γ
+3 vS
+2
+∂Γ
+1 vS
+3
+∂Γ
+2 vS
+3
+∂Γ
+3 vS
+3
+�
+� .
+We often call πSHΓnΓ surface tension.
+(ii) If the fluids in ΩA,T , ΩB,T , ΓT are barotropic fluids, then we can write
+�
+�
+�
+�
+�
+πA = πA(ρA) = ρAp′
+A(ρA) − pA(ρA),
+πB = πB(ρB) = ρBp′
+B(ρB) − pB(ρB),
+πS = πS(ρS) = ρSp′
+S(ρS) − pS(ρS).
+Here pA, pB, pS are three C1-functions, p′ = p′(r) = dp/dr(r). See Theorem 2.4,
+Remark 2.5, and Section 4 for details.
+The second purpose is to study the conservative forms and conservation laws of
+system ( 1.2)-( 1.4). In fact, if we set DN
+t f = ∂tf +(vS ·nΓ)(nΓ ·∇)f, and the total
+energy E♯ = E♯(x, t) by E♯ = ρ♯|v♯|2/2 + ρ♯e♯, then we can write our system as the
+conservative form:
+(1.9)
+�
+�
+�
+�
+�
+∂tρA + div(ρAvA) = 0
+in ΩA,T ,
+∂tρB + div(ρBvB) = 0
+in ΩB,T ,
+DN
+t ρS + divΓ(ρSvS) = 0
+on ΓT ,
+
+MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW
+5
+(1.10)
+�
+�
+�
+�
+�
+∂tEA + div(EAvA − qA − TAvA) = 0
+in ΩA,T ,
+∂tEB + div(EBvB − qB − TBvB) = 0
+in ΩB,T ,
+DN
+t ES + divΓ(ESvS − qS − TSvS) = ES
+on ΓT ,
+(1.11)
+�
+�
+�
+�
+�
+∂t(ρAvA) + div(ρAvA ⊗ vA − TA) = t(0, 0, 0)
+in ΩA,T ,
+∂t(ρBvB) + div(ρBvB ⊗ vB − TB) = t(0, 0, 0)
+in ΩB,T ,
+DN
+t (ρSvS) + divΓ(ρSvS ⊗ vS − TS) = �TBnΓ − �TAnΓ
+on ΓT ,
+where
+ES := �TBnΓ · vS − �TAnΓ · vS + qB · nΓ − qA · nΓ.
+Moreover, any solution to system ( 1.1)-( 1.4) satisfies that for t1 < t2,
+(1.12)
+�
+ΩA(t2)
+ρA(x, t2) dx +
+�
+ΩB(t2)
+ρB(x, t2) dx +
+�
+Γ(t2)
+ρS(x, t2) dH2
+x
+=
+�
+ΩA(t1)
+ρA(x, t1) dx +
+�
+ΩB(t1)
+ρB(x, t1) dx +
+�
+Γ(t1)
+ρS(x, t1) dH2
+x,
+(1.13)
+�
+ΩA(t2)
+EA dx +
+�
+ΩB(t2)
+EB dx +
+�
+Γ(t2)
+ES dH2
+x
+=
+�
+ΩA(t1)
+EA dx +
+�
+ΩB(t1)
+EB dx +
+�
+Γ(t1)
+ES dH2
+x,
+(1.14)
+�
+ΩA(t2)
+1
+2ρA|vA|2 dx +
+�
+ΩB(t2)
+1
+2ρB|vB|2 dx +
+�
+Γ(t2)
+1
+2ρS|vS|2 dH2
+x
++
+� t2
+t1
+�
+ΩA(t)
+eDA dxdt +
+� t2
+t1
+�
+ΩB(t)
+eDB dxdt +
+� t2
+t1
+�
+Γ(t)
+eDS dH2
+xdt
+=
+�
+ΩA(t1)
+1
+2ρA|vA|2 dx +
+�
+ΩB(t1)
+1
+2ρB|vB|2 dx +
+�
+Γ(t1)
+1
+2ρS|vS|2 dH2
+x
++
+� t2
+t1
+�
+ΩA(t)
+(divvA)πA dxdt +
+� t2
+t1
+�
+ΩB(t)
+(divvB)πB dxdt
++
+� t2
+t1
+�
+Γ(t)
+(divΓvS)πS dH2
+xdt.
+Here dH2
+x denotes the 2-dimensional Hausdorff measure. Under some assumptions
+(see Theorem 2.8), any solution to system ( 1.1)-( 1.3) satisfies that for t1 < t2
+(1.15)
+�
+ΩA(t2)
+ρAvA dx +
+�
+ΩB(t2)
+ρBvB dx +
+�
+Γ(t2)
+ρSvS dH2
+x
+=
+�
+ΩA(t1)
+ρAvA dx +
+�
+ΩB(t1)
+ρBvB dx +
+�
+Γ(t1)
+ρSvS dH2
+x.
+We often call ( 1.12), ( 1.13), ( 1.14), and ( 1.15), the law of conservation of mass,
+the law of conservation of total energy, the energy law of our system, and the law
+of conservation of momentum, respectively. See Theorem 2.8 and Section 5 for
+details.
+
+6
+HAJIME KOBA
+Remark 1.3. From DS
+t f = DN
+t f + (vS · ∇Γ)f for f ∈ C1(R4), we see that
+DN
+t (ρSf) + divΓ(ρSfvS) = ρSDS
+t f,
+DN
+t (ρSvS) + divΓ(ρSvS ⊗ vS) = ρSDS
+t vS.
+Since ρS, vS ∈ C1(R4) in this paper, we can define DN
+t
+for ρS, ρSvS.
+The third purpose is to investigate the thermodynamic potential such as the
+enthalpy h♯, the entropy ς♯, the Helmholtz free energy F H
+♯ , and the Gibbs free
+energy F G
+♯ of the fluid in Ω♯(t), where ♯ = A, B, S. Assume that (ρ♯, θ♯) are positive
+functions. Set the enthalpy h♯ by h♯ = e♯ + π♯/ρ♯. Then
+(1.16)
+�
+�
+�
+�
+�
+∂t(ρAhA) + div(ρAhAvA − qA) = eDA + DA
+t πA,
+∂t(ρBhB) + div(ρBhBvB − qB) = eDB + DB
+t πB,
+DN
+t (ρShS) + divΓ(ρShSvS − qS) = eDS + DS
+t πS + qB · nΓ − qA · nΓ.
+Suppose that the thermodynamic identity (Gibbs [9]): D♯
+te♯ = θ♯D♯
+tς♯ −π♯D♯
+t(1/ρ♯)
+holds. Then
+(1.17)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂t(ρAςA) + div
+�
+ρAςAvA − qA
+θA
+�
+=
+eDA
+θA + qA·gradθA
+θ2
+A
+,
+∂t(ρBςB) + div
+�
+ρBςBvB − qB
+θB
+�
+=
+eDB
+θB + qB·gradθB
+θ2
+B
+,
+DN
+t (ρSςS) + divΓ
+�
+ρSςSvS − qS
+θS
+�
+=
+eDS
+θS + qS·gradΓθS
+θ2
+S
++ qB·nΓ−qA·nΓ
+θS
+.
+Set the Helmholtz free energy F H
+♯
+by F H
+♯
+= e♯ − θ♯ς♯. Then
+(1.18)
+�
+�
+�
+�
+�
+ρADA
+t F H
+A + ρAςADA
+t θA = −(divvA)πA,
+ρBDB
+t F H
+B + ρBςBDB
+t θB = −(divvB)πB,
+ρSDS
+t F H
+S + ρSςSDS
+t θS = −(divΓvS)πS.
+Set the Gibbs free energy F G
+♯
+by F G
+♯ = h♯ − θ♯ς♯. Then
+(1.19)
+�
+�
+�
+�
+�
+ρADA
+t F G
+A + ρAςADA
+t θA = DA
+t πA,
+ρBDB
+t F G
+B + ρBςBDB
+t θB = DB
+t πB,
+ρSDS
+t F G
+S + ρSςSDS
+t θS = DS
+t πS.
+See Theorem 2.9 and Section 6 for details.
+Remark 1.4. (i) Since
+TA(vA, πA) : D(vA) = eDA − (divvA)πA,
+TB(vB, πB) : D(vB) = eDB − (divvB)πB,
+TS(vS, πS) : DΓ(vS) = eDS − (divΓvS)πS,
+it follows from ( 1.18) to see that
+�
+�
+�
+�
+�
+ρADA
+t F H
+A + ρAςADA
+t θA − TA(vA, πA) : D(vA) = −eDA,
+ρBDB
+t F H
+B + ρBςBDB
+t θB − TB(vB, πB) : D(vB) = −eDB,
+ρSDS
+t F H
+S + ρSςSDS
+t θS − TS(vS, πS) : DΓ(vS) = −eDS.
+
+MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW
+7
+Note that PΓ : DΓ(vS) = divΓvS.
+(ii) Set DAf := ρADA
+t f, DBf := ρBDB
+t f, DSf := ρSDS
+t f. Then
+�
+�
+�
+�
+�
+DAeA = θADAςA − πADA(1/ρA),
+DBeB = θBDBςB − πBDB(1/ρB),
+DSeS = θSDSςS − πSDS(1/ρS),
+�
+�
+�
+�
+�
+DAhA = θADAςA + (1/ρA)DAπA,
+DBhB = θBDBςB + (1/ρB)DBπB,
+DShS = θSDSςS + (1/ρS)DSπS,
+�
+�
+�
+�
+�
+DAF H
+A = −ςADAθA − πADA(1/ρA),
+DBF H
+B = −ςBDBθB − πBDB(1/ρB),
+DSF H
+S = −ςSDSθS − πSDS(1/ρS),
+�
+�
+�
+�
+�
+DAF G
+A = −ςADAθA + (1/ρA)DAπA,
+DBF G
+B = −ςBDBθB + (1/ρB)DBπB,
+DSF G
+S = −ςSDSθS + (1/ρS)DSπS.
+We often call 1/ρ♯ a specific volume.
+Let us explain the main difficulties in the derivation of our multiphase flow system
+with surface tension and flow, and the key ideas to overcome these difficulties. The
+main difficulties are to derive the viscous terms of the system, to derive the surface
+tension from a theoretical point of view, and to derive the dependencies among
+fluid-flows in two moving domains and surface flow. To overcome these difficulties,
+we apply the first law of thermodynamics (Theorem 2.4), our energy densities
+(Definition 2.2), and the conservation law of total energy to derive equations ( 1.3)
+and ( 1.4). See Section 4 for details.
+Let us mention the study of surface flow (interfacial flow). Boussinesq [5] first
+discovered the existence of surface flow. Scriven [22] considered their surface stress
+tensor. Slattery [23] investigated some properties of the surface stress tensor de-
+termined by the Boussinesq-Scriven law (see TS in ( 1.7)). Then many researchers
+have studied surface flow (see Slattery-Sagis-Oh [24] and Gatignol-Prud’homme [8]
+for the study of interfacial phenomena).
+Let us state derivations of the governing equations for the motion of the viscous
+fluid on manifolds and surfaces. Taylor [26] introduced their surface stress tensor
+to make their incompressible viscous fluid system on a manifold. Mitsumatsu-Yano
+[20] applied their energetic variational approach to derive their incompressible vis-
+cous fluid system on a manifold. Arnaudon-Cruzeiro [2] made use of their stochastic
+variational approach to derive their incompressible viscous fluid system on a man-
+ifold. Koba-Liu-Giga [18] employed their energetic variational approach and the
+generalized Helmholtz-Weyl decomposition on a closed surface to derive their in-
+compressible fluid systems on an evolving closed surface. Koba [14, 15] applied their
+energetic variational approaches and the first law of thermodynamics to derive their
+compressible fluid flow systems on an evolving closed surface and an evolving sur-
+face with a boundary. This paper modifies and improves the methods in [14, 15] to
+derive our multiphase flow system.
+Now we mention results for modeling of multiphase flow system with surface
+flow.
+Bothe-Pr¨uss [4] made their multiphase flow system with surface flow by
+using the surface stress tensor determined by the Boussinesq-Scriven law. Koba
+[17] derived the inviscid multiphase flow system with surface flow by applying a
+geometric variational approach.
+This paper derives our multiphase flow system
+from a thermodynamic point of view. Therefore, our modeling methods are different
+from ones in [4] and [17].
+Finally, we introduce some results and textbooks related to this paper. Hyon-
+Kwak-Liu [13] and Koba-Sato [19] applied their energetic variational approaches
+to derive and study their complex and non-Newtonian fluid systems in domains.
+
+8
+HAJIME KOBA
+Feireisl [7] studied the motion of the viscous fluid in a domain from a thermody-
+namic point of view. We refer the readers to Gyarmati [12] and Gurtin-Fried-Anand
+[11] for the theory of thermodynamics, Chapter XIII in Angel [1] for thermody-
+namical potential such as internal energy, enthalpy, entropy, and free energies, and
+Pr¨uss-Simonett [21] for several elliptic and parabolic equations on hypersurfaces.
+The outline of this paper is as follows: In Section 2, we first introduce the
+transport theorems and the energy densities for our model, and then we state the
+main results of this paper. In Section 3, we make use of the transport theorems to
+derive the first law of thermodynamics, and apply integration by parts to calculate
+variations of our dissipation energies. In Section 4, we apply our thermodynamic
+approaches to make mathematical models for multiphase flow with surface tension
+and flow. In Section 5, we study the conservation and energy laws of our system.
+In Section 6, we investigate the thermodynamic potential for our system.
+2. Main Results
+We first introduce the transport theorems and the energy densities for our mul-
+tiphase flow system. Then we state the main results.
+Definition 2.1 (ΩT is flowed by the velocity fields (vA, vB, vS)). We say that ΩT
+is flowed by the velocity fields (vA, vB, vS) if for each 0 < t < T, f ∈ C1(R4), and
+Λ ⊂ Ω,
+d
+dt
+�
+ΩA(t)∩Λ
+f(x, t) dx =
+�
+ΩA(t)∩Λ
+{DA
+t f + (divvA)f} dx,
+(2.1)
+d
+dt
+�
+ΩB(t)∩Λ
+f(x, t) dx =
+�
+ΩB(t)∩Λ
+{DB
+t f + (divvB)f} dx,
+(2.2)
+d
+dt
+�
+Γ(t)∩Λ
+f(x, t) dH2
+x =
+�
+Γ(t)∩Λ
+{DS
+t f + (divΓvS)f} dH2
+x.
+(2.3)
+Here D♯
+tf = ∂tf +(v♯·∇)f, divΓvS = ∂Γ
+1 vS
+1 +∂Γ
+2 vS
+2 +∂Γ
+3 vS
+3 , ∂Γ
+j f = ∂jf −nΓ
+j (nΓ·∇)f,
+where ♯ = A, B, S, and j = 1, 2, 3.
+We often call ( 2.1), ( 2.2) the transport theorems, and ( 2.2) the surface transport
+theorem. The derivation of the surface transport theorem can be founded in [3, 10,
+6, 18]. Throughout this paper we assume that ΩT is flowed by the velocity fields
+(vA, vB, vS).
+Definition 2.2 (Energy densities). Set
+�
+�
+�
+�
+�
+eKA = ρA|vA|2/2,
+eKB = ρB|vB|2/2,
+eKS = ρS|vS|2/2,
+�
+�
+�
+�
+�
+eDA = µA|D(vA)|2 + λA|divvA|2,
+eDB = µB|D(vB)|2 + λB|divvB|2,
+eDS = µS|DΓ(vS)|2 + λS|divΓvS|2,
+�
+�
+�
+�
+�
+eWA = (divvA)πA,
+eWB = (divvB)πB,
+eWS = (divΓvS)πS,
+�
+�
+�
+�
+�
+eQA = κA|gradθA|2,
+eQB = κB|gradθB|2,
+eQS = κS|gradΓθS|2.
+We call eK♯ the kinetic energy, eD♯ the energy density for the energy dissipation due
+to the viscosities (µ♯, λ♯), eW♯ the power density for the work done by the pressure
+π♯, and eQ♯ the energy density for the energy dissipation due to thermal diffusion.
+
+MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW
+9
+See [19], [14], and Remark 2.5 in [15] for mathematical validity of the energy densi-
+ties. Applying the energy densities, the restrictions ( 1.1), and our thermodynamic
+approaches, we derive system ( 1.2)-( 1.4) in Section 4.
+We now state the main results of this paper. From Definition 2.1, we have
+Proposition 2.3 (Continuity equations). Assume that for each 0 < t < T and
+Λ ⊂ Ω,
+d
+dt
+� �
+ΩA(t)∩Λ
+ρA(x, t) dx +
+�
+ΩB(t)∩Λ
+ρB(x, t) dx +
+�
+Γ(t)∩Λ
+ρS(x, t) dH2
+x
+�
+= 0.
+Then (ρA, ρB, ρS) satisfies ( 1.2).
+The proof of Proposition 2.3 is left for the readers.
+Theorem 2.4 (First law of thermodynamics). Let �QA, �QB, �QS ∈ C(R4). Assume
+that (ρA, ρB, ρS) satisfies ( 1.2). Then
+(i) Suppose that for every 0 < t < T and Λ ⊂ Ω,
+d
+dt
+�
+ΩA(t)∩Λ
+ρAeA dx =
+�
+ΩA(t)∩Λ
+{ �QA − (divvA)πA} dx,
+d
+dt
+�
+ΩB(t)∩Λ
+ρBeB dx =
+�
+ΩB(t)∩Λ
+{ �QB − (divvB)πB} dx,
+d
+dt
+�
+Γ(t)∩Λ
+ρSeS dH2
+x =
+�
+Γ(t)∩Λ
+{ �QS − (divΓvS)πS} dH2
+x.
+Then
+(2.4)
+�
+�
+�
+�
+�
+ρADA
+t eA + (divvA)πA = �QA in ΩA,T ,
+ρBDB
+t eB + (divvB)πB = �QB in ΩB,T ,
+ρSDS
+t eS + (divΓvS)πS = �QS on ΓT .
+(ii) Let pA, pB, pS ∈ C1(R). Suppose that for every 0 < t < T and Λ ⊂ Ω,
+d
+dt
+�
+ΩA(t)∩Λ
+{ρAeA − pA(ρA)} dx =
+�
+ΩA(t)∩Λ
+�QA dx,
+d
+dt
+�
+ΩB(t)∩Λ
+{ρBeB − pB(ρB)} dx =
+�
+ΩB(t)∩Λ
+�QB dx,
+d
+dt
+�
+Γ(t)∩Λ
+{ρSeS − pS(ρS)} dH2
+x =
+�
+Γ(t)∩Λ
+�QS dH2
+x.
+Then
+(2.5)
+�
+�
+�
+�
+�
+ρADA
+t eA + (divvA)ΠA = �QA in ΩA,T ,
+ρBDB
+t eB + (divvB)ΠB = �QB in ΩB,T ,
+ρSDS
+t eS + (divΓvS)ΠS = �QS on ΓT .
+Here
+(2.6)
+�
+�
+�
+�
+�
+ΠA = ΠA(ρA) = ρAp′
+A(ρA) − pA(ρA),
+ΠB = ΠB(ρB) = ρBp′
+B(ρB) − pB(ρB),
+ΠS = ΠS(ρS) = ρSp′
+S(ρS) − pS(ρS).
+
+10
+HAJIME KOBA
+Remark 2.5. (i) We can write system ( 2.4) as follows:
+�
+�
+�
+�
+�
+DAeA = �QA − πADA(1/ρA),
+DBeB = �QB − πBDB(1/ρB),
+DSeS = �QS − πSDS(1/ρS),
+where D♯f = ρ♯D♯
+tf. Therefore, we call Theorem 2.4 the first law of thermody-
+namics in this paper. See also (ii) in Remark 1.4.
+(ii) The pressures (ΠA, ΠB, ΠS) derived from the assertion (ii) of Theorem 2.4 cor-
+respond to the pressures derived from an energetic variational approach (see [17]).
+Next we consider the variation of our dissipation energies. Let r ∈ {0, 1} and 0 <
+t < T. Let ϕA, ϕB, ϕS ∈ [C∞(R3)]3 and ψA, ψB, ψS ∈ C∞(R3). For −1 < ε < 1,
+vε
+A := vA +εϕA, vε
+B := vB +εϕB, vε
+S := vS +εϕS, θε
+A := θA +εψA, θε
+B := θB +εψB,
+and θε
+S := θS + εψS. We call (vε
+A, vε
+B, vε
+S, θε
+A, θε
+B, θε
+S) variations of (vA, vB, vS, θA,
+θB, θS). From ( 1.1), for −1 < ε < 1, we assume that
+(2.7)
+�
+�
+�
+�
+�
+vε
+B = t(0, 0, 0)
+on ∂Ω,
+vε
+A · nΓ = vε
+B · nΓ = vε
+S · nΓ
+on Γ(t),
+PΓvε
+A = PΓvε
+B = rPΓvε
+S
+on Γ(t),
+�
+(nΩ · ∇)θε
+B = 0
+on ∂Ω,
+θε
+A = θε
+B = θε
+S
+on Γ(t).
+Then we have
+(2.8)
+�
+�
+�
+�
+�
+ϕB = t(0, 0, 0)
+on ∂Ω,
+ϕA · nΓ = ϕB · nΓ = ϕS · nΓ
+on Γ(t),
+PΓϕA = PΓϕB = rPΓϕS
+on Γ(t),
+and
+(2.9)
+�
+(nΩ · ∇)ψB = 0
+on ∂Ω,
+ψA = ψB = ψS
+on Γ(t).
+For each variation (vε
+A, vε
+B, vε
+S, θε
+A, θε
+B, θε
+S),
+ED[vε
+A, vε
+B, vε
+S] :=
+�
+ΩA(t)
+�
+− µA
+2 |D(vε
+A)|2 − λA
+2 |divvε
+A|2
+�
+dx
++
+�
+ΩB(t)
+�
+− µB
+2 |D(vε
+B)|2 − λB
+2 |divvε
+B|2
+�
+dx
++
+�
+Γ(t)
+�
+− µS
+2 |DΓ(vε
+S)|2 − λS
+2 |divΓvε
+S|2
+�
+dH2
+x,
+EW [vε
+A, vε
+B, vε
+S] :=
+�
+ΩA(t)
+(divvε
+A)πA dx
++
+�
+ΩB(t)
+(divvε
+B)πB dx +
+�
+Γ(t)
+(divΓvε
+S)πS dH2
+x,
+and
+ET D[θε
+A, θε
+B, θε
+S] :=
+�
+ΩA(t)
+�
+− κA
+2 |gradθε
+A|2
+�
+dx
++
+�
+ΩB(t)
+�
+− κB
+2 |gradθε
+B|2
+�
+dx +
+�
+Γ(t)
+�
+− κS
+2 |gradΓθε
+S|2
+�
+dH2
+x.
+
+MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW
+11
+Set ED+W [·] = ED[·] + EW [·]. We call ED the energy dissipation due to viscosities,
+EW the work done by pressures, and ET D the energy dissipation due to thermal
+diffusion.
+Theorem 2.6 (Forces derived from variation of dissipation energies). Let r ∈
+{0, 1}, 0 < t < T, and FA, FB, FS ∈ [C(R3)]3. Assume that for every ϕA, ϕB, ϕS ∈
+[C∞(R3)]3 satisfying ( 2.8),
+d
+dε
+����
+ε=0
+ED+W [vε
+A, vε
+B, vε
+S] =
+�
+ΩA(t)
+FA·ϕA dx+
+�
+ΩB(t)
+FB·ϕB dx+
+�
+Γ(t)
+FS·ϕS dH2
+x.
+Then
+(2.10)
+�
+�
+�
+�
+�
+FA = divTA(vA, πA)
+in ΩA(t),
+FB = divTB(vB, πB)
+in ΩB(t),
+FS = divΓTS(vS, πS) + �TB(vB, πB)nΓ − �TA(vA, πA)nΓ
+on Γ(t),
+where (TA, TB, TS) and ( �TA, �TB) are defined by ( 1.7) and ( 1.8), respectively.
+Theorem 2.7 (Endothermic energies derived from thermal diffusion). Let 0 <
+t < T, and QA, QB, QS ∈ C(R3). Assume that for every ψA, ψB, ψS ∈ C∞(R3)
+satisfying ( 2.9),
+d
+dε
+����
+ε=0
+ET D[θε
+A, θε
+B, θε
+S] =
+�
+ΩA(t)
+QAψA dx +
+�
+ΩB(t)
+QBψB dx +
+�
+Γ(t)
+QSψS dH2
+x.
+Then
+(2.11)
+�
+�
+�
+�
+�
+QA = div(κA∇θA)
+in ΩA(t),
+QB = div(κB∇θB)
+in ΩB(t),
+QS = divΓ(κS∇ΓθS) + κB(nΓ · ∇)θB − κA(nΓ · ∇)θA
+on Γ(t).
+Finally, we state the conservation laws and thermodynamic potential for our
+multiphase flow system.
+Theorem 2.8 (Conservative forms and conservation laws). Let r ∈ {0, 1}. Then
+(i) Any solution to system ( 1.2)-( 1.4) satisfies ( 1.9)-( 1.11).
+(ii) Any solution to system ( 1.1)-( 1.4) satisfies ( 1.12)-( 1.14).
+(iii) Assume that r = 1, and that for 0 < t < T
+(2.12)
+�
+∂Ω
+{µBD(vB)nΩ + λB(divvB)nΩ − πBnΩ} dH2
+x = t(0, 0, 0).
+Then any solution to ( 1.1)-( 1.4) satisfies ( 1.15).
+Theorem 2.9 (Thermodynamic potential). Let ♯ = A, B, S, and r ∈ {0, 1}. Sup-
+pose that ρ♯ and θ♯ are positive functions. Set h♯ = e♯ + π♯/ρ♯, F H
+♯
+= e♯ − θ♯ς♯,
+F G
+♯
+= h♯ − θ♯ς♯.
+Assume that e♯ satisfies D♯
+te♯ = θ♯D♯
+tς♯ − π♯D♯
+t(1/ρ♯).
+Then
+( 1.16)-( 1.19) hold.
+We prove Theorems 2.4, 2.6, 2.7 in Section 3, Theorem 2.8 in Section 5, and
+Theorem 2.9 in Section 6. In Section 4 we derive system ( 1.2)-( 1.4) by our ther-
+modynamic approaches.
+
+12
+HAJIME KOBA
+3. Application of Transport Theorems and Integration by Parts
+We apply the transport theorems (Definition 2.1) and several formulas for inte-
+gration by parts (Lemma 3.1) to prove Theorems 2.4, 2.6, and 2.7.
+Lemma 3.1 (Formulas for integration by parts).
+Fix 0 ≤ t < T and j = 1, 2, 3.
+Then for every f, g ∈ C1(R3),
+�
+ΩA(t)
+(∂jf)g dx = −
+�
+ΩA(t)
+f(∂jg) dx +
+�
+Γ(t)
+fgnΓ
+j dH2
+x,
+�
+ΩB(t)
+(∂jf)g dx = −
+�
+ΩB(t)
+f(∂jg) dx +
+�
+∂Ω
+fgnΩ
+j dH2
+x −
+�
+Γ(t)
+fgnΓ
+j dH2
+x,
+�
+Γ(t)
+(∂Γ
+j f)g dH2
+x = −
+�
+Γ(t)
+f(∂Γ
+j g) dH2
+x −
+�
+Γ(t)
+HΓfgnΓ
+j dH2
+x,
+where HΓ = −divΓnΓ and ∂Γ
+j f = ∂jf − nΓ
+j (nΓ · ∇)f.
+Here nΓ = nΓ(x, t) =
+t(nΓ
+1, nΓ
+2, nΓ
+3) denotes the unit outer normal vector at x ∈ Γ(t) and nΩ = nΩ(x) =
+t(nΩ
+1 , nΩ
+2 , nΩ
+3 ) the unit outer normal vector at x ∈ ∂Ω (see Figure 1).
+Applying the Gauss divergence theorem and the surface divergence theorem (Sec-
+tion 9 in [25], Theorem 2.3 in [16]), we can prove Lemma 3.1.
+We now make use of the transport theorems to prove Theorem 2.4.
+Proof of Theorem 2.4. We only prove (ii) since the proof of (i) is similar. We as-
+sume that (ρA, ρB, ρS) satisfies ( 1.2). Let �QA, �QB, �QS ∈ C(R4) and pA, pB, pS ∈
+C1(R). We first show that for every 0 < t < T and Λ ⊂ Ω,
+d
+dt
+�
+ΩA(t)∩Λ
+{ρAeA − pA(ρA)} dx =
+�
+ΩA(t)∩Λ
+{ρADA
+t eA + (divvA)ΠA} dx,
+(3.1)
+d
+dt
+�
+ΩB(t)∩Λ
+{ρBeB − pB(ρB)} dx =
+�
+ΩB(t)∩Λ
+{ρBDB
+t eB + (divvB)ΠB} dx,
+(3.2)
+d
+dt
+�
+Γ(t)∩Λ
+{ρSeS − pS(ρS)} dH2
+x =
+�
+Γ(t)∩Λ
+{ρSDS
+t eS + (divΓvS)ΠS} dH2
+x,
+(3.3)
+where (ΠA, ΠB, ΠS) is defined by ( 2.6). We only derive ( 3.3). Using the surface
+transport theorem ( 2.3) and ( 1.2), we check that for 0 < t < T and Λ ⊂ Ω
+d
+dt
+�
+Γ(t)∩Λ
+{ρSeS−pS(ρS)} dH2
+x =
+�
+Γ(t)∩Λ
+{(DS
+t ρS)eS+ρSDteS+ρSeS(divΓvS)} dH2
+x
++
+�
+Γ(t)∩Λ
+{−DS
+t ρSp′
+S(ρS) − pS(ρS)(divΓvS)} dH2
+x
+=
+�
+Γ(t)∩Λ
+{ρSDS
+t eS + (ρSp′
+S(ρS) − pS(ρS))(divΓvS)} dH2
+x.
+
+MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW
+13
+Thus, we see ( 3.3). We assume that for every 0 < t < T and Λ ⊂ Ω,
+d
+dt
+�
+ΩA(t)∩Λ
+{ρAeA − pA(ρA)} dx =
+�
+ΩA(t)∩Λ
+�QA dx,
+d
+dt
+�
+ΩB(t)∩Λ
+{ρBeB − pB(ρB)} dx =
+�
+ΩB(t)∩Λ
+�QB dx,
+d
+dt
+��
+Γ(t)∩Λ
+{ρSeS − pS(ρS)} dH2
+x =
+�
+Γ(t)∩Λ
+�QS dH2
+x.
+Applying ( 3.1)-( 3.3), we have ( 2.5). Therefore, Theorem 2.4 is proved.
+□
+Let us apply integration by parts to prove Theorems 2.6 and 2.7.
+Proof of Theorem 2.6. Let r ∈ {0, 1} and 0 < t < T. Let ϕA, ϕB, ϕS ∈ [C∞(R3)]3
+satisfying ( 2.8). A direct calculation gives
+d
+dε
+����
+ε=0
+ED[vε
+A, vε
+B, vε
+S] = −
+�
+ΩA(t)
+{µAD(vA) : D(ϕA) + λA(divvA)(divϕA)} dx
+−
+�
+ΩB(t)
+{µBD(vB) : D(ϕB) + λB(divvB)(divϕB)} dx
+−
+�
+Γ(t)
+{µSDΓ(vS) : DΓ(ϕS) + λS(divΓvS)(divΓϕS)} dH2
+x,
+and
+d
+dε
+����
+ε=0
+EW [vε
+A, vε
+B, vε
+S]
+=
+�
+ΩA(t)
+(divϕA)πA dx +
+�
+ΩB(t)
+(divϕB)πB dx +
+�
+Γ(t)
+(divΓϕS)πS dH2
+x.
+Using integration by parts (Lemma 3.1), we find that
+(3.4)
+d
+dε
+����
+ε=0
+ED+W [vε
+A, vε
+B, vε
+S] =
+�
+ΩA(t)
+divTA(vA, πA) · ϕA dx
++
+�
+ΩB(t)
+divTB(vB, πB) · ϕB dx +
+�
+Γ(t)
+divΓTS(vS, πS) · ϕS dH2
+x
+−
+�
+Γ(t)
+{TA(vA, πA)nΓ} · ϕA dH2
+x +
+�
+Γ(t)
+{TB(vB, πB)nΓ} · ϕB dH2
+x
+−
+�
+∂Ω
+{TB(vB, πB)nΩ} · ϕB dH2
+x,
+where (TA, TB, TS) and (�TA, �TB) are defined by ( 1.7) and ( 1.8). Applying ( 2.8),
+we check that
+(R.H.S) of ( 3.4) =
+�
+ΩA(t)
+divTA(vA, πA) · ϕA dx +
+�
+ΩB(t)
+divTB(vB, πB) · ϕB dx
++
+�
+Γ(t)
+{divΓTS(vS, πS) + �TB(vB, πB)nΓ − �TA(vA, πA)nΓ} · ϕS dH2
+x.
+
+14
+HAJIME KOBA
+Here we used the facts that
+D(vA)nΓ · ϕA = (nΓ · {(nΓ · ∇)vA})(nΓ · ϕS) if r = 0,
+D(vB)nΓ · ϕB = (nΓ · {(nΓ · ∇)vB})(nΓ · ϕS) if r = 0.
+From fundamental lemma of calculation of variations, we have ( 2.10). Therefore,
+Theorem 2.6 is proved.
+□
+Proof of Theorem 2.7. Fix 0 < t < T. Let ψA, ψB, ψS ∈ C∞(R3) satisfying ( 2.9).
+Since
+d
+dε
+����
+ε=0
+ET D[θε
+A, θε
+B, θε
+S]
+= −
+�
+ΩA(t)
+κA∇θA·∇ψA dx−
+�
+ΩB(t)
+κB∇θB·∇ψB dx−
+�
+Γ(t)
+κS∇ΓθS·∇ΓψS dH2
+x,
+we apply the integration by parts (Lemma 3.1), ( 2.9), and ( 1.1) to see that
+d
+dε
+����
+ε=0
+ET D[θε
+A, θε
+B, θε
+S] =
+�
+ΩA(t)
+div(κA∇θA)ψA dx +
+�
+ΩB(t)
+div(κB∇θB)ψB dx
++
+�
+Γ(t)
+{divΓ(κS∇ΓθS) + κB(nΓ · ∇)θB − κA(nΓ · ∇)θA}ψS dH2
+x.
+From fundamental lemma of calculation of variations, we have ( 2.11). Therefore,
+Theorem 2.7 is proved.
+□
+4. Thermodynamical Modeling
+In this section we make mathematical models for multiphase flow with surface
+tension and flow by our thermodynamic approaches. Under the restrictions ( 1.1),
+we apply Proposition 2.3, the first law of thermodynamics (Theorem 2.4), and our
+energy densities (Definition 2.2) to derive equations ( 1.2)-( 1.4). In this section we
+consider the case when the fluids in ΩA,T , ΩB,T , ΓT are barotropic fluids.
+Let r ∈ {0, 1}, and pA, pB, pB ∈ C1(R). We assume that (vA, vB, vS, θA, θB, θS)
+satisfies ( 1.1). We consider the energy densities defined in Definition 2.2 as the
+energy densities for multiphase flow with surface flow.
+From Proposition 2.3, we admit that system ( 1.2) is the continuity equations of
+our system, that is, we assume that (ρA, ρB, ρS) satisfies ( 1.2).
+Let (QA, QB, QS) be the endothermic energies derived from energies dissipation
+due to thermal diffusion. From Theorem 2.7, we set
+�
+�
+�
+�
+�
+QA = div(κA∇θA)
+in ΩA,T ,
+QB = div(κB∇θB)
+in ΩB,T ,
+QS = divΓ(κS∇ΓθS) + κB(nΓ · ∇)θB − κA(nΓ · ∇)θA
+on ΓT .
+Let �Q♯ = �Q♯(x, t) be the quantity of heat supplied the fluid in Ω♯(t), where ♯ =
+A, B, S, and ΩS(t) := Γ(t). Since eD♯ is the energy density for energy dissipation
+due to the viscosities (µ♯, λ♯), we set �Q♯ = Q♯ + eD♯. Now we admit the first law of
+
+MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW
+15
+thermodynamics, that is, suppose that for every 0 < t < T and Λ ⊂ Ω,
+d
+dt
+�
+ΩA(t)∩Λ
+{ρAeA − pA(ρA)} dx =
+�
+ΩA(t)∩Λ
+(QA + eDA) dx,
+d
+dt
+�
+ΩB(t)∩Λ
+{ρBeB − pB(ρB)} dx =
+�
+ΩB(t)∩Λ
+(QB + eDB) dx,
+d
+dt
+�
+Γ(t)∩Λ
+{ρSeS − pS(ρS)} dH2
+x =
+�
+Γ(t)∩Λ
+(QS + eDS) dH2
+x.
+From Theorem 2.4, we have
+(4.1)
+�
+�
+�
+�
+�
+ρADA
+t eA + (divvA)ΠA = divqA + eDA in ΩA,T ,
+ρBDB
+t eB + (divvB)ΠB = divqB + eDB in ΩB,T ,
+ρSDS
+t eS + (divΓvS)ΠS = divΓqS + eDS + qB · nΓ − qA · nΓ on ΓT ,
+where (qA, qB, qS) and (ΠA, ΠB, ΠS) are defined by ( 1.5) and ( 2.6).
+Let FA, FB, FS ∈ [C(R4)]3. We assume that the momentum equations of our
+system are written by
+(4.2)
+�
+�
+�
+�
+�
+ρADA
+t vA = FA
+in ΩA,T ,
+ρBDB
+t vB = FB
+in ΩB,T ,
+ρSDS
+t vS = FS
+on ΓT .
+From a thermodynamic point of view we assume that our system satisfies the con-
+servation law of total energy, that is, (FA, FB, FS) satisfies that for each 0 < t < T,
+(4.3)
+d
+dt
+� �
+ΩA(t)
+�1
+2ρA|vA|2 + ρAeA
+�
+dx +
+�
+ΩB(t)
+�1
+2ρB|vB|2 + ρBeB
+�
+dx
++
+�
+Γ(t)
+�1
+2ρS|vS|2 + ρSeS
+�
+dH2
+x
+�
+= 0.
+Using the transport theorems with ( 1.2), ( 4.1), ( 4.2), we see that
+(L.H.S) of ( 4.3) =
+�
+ΩA(t)
+(ρADA
+t vA · vA + ρADA
+t eA) dx
++
+�
+ΩB(t)
+(ρBDB
+t vB · vB + ρBDB
+t eB) dx +
+�
+Γ(t)
+(ρSDS
+t vS · vS + ρSDS
+t eS) dH2
+x
+=
+�
+ΩA(t)
+{FA · vA + divqA + eDA − (divvA)ΠA} dx
++
+�
+ΩB(t)
+{FB · vB + divqB + eDB − (divvB)ΠB} dx
++
+�
+Γ(t)
+{FS · vS + divΓqS + eDS − (divΓvS)ΠS + qB · nΓ − qA · nΓ} dH2
+x.
+
+16
+HAJIME KOBA
+Applying the integration by parts with ( 1.1), we observe that
+(L.H.S) of ( 4.3) =
+�
+ΩA(t)
+{FA − divTA(vA, ΠA)} · vA dx
++
+�
+ΩB(t)
+{FB − divTB(vB, ΠB)} · vB dx +
+�
+Γ(t)
+{FS − divΓTS(vS, ΠS)} · vS dH2
+x
++
+�
+Γ(t)
+�TA(vA, ΠA)nΓ · vS dH2
+x −
+�
+Γ(t)
+�TB(vB, ΠB)nΓ · vS dH2
+x,
+where (TA, TB, TS) and (�TA, �TB) are defined by ( 1.7) and ( 1.8). Here we used the
+facts that
+�
+Γ(t)
+{TA(vA, ΠA)nΓ} · vA dH2
+x =
+�
+Γ(t)
+�TA(vA, ΠA)nΓ · vS dH2
+x,
+�
+Γ(t)
+{TB(vB, ΠB)nΓ} · vB dH2
+x =
+�
+Γ(t)
+�TB(vB, ΠB)nΓ · vS dH2
+x.
+Thus, we set
+(4.4)
+�
+�
+�
+�
+�
+FA = divTA(vA, ΠA),
+FB = divTB(vB, ΠB),
+FS = divΓTS(vS, ΠS) + �TB(vB, ΠB)nΓ − �TA(vA, ΠA)nΓ
+to see that (L.H.S) of ( 4.3) equals to zero. Combining ( 4.1), ( 4.2), ( 4.4), we
+have ( 1.3) and ( 1.4). Therefore, we derive ( 1.2)-( 1.4) by our thermodynamic
+approach.
+Finally, we introduce another approach to derive the momentum equations ( 1.4).
+We assume that the time rate of change of the momentum equals to the forces
+derived from the variation of energies dissipation due to the viscosities, that is,
+suppose that for every 0 < t < T and Λ ⊂ Ω,
+d
+dt
+�
+ΩA(t)∩Λ
+ρAvA dx =
+�
+ΩA(t)∩Λ
+FA dx,
+d
+dt
+�
+ΩB(t)∩Λ
+ρBvB dx =
+�
+ΩB(t)∩Λ
+FB dx,
+d
+dt
+�
+Γ(t)∩Λ
+ρSvS dH2
+x =
+�
+Γ(t)∩Λ
+FS dH2
+x,
+where (FA, FB, FS) is defined by ( 2.10). Using the transport theorems with ( 1.2),
+we have ( 1.4).
+5. Conservative Forms and Conservation Laws
+We study the conservation laws of our model to prove Theorem 2.8.
+Proof of Theorem 2.8. Let r ∈ {0, 1}. Direct calculations give (i) (see Remark 1.3).
+We now prove (ii). From Proposition 2.3 and the arguments in Section 4, we see
+( 1.12) and ( 1.13). Using the transport theorems (Definition 2.1) with ( 1.2) and
+
+MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW
+17
+( 1.4), we see that
+(5.1)
+d
+dt
+� �
+ΩA(t)
+1
+2ρA|vA|2 dx +
+�
+ΩB(t)
+1
+2ρB|vB|2 dx +
+�
+Γ(t)
+1
+2ρS|vS|2 dH2
+x
+�
+=
+�
+ΩA(t)
+ρADA
+t vA · vA dx +
+�
+ΩB(t)
+ρBDB
+t vB · vB dx +
+�
+Γ(t)
+ρSDS
+t vS · vS dH2
+x
+=
+�
+ΩA(t)
+divTA(vA, πA) · vA dx +
+�
+ΩB(t)
+divTB(vB, πB) · vB dx
++
+�
+Γ(t)
+{divΓTS(vS, πS) + �TB(vB, πB)nΓ − �TA(vA, πA)nΓ} · vS dH2
+x.
+Applying integration by parts (Lemma 3.1) and ( 1.1), we check that
+(R.H.S) of ( 5.1) =
+�
+ΩA(t)
+{−eDA + (divvA)πA} dx
++
+�
+ΩB(t)
+{−eDB + (divvB)πB} dx +
+�
+Γ(t)
+{−eDS + (divΓvS)πS} dH2
+x,
+where (eDA, eDB, eDS) is defined by ( 1.6). Integrating with respect to t, we have
+( 1.14).
+Finally, we show (iii). Assume that r = 1. Using the transport and divergence
+theorems (Definition 2.1 and Lemma 3.1) with ( 2.12), we see that
+d
+dt
+� �
+ΩA(t)
+ρAvA dx +
+�
+ΩB(t)
+ρBvB dx +
+�
+Γ(t)
+ρSvS dH2
+x
+�
+=
+�
+ΩA(t)
+divTA(vA, πA) dx +
+�
+ΩB(t)
+divTB(vB, πB) dx
++
+�
+Γ(t)
+{divΓTS(vS, πS) + TB(vB, πB)nΓ − TA(vA, πA)nΓ} dH2
+x = t(0, 0, 0).
+Integrating with respect to t, we have ( 1.15). Therefore, Theorem 2.8 is proved.
+□
+6. Thermodynamic Potential
+We investigate thermodynamic potential for our model to prove Theorem 2.9.
+Proof of Theorem 2.9. We only prove the case when ♯ = S. Let r ∈ {0, 1}. Assume
+that ρS and θS are positive functions. Set hS = eS + πS/ρS, F H
+S
+= eS − θSςS,
+F G
+S = hS − θSςS. Assume that eS satisfies the thermodynamic identity:
+(6.1)
+DS
+t eS = θSDS
+t ςS − πSDS
+t
+� 1
+ρS
+�
+.
+We first derive ( 1.16). By ( 1.2) and ( 1.3), we see that
+ρSDS
+t hS = ρSDS
+t eS + ρSDS
+t
+�πS
+ρS
+�
+= divΓqS + eDS + qB · nΓ − qA · nΓ + DS
+t πS.
+
+18
+HAJIME KOBA
+This shows that
+DN
+t (ρShS) + divΓ(ρShSvS − qS) = ρSDS
+t hS − divΓqS
+= eDS + DS
+t πS + qB · nΓ − qA · nΓ,
+which is ( 1.16).
+Next we show ( 1.17). From ( 1.3) and ( 6.1), we find that
+θSρSDS
+t ςS = ρSDS
+t eS + ρSπSDS
+t
+� 1
+ρS
+�
+= divΓqS + eDS + qB · nΓ − qA · nΓ.
+Using the above equality and ( 1.16), we check that
+DN
+t (ρSςS) + divΓ
+�
+ρSςSvS − qS
+θS
+�
+= ρSDS
+t hS − divΓ
+�qS
+θS
+�
+= eDS
+θS
++ qS · gradΓθS
+θ2
+S
++ qB · nΓ − qA · nΓ
+θS
+.
+Thus, we have ( 1.17).
+Finally, we derive ( 1.18) and ( 1.19). Applying ( 6.1) and ( 1.2), we see that
+ρSDS
+t F H
+S + ρSςSDS
+t θS = ρSDS
+t eS − ρSθSDS
+t ςS
+= −(divΓvS)πS,
+and that
+ρSDS
+t F G
+S + ρSςSDS
+t θS = ρSDS
+t hS − ρSθSDS
+t ςS
+= DS
+t πS.
+Therefore, Theorem 2.9 is proved.
+□
+Data Availability : The author declares that data sharing not applicable to this
+article as no datasets were generated or analyzed during the current study.
+Conflict of interest : The author declares no conflict of interest associated with
+this manuscript.
+Acknowledgments : This work was partly supported by the Japan Society for
+the Promotion of Science (JSPS) KAKENHI Grant Number JP21K03326.
+References
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+MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW
+19
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+interfaces. World Scientific, Singapore, 2001. xviii,+248 pp. ISBN=9810243057.
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+tinua. Cambridge University Press, Cambridge, 2010. xxii+694 pp. ISBN: 978-0-521-40598-0
+MR2884384
+[12] Istv´an Gyarmati. Non-equilibrium Thermodynamics. Springer, 1970. ISBN:978-3-642-51067-0
+[13] Yunkyong Hyon, Do Y. Kwak, and Chun Liu, Energetic variational approach in complex
+fluids: maximum dissipation principle. Discrete Contin. Dyn. Syst. 26 (2010), no. 4, 1291–
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+Appl. Math. 76 (2018), no. 2, 303–359.
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+Boundary, preprint. arXiv:1810.07909. to appear in Quart. Appl. Math.
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+Boundary, Quart. Appl. Math. 78 (2020), 617-640
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+surface flow and tension, preprint. arXiv:2211.06672
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+pressible fluid systems on an evolving surface, Quart. Appl. Math. 75 (2017), no 2, 359–389.
+MR3614501. Errata to Energetic variational approaches for incompressible fluid systems on
+an evolving surface. Quart. Appl. Math. 76 (2018), no 1, 147–152.
+[19] Hajime Koba and Kazuki Sato, Energetic variational approaches for non-Newtonian fluid
+systems. Z. Angew. Math. Phys (2018) 69: 143. https://doi.org/10.1007/s00033-018-1039-1.
+[20] Yoshihiko Mitsumatsu and Yasuhisa Yano,
+Geometry of an incompressible fluid on
+a Riemannian manifold. (Japanese) Geometric mechanics (Japanese) (Kyoto,
+2002).
+Surikaisekikenkyusho Kokyuroku No. 1260 (2002), 33–47.
+[21] Jan Pr¨uss and Gieri Simonett, Moving interfaces and quasilinear parabolic evolution equa-
+tions. Monographs in Mathematics, 105. Birkh¨auser/Springer, [Cham], 2016. xix+609 pp.
+ISBN: 978-3-319-27697-7; 978-3-319-27698-4 MR3524106
+[22] L.E. Scriven, Dynamics of a fluid interface Equation of motion for Newtonian surface fluids.
+Chem. Eng. Sci. 12 (1960), 98–108.
+[23] John
+C.
+Slattery,
+Momentum
+and
+moment-of-momentum
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+sur-
+facesChemical Engineering Science, Volume 19, 1964, Pages 379–385.
+[24] John C. Slattery, Leonard. Sagis, and Eun-Suok Oh, Interfacial transport phenomena. Second
+edition. Springer, New York, 2007. xviii+827 pp. ISBN: 978-0-387-38438-2; 0-387-38438-3
+MR2284654.
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+matical Analysis, Australian National University, 3. Australian National University, Centre
+for Mathematical Analysis, Canberra, 1983. vii+272 pp. ISBN: 0-86784-429-9 MR0756417.
+[26] Michael E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other
+evolution equations. Comm. Partial Differential Equations 17 (1992), no. 9-10, 1407–1456.
+MR1187618.
+Graduate School of Engineering Science, Osaka University,, 1-3 Machikaneyamacho,
+Toyonaka, Osaka, 560-8531, Japan
+Email address: [email protected]
+

8dAyT4oBgHgl3EQf2_nF/content/tmp_files/2301.00762v1.pdf.txt

@@ -0,0 +1,881 @@
+ 
+High Altitude Platform Station (HAPS)-Aided 
+GNSS for Urban Areas 
+Hongzhao Zheng, Mohamed Atia, Halim Yanikomeroglu  
+Department of Systems and Computer Engineering, Carleton University, Ottawa, Canada 
+[email protected]
+Abstract—Today the global averaged civilian positioning 
+accuracy is still at meter level for all existing Global Navigation 
+Satellite Systems (GNSSs), and the civilian positioning 
+performance is even worse in regions such as the Arctic region 
+and the urban areas. In this work, we examine the positioning 
+performance of the High Altitude Platform Station (HAPS)-
+aided GPS system in an urban area via both simulation and 
+physical experiment. HAPS can support GNSS in many ways, 
+herein we treat the HAPS as an additional ranging source. From 
+both simulation and experiment results, we can observe that 
+HAPS can improve the horizontal dilution of precision (HDOP) 
+and the 3D positioning accuracy. The simulated positioning 
+performance of the HAPS-aided GPS system is subject to the 
+estimation accuracy of the receiver clock offset. This work also 
+presents the future work and challenges in modelling the 
+pseudorange of HAPS.  
+Keywords—High Altitude Platform Station (HAPS), Global 
+Navigation Satellite System (GNSS), pseudorange, horizontal 
+dilution of precision (HDOP) 
+I. INTRODUCTION  
+The global navigation satellite system (GNSS) has been 
+around for decades. Since the first launch of a legacy GNSS 
+in 1978, the global positioning system (GPS) owned by the 
+US, the positioning accuracy brought by satellites has been 
+improving thanks to the ongoing research in the associated 
+scientific fields. Depending on the application, centimeter 
+level accuracy can be obtained by techniques such as 
+differential GPS (DGPS), real-time kinematic (RTK), multi-
+constellation GNSS and so forth. For example, the multi-
+constellation GNSS (BeiDou + Galileo + GLONASS + GPS) 
+has been shown to not only shorten the convergence time, but 
+also to provide centimeter-level positioning accuracy even 
+with 40° cut-off elevation using the precise positioning 
+algorithm [1]. Although numerous techniques have been 
+developed to achieve centimeter-level positioning accuracy, 
+many of which are not suitable for civilian applications such 
+as smartphones, smartwatches, bikes, and so on. Most civilian 
+applications use single frequency and low cost receivers for 
+navigation and positioning, hence precise positioning is not 
+applicable due to reasons such as the incomplete elimination 
+of the ionospheric delay which appears to be one of the largest 
+error sources in the pseudorange measurement. For similar 
+reasons, the most common algorithm used in the civilian 
+applications is therefore the single point positioning algorithm 
+which only requires a single frequency in localization. 
+However the global averaged positioning accuracy of using 
+the single point positioning algorithm is still at meter level. 
+For example, the 95% global averaged horizontal error is less 
+than or equal to 8 m and the 95% global averaged vertical error 
+is less than or equal to 13 m for the GPS system [2]; the 95% 
+global averaged horizontal error is less than or equal to 9 m 
+and the 95 % global averaged vertical error is less than or 
+equal to 10 m for the BeiDou Navigation Satellite System 
+(BDS) [3]; the 95% global averaged horizontal error is less 
+than or equal to 5 m and the 95% global averaged   vertical 
+error is less than or equal to 9 m for GLONASS [4]; the 95% 
+global averaged positioning error is less than or equal to 7 m 
+for Galileo [5]. The positioning performance of the GNSS is 
+even worse in the urban area.  
+Today there are many low Earth orbit (LEO) satellites 
+launched into space, people are also interested in utilizing 
+LEO satellites to aid positioning service. For instance, Li et al. 
+prove that the LEO enhanced GNSS can provide centimeter 
+level Signal-In-Space Ranging Error (SISRE) in real-time 
+precise point positioning (PPP) application [6]. Furthermore, 
+researchers have also been investigating the navigation 
+performance which relies exclusively on the LEO satellite 
+signals in case the GNSS signals are unavailable. Khalife et 
+al. have shown that a position root mean squared error 
+(RMSE) of 14.8 m for an unmanned aerial vehicle (UAV) can 
+be achieved with only two Orbcomm LEO satellites using the 
+carrier phase differential algorithm [7]. Compare LEO 
+satellites with the typical satellites used in the traditional 
+GNSS which are the medium Earth orbit (MEO) satellites, 
+LEO exhibits the advantages including but not limited to 
+shorter propagation delay and lower pathloss due to shorter 
+distance to the ground user, wider coverage and higher 
+availability due to the enormous number of satellites 
+simultaneously visible/available for positioning. To further 
+enhance high bandwidth networking coverage in challenging 
+areas, High Altitude Platform Stations (HAPS), which resides 
+in the stratosphere with a typical altitude of about 20 km, can 
+be introduced. As urban area is the region where the GNSS 
+positioning performance degrades most severely, we could 
+utilize HAPS as another ranging source by equipping it with a 
+satellite-grade atomic clock on top of a metro city. Since 
+HAPS is only 20 km above the ground, the pathloss of the 
+HAPS signal is expected to be much less than that for any 
+satellites, making the received signal power of HAPS stronger 
+than that of satellite, which likely renders less estimation error 
+in the multipath mitigation of the HAPS signal. HAPS is 
+quasi-stationary as it does not orbit around the globe, this can 
+reduce the number of handovers during the course of 
+positioning. Moreover the HAPS signal does not suffer from 
+the ionospheric effect since it is transmitted from below the 
+ionosphere. Therefore the pseudorange from HAPS can likely 
+be estimated with less error compared with that from satellites. 
+Similar to the pseudorange from satellites which incorporates 
+satellite position error, we should also consider the position 
+error in the pseudorange measurement for HAPS. Fortunately, 
+there are ongoing research in the literature investigating the 
+positioning of HAPS and showing HAPS positioning error 
+can be comparable or even better than the satellite orbit error. 
+For example, a 0.5 m positioning accuracy (circular error 
+probable [CEP] 68 percent) for HAPS has been shown 
+achievable using the modified RTK method [8]. In fact, there 
+are a handful of papers in the literature investigating the 
+HAPS-aided GNSS positioning performance [9]-[12], but 
+none of which considers utilizing HAPS for the sole mission 
+of improving the GNSS positioning performance in the urban 
+area. In this work we examine the HAPS-aided GNSS 
+positioning performance in the urban area via both simulation 
+and physical experiment. For simplicity, the GNSS signal only 
+
+involves the GPS C/A L1 signal, and the single point 
+positioning algorithm is used to compute the position solution. 
+II. SYSTEM MODEL 
+The general system model is depicted in Fig. 1. The HAPS 
+is situated at 20 km above ground in the stratosphere which is 
+below the ionosphere. There are four satellites shown in Fig. 
+1, this is just a reminder that at least four satellites are required 
+to perform precise 3D localization using GNSS. Although 
+only a selection of visible satellites is used in position solution 
+calculation in reality, in this work all available satellites are 
+used in position solution calculation for simplicity. The 
+elevation masks for both satellite and HAPS are chosen to be 
+15 degrees. The pseudorange equation for satellite is given by 
+ 
+ 
+𝑝𝑆𝐴𝑇 = 𝜌𝑆𝐴𝑇 + 𝑑𝑆𝐴𝑇 + 𝑐(𝑑𝑡 − 𝑑𝑇𝑆𝐴𝑇) + 𝑑𝑖𝑜𝑛,𝑆𝐴𝑇
++ 𝑑𝑡𝑟𝑜𝑝,𝑆𝐴𝑇 + 𝜖𝑚𝑝,𝑆𝐴𝑇 + 𝜖𝑝 
+ 
+ 
+ 
+(1) 
+where 𝑝𝑆𝐴𝑇 denotes the satellite pseudorange measurement, 
+𝜌𝑆𝐴𝑇 is the geometric range between the satellite and receiver, 
+𝑑𝑆𝐴𝑇 represents the satellite orbit error, 𝑐 is the speed of light, 
+𝑑𝑡 is the receiver clock offset from GPS time, 𝑑𝑇𝑆𝐴𝑇 is the 
+satellite clock offset from GPS time, 𝑑𝑖𝑜𝑛,𝑆𝐴𝑇  denotes the 
+ionospheric delay for satellite signals, 𝑑𝑡𝑟𝑜𝑝,𝑆𝐴𝑇 denotes the 
+tropospheric delay for satellite signals, 𝜖𝑚𝑝,𝑆𝐴𝑇 is the delay 
+caused by the multipath for satellite signals and 𝜖𝑝 is the 
+delay caused by the receiver noise. The pseudorange equation 
+for HAPS is described by 
+ 
+ 
+𝑝𝐻𝐴𝑃𝑆 = 𝜌𝐻𝐴𝑃𝑆 + 𝑑𝐻𝐴𝑃𝑆 + 𝑐(𝑑𝑡 − 𝑑𝑇𝐻𝐴𝑃𝑆) + 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆
++ 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 + 𝜖𝑝 
+(2)  
+where 𝑝𝐻𝐴𝑃𝑆 denotes the HAPS pseudorange measurement, 
+𝜌𝐻𝐴𝑃𝑆 represents the geometric range between the HAPS and 
+the receiver, 𝑑𝐻𝐴𝑃𝑆  represents the HAPS position error, 
+𝑑𝑇𝐻𝐴𝑃𝑆 is the HAPS clock offset from GPS time, 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆 
+denotes the tropospheric delay for HAPS signals, 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 is 
+the delay caused by the multipath for HAPS signals. In this 
+work, the satellite orbit error, the HAPS position error, and 
+the HAPS clock offset are assumed to be zero for simplicity. 
+The simulated vehicle trajectory originates from Carleton 
+University in the suburban area and ends at the Rideau Street 
+of Ottawa in the dense urban area (see Fig. 2). There are four 
+simulated HAPS where one HAPS is following a circular 
+trajectory on top of the downtown Ottawa area, and the other 
+three HAPS are following similar circular trajectories on top 
+of three populated regions near Ottawa. Note that HAPS is 
+quasi-stationary due to factors such as wind, it can move 
+within a confined space. Fig. 3 shows the flowchart of the 
+single point positioning algorithm. Since the HAPS clock 
+offset in this work is assumed zero, we simply use 𝑑𝑇 to 
+denote the satellite clock offset. From the data collected by 
+the GNSS receiver, we shall obtain both the receiver 
+independent exchange (RINEX) format observation file and 
+the RINEX navigation file, which contains the satellite 
+information such as the pseudorange, the ionospheric 
+parameters, 𝜶, the Keplerian parameters, and so on. With that 
+information, we know the pseudo-random noise (𝑷𝑹𝑵) code 
+which represents the unique number of each satellite, the day 
+of year (𝐷𝑂𝑌) which represents the day of year at the time of 
+measurement. Note that 𝑷𝑹𝑵 is in bold to represent a vector  
+ 
+Fig. 1: System model of the HAPS-aided GPS system. 
+ 
+Fig. 2: Vehicle trajectory. 
+containing the pseudo-random noise code of all visible 
+satellites at the current epoch and the current iteration of 
+estimation. We can compute the satellite positions, 𝑷𝑺𝑨𝑻, and 
+satellite clock offset, 𝒅𝑻, using the Keplerian parameters 
+contained in the navigation file. 𝑷𝑯𝑨𝑷𝑺  denotes a vector 
+containing the positions of all HAPS which are generated 
+using the Skydel GNSS simulator [13], and 𝒑𝑯𝑨𝑷𝑺 denotes a 
+vector containing the HAPS pseudorange which will be 
+explained in Section III. To compute the position solution, 𝒙, 
+we firstly initialize the receiver position to the center of the 
+Earth, and the receiver clock offset is initialized to zero. The 
+change in estimate, 𝒅𝒙, is initialized to infinity. For each 
+epoch of measurement, we will first check if the number of 
+available ranging sources is more than three as at least four 
+ranging sources are required to perform precise 3D 
+localization. Since the receiver position is iteratively 
+estimated, we calculate the elevation angles for both satellites 
+and HAPS with respect to the recently estimated receiver 
+position. Since both the tropospheric delay and the 
+ionospheric delay are functions of the receiver position, these 
+two atmospheric delays are estimated iteratively as well. The  
+
+Ionosphere
+HAPS
+HAPS
+20km
+HAPSfootprint
+HAPSfootprint
+15°
+cell.
+cellOSM+ relief shading
+V
+Tracks:
+nelByDrivi
+Dense
+urban Areas
+OldOrtowa
+Huram
+tonbu
+neGleb
+Suburban
+Are
+eas
+Ottawa
+enEza
+e45.40751.-75.60857
+Googe
+Map created at GpSVisualiz
+Madutn
+con 
+Fig. 3: Flow chart of the single point positioning algorithm. 
+elevation angle, satellite pseudorange, HAPS pseudorange, 
+satellite position, satellite clock offset, the tropospheric 
+delay, 𝒅𝒕𝒓𝒐𝒑, the ionospheric delay, 𝒅𝒊𝒐𝒏, and the pseudo-
+random noise (𝑷𝑹𝑵) code are modified iteratively based on 
+the re-computed elevation angles for both satellites and 
+HAPS. To prepare the parameters needed for the least square 
+methods, the corrected pseudorange needs to be computed as 
+follows: 
+𝒑𝑺𝑨𝑻
+𝒄
+= 𝒑𝑺𝑨𝑻 + 𝑐 ∙ 𝒅𝑻 − 𝒅𝒕𝒓𝒐𝒑,𝑺𝑨𝑻 − 𝒅𝒊𝒐𝒏,𝑺𝑨𝑻         (3) 
+where 𝒑𝑺𝑨𝑻
+𝒄
+ represents the corrected pseudorange for 
+satellite, 𝒑𝑺𝑨𝑻  represents the uncorrected pseudorange for 
+satellite.  Since the pseudorange error of HAPS is modeled as 
+Gaussian noise representing the estimation residual, the 
+HAPS pseudorange does not need to be corrected. Due to the 
+Earth rotation, the positions of satellites and HAPS at the 
+signal emission time are different from that at the signal 
+reception time, this is known as the Sagnac effect [14]. The 
+coordinates of satellite/HAPS can be transformed from the 
+signal emission time to the signal reception time by [14] 
+∆𝑡𝑅𝑂𝑇 = 𝑡𝑟𝑥 − 𝑡𝑡𝑥                             (4) 
+𝑃𝑖,𝑟𝑥 = 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)𝑃𝑖,𝑡𝑥                 (5) 
+where ∆𝑡𝑅𝑂𝑇  denotes the signal propagation time, 𝑡𝑟𝑥 
+represents the signal reception time, 𝑡𝑡𝑥 represents the signal 
+emission time, 𝑃𝑖,𝑟𝑥 is the 𝑖𝑡ℎ satellite/HAPS coordinates at 
+the signal reception time, 𝑃𝑖,𝑡𝑥  is the 𝑖𝑡ℎ  satellite/HAPS 
+coordinates at the signal emission time, 𝜔𝐸  denotes the 
+Earth’s rotation rate, and 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) is known as the 
+rotation matrix which is described by 
+𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)
+= [
+cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)
+sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)
+0
+− sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)
+0
+cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)
+0
+0
+1
+] . 
+       (6) 
+The line-of-sight vector, 𝒗, and the geometric range between 
+ranging sources and receiver, 𝝆 , are then calculated to 
+compute the a-priori range residual vector 𝒃 and the design 
+matrix 𝑯, where 
+𝒃 = 𝒑𝒄 − 𝝆                                 (7) 
+𝑯 = [𝒗, 𝟏𝑙𝑒𝑛𝑔𝑡ℎ(𝒑𝐜)×1)]                      (8) 
+where 𝒑𝒄  is the corrected satellite pseudorange combined 
+with the corrected HAPS pseudorange. At last, the least 
+square solution is computed as 
+𝑸 = (𝑯′𝑯)−1                            (9) 
+𝒅𝒙 = 𝑸𝑯′𝒃                             (10) 
+𝑑𝑡 = 𝒅𝒙(4)/𝑐                           (11) 
+where Q is known as the covariance matrix, 𝒅𝒙(4) means the 
+fourth element in the vector 𝒅𝒙. The covariance matrix, 𝑸, is 
+described by 
+𝑸 =
+[
+ 
+ 
+ 
+ 𝜎𝑥
+2
+𝜎𝑥𝑦
+𝜎𝑥𝑧
+𝜎𝑥𝑡
+𝜎𝑥𝑦
+𝜎𝑦
+2
+𝜎𝑦𝑧
+𝜎𝑦𝑡
+𝜎𝑥𝑧
+𝜎𝑦𝑧
+𝜎𝑧
+2
+𝜎𝑧𝑡
+𝜎𝑥𝑡
+𝜎𝑦𝑡
+𝜎𝑧𝑡
+𝜎𝑡
+2 ]
+ 
+ 
+ 
+ 
+                   (12) 
+where 𝜎𝑥, 𝜎𝑦, 𝜎𝑧 and 𝜎𝑡 represent the standard deviations of 
+the receiver coordinates x, y, z in the Earth-centered Earth-
+fixed (ECEF) coordinate frame and the receiver clock offset, 
+respectively. The least square solution shall be found when 
+the norm of the change in receiver position, 𝒅𝒙(1: 3), is 
+sufficiently small. In this work, this threshold is chosen to be 
+0.01 m. We use the horizontal dilution of precision (HDOP) 
+and the 3D positioning accuracy as the metrics to examine the 
+positioning performance of the proposed HAPS-aided GPS 
+system. To compute the HDOP, we must convert the 
+covariance matrix into the local north-east-down (NED) 
+coordinate frame, which can be done with the following 
+equation [15]: 
+𝑸𝑵𝑬𝑫 = 𝑹′𝑸̃𝑹                          (13) 
+where 𝑸̃ and 𝑹 are defined as 
+𝑸̃ = [
+𝜎𝑥
+2
+𝜎𝑥𝑦
+𝜎𝑥𝑧
+𝜎𝑥𝑦
+𝜎𝑦
+2
+𝜎𝑦𝑧
+𝜎𝑥𝑧
+𝜎𝑦𝑧
+𝜎𝑧
+2
+]                           (14) 
+
+Initialization
+PsAT,PHAPS,PRN,DOY
+x = 04x1
+Input
+dt = x(4)/c
+PHAPs,PsAT,dT,α
+dx =x+Inf
+stop = 0
+No
+Exit 4
+NsAT + NHAPs ≥ 4
+ Yes
+No
+Idx(1:3)I>0.01
+Yes
+Finding parameters
+For satellites
+For HAPS
+sAT,dtrop,dion
+OHAPS
+Applying elevation mask
+For satellites
+For HAPS
+sAT,dtrop,dion,PRN,dT,PsAT
+HAPS,PHAPS
+Pseudorangecorrection
+PSAT,PHAPS
+Combining the
+corrected
+pseudoranges
+P'=[PSAT,PHAPS]
+Correcting for the Sagnac effect
+(i.e., Earth rotation)
+PSAT,PHAPS
+Combiningthecorrected
+ranging source positions
+P° = [PSAT, PHAPS]
+Finding parameters
+V,P,b,H,Q
+Output
+Computingtheposition solution
+using the Least Square method
+x,dt
+x,dt𝑹 = [
+−sin 𝜆
+cos 𝜆
+0
+− cos 𝜆 sin 𝜑
+− sin 𝜆 sin 𝜑
+cos 𝜑
+cos 𝜆 cos 𝜑
+sin 𝜆 cos 𝜑
+sin 𝜑
+]        (15) 
+where 𝜆 and 𝜑 represent the longitude and latitude of the 
+receiver, respectively. Then the HDOP is described by 
+𝐻𝐷𝑂𝑃 = √𝜎𝑛2 + 𝜎𝑒2                         (16) 
+where 𝜎𝑛, 𝜎𝑒, and 𝜎𝑑 represent the receiver position errors in 
+the local north, east and down directions, respectively. 
+III. SIMULATION OF THE HAPS-AIDED GPS SYSTEM 
+A. Simulation Setup 
+The system model is established using the default Earth 
+orientation parameters of the Skydel GNSS simulation 
+software [13] which considers all GPS satellites orbiting 
+around the Earth and transmitting the L1 C/A code. The 
+Saastamoinen model is chosen to emulate the tropospheric 
+effect and the Klobuchar model is chosen to emulate the 
+ionospheric effect along with the software default Klobuchar 
+parameters (i.e., alpha and beta). The output from Skydel 
+contains the ECEF coordinates of satellites at the signal 
+emission time, the ionospheric corrections, the tropospheric 
+corrections, the satellite clock offsets, the ECEF coordinates 
+of the receiver, the signal emission time, and so forth, at each 
+time stamp from the start of the simulation. The receiver clock 
+offset in the simulation is zero by default. The correction terms 
+in the pseudorange equation of satellite including the satellite 
+orbit error, the multipath and the receiver noise are not 
+separately considered in the simulation, instead a pseudorange 
+error is introduced to reflect the presence of those effect. The 
+pseudorange error of satellite is featured using the built-in first 
+order Gauss-Markov process with the default time constant of 
+10 s and the standard deviation of 6 m. The continuous model 
+for the first order Gauss-Markov process is described by [16] 
+𝑥̇ = −
+1
+𝑇𝑐 𝑥 + 𝑤                                   (17) 
+where 𝑥  represents a random process with zero mean, 
+correlation time 𝑇𝑐, and noise 𝑤. The autocorrelation of the 
+first order Gauss-Markov process is described by [17] 
+𝑅(∆𝑡) = 𝜎2𝑒−|∆𝑡|
+𝜏                                 (18) 
+where ∆𝑡 represents the sampling interval, 𝜎 and 𝜏 denote the 
+standard deviation and the time constant of the first order 
+Gauss-Markov process, respectively. The pseudorange of 
+HAPS is simulated by adding Gaussian noise to the geometric 
+range between HAPS and receiver, where the Gaussian noise 
+represents the sum of all kinds of estimation residuals 
+including the HAPS position, the HAPS clock offset, the 
+tropospheric delay, the multipath and the receiver noise. The 
+pseudorange error for HAPS is modelled using the Gaussian 
+noise with standard deviations of 2 m and 5 m representing the 
+suburban and the dense urban scenario, respectively. The 
+characteristics of the pseudorange errors for the suburban 
+scenario and the dense urban scenario are set to be the same 
+for satellites. Note that by doing this, the positioning 
+performance of the GPS-only system stays the same in both 
+suburban scenario and dense urban scenario. The standard 
+deviation for the HAPS pseudorange error is enforced to be 
+smaller than that for the satellite pseudorange error in both 
+suburban scenario and dense urban scenario. All the available 
+satellites (i.e., satellites with elevation angles greater than the 
+predefined elevation mask) are simultaneously utilized for 
+positioning as if all satellites above the elevation mask are in 
+line of sight (LOS) with the receiver. Under this setting, we 
+examine the 3D positioning performance for the GPS-only 
+system, the one-HAPS with GPS system, the four-HAPS with 
+GPS system and the four-HAPS-only system. For the one-
+HAPS with GPS system, we use the HAPS on top of the 
+downtown Ottawa area which elevation is above 80°. 
+B. Simulation Results 
+The cumulative distribution functions of the 3D 
+positioning accuracy for different systems with the standard 
+deviations of the HAPS pseudorange error being 2 m and 5 m 
+are shown in Fig. 4 and Fig. 5, respectively. From Fig. 4, we 
+can observe that with much less pseudorange error for HAPS, 
+the four-HAPS with GPS system achieves the best 
+positioning performance, the one-HAPS with GPS system 
+achieves almost the same positioning performance as the 
+GPS-only system, and the four-HAPS-only system achieves 
+slightly worse performance than the four-HAPS with GPS 
+system. The reasons why the four-HAPS-only system does 
+not achieve the best positioning performance is potentially 
+due to the following reasons 1) it has much fewer ranging 
+sources in receiver position computation; 2) the ranging 
+source geometry is poor as the elevation angles for all four 
+HAPS at any given time are above 40° with one even above 
+80°. From Fig. 5, we see that with the HAPS pseudorange 
+error similar but slightly smaller than the satellites’ 
+pseudorange error, the four-HAPS-only system achieves the 
+worst positioning performance but the four- HAPS with GPS  
+ 
+Fig. 4: CDF for 3D position accuracy (suburban scenario). 
+ 
+Fig. 5: CDF for 3D position accuracy (dense urban scenario). 
+
+Denseurbanscenario(HAPSprstd=5m)
+0.9
+0.8
+0.7
+0.6
+DF
+0.5
+0.4
+0.3
+0.2
+GPS-onlysystem
+One-HAPSwithGPSsystem
+0.1
+Four-HAPSwithGPSsystem
+Four-HAPS-only system
+0
+0
+5
+10
+15
+20
+25
+30
+35
+3Dpositionalaccuracy(m)inlocalNEDframeSuburbanscenario(HAPSprstd=2m)
+0.9
+0.8
+0.7
+0.6
+DF
+0.5
+C
+0.4
+0.3
+0.2
+GPS-onlysystem
+One-HAPSwithGPSsystem
+0.1
+Four-HAPS with GPS system
+Four-HAPS-only system
+0
+0
+5
+10
+15
+20
+25
+30
+35
+3Dpositionalaccuracy(m)inlocalNEDframesystem still outperforms the other systems considered. 
+IV. FIELD EXPERIMENTS 
+A. Experiment Setup 
+To verify and support the simulation results, we also 
+process the raw GNSS data collected along the vehicle 
+trajectory which is similar to the one shown in Fig. 2 with a 
+slight difference due to partial road closure on the day of data 
+collection. The raw GNSS data are collected using the Ublox 
+EVK-M8T GNSS unit and processed using the single point 
+positioning package developed by Napat Tongkasem [18] 
+with proper modification so that HAPS can be incorporated 
+in the single point positioning algorithm. Table I gives the 
+specifications of the EVK-M8T GNSS unit. To reflect 
+realistic LOS conditions for HAPS, the LOS probability with 
+respect to the HAPS elevation angle in the urban area is 
+implemented based on [19] and [20]. Note that the LOS 
+probability for HAPS provided by [19] is generated based on 
+the city of Chicago and enforcing the LOS probability on 
+HAPS in the dense urban area in Ottawa might be too harsh 
+considering the incompatible city scale. The pseudorange of 
+HAPS in the experiment is modeled as the addition of the 
+geometric range between the satellite and receiver, the 
+receiver clock offset multiplied by the speed of light and the 
+pseudorange error representing the sum of all kinds of 
+estimation residuals. The pseudorange errors for HAPS in the 
+suburban area and in the dense urban area are simulated as 
+Gaussian noise with standard deviations of 2 m and 5 m, 
+respectively. Since the vehicle trajectory involves both 
+suburban area and dense urban area, the entire route is 
+divided into two parts where the first part is considered as the 
+suburban scenario and the second part is considered as the 
+dense urban scenario (see Fig. 2). By observing the 
+positioning performance of the GPS-only system using the 
+real GPS data, the LOS probability for the suburban area is 
+applied to HAPS for epochs less than 380 s, and the LOS 
+probability for the dense urban area is applied to HAPS for 
+epochs greater than or equal to 380 s (refer to Fig. 6). Since 
+the GNSS receiver does not provide accurate receiver clock 
+offset with respect to the GPS time, the receiver clock offset 
+in each epoch is estimated by making use of the ground truth 
+receiver position. The ground truth data is provided by Ublox 
+EVK-M8U 
+GNSS 
+unit, 
+which 
+is 
+equipped 
+with 
+accelerometer and gyroscope, hence it can perform sensor 
+fusion to get better positioning performance and dead 
+reckoning when the signal quality degrades. 
+TABLE I.  
+EVK-M8T GNSS UNIT SPECIFICATIONS [21] 
+Parameter 
+Specification 
+Serial Interfaces 
+1 USB V2.0 
+1 RS232, max.baud rate 921,6 kBd 
+DB9 +/- 12 V level 
+14 pin – 3.3 V logic 
+1 DDC (I2C compatible) max. 400 kHz 
+1 SPI-clock signal max. 5,5 MHz – SPI DATA 
+max. 1 Mbit/s 
+Timing Interfaces 
+2 Time-pulse outputs 
+1 Time-mark input 
+Dimensions 
+105 × 64 × 26 mm 
+Power Supply 
+5 V via USB or external powered via extra power 
+supply pin 14 (V5_IN) 13 (GND) 
+Normal Operating 
+Temperature 
+−40℃ to +65℃ 
+ 
+Fig. 6: HDOP (top) and 3D position accuracy (bottom). 
+B. Experiment Results 
+Fig. 6 shows the HDOP, and the 3D positioning accuracy 
+overlapped with the number of visible HAPS at each epoch. 
+As we can see from Fig. 6, the HDOP and 3D positioning 
+accuracy of the HAPS-aided GPS system are better than that 
+of the GPS-only system in both suburban area and dense 
+urban area. Moreover, we can observe that the positioning 
+performance of the HAPS-aided GPS system is more stable 
+than the GPS-only system as there are less spikes for the 
+HAPS-aided GPS system. Note that, the pseudorange of 
+HAPS in the experiment is modeled as a function of the 
+receiver clock offset, which is estimated with the best effort, 
+additional error should be expected in the pseudorange of 
+HAPS with the magnitude depending on the quality of all 
+visible satellite signals and the ground truth receiver position. 
+As we would expect the quality of the satellite signals in the 
+suburban area is better compared to that in the dense urban 
+area, the receiver clock offset would also be expected to be  
+ 
+Fig. 7: CDF of 3D position accuracy in the suburban area. 
+ 
+Fig. 8:  CDF of 3D position accuracy in the dense urban area. 
+
+Suburbanarea
+1
+0.9
+0.8
+0.7
+0.6
+CDF
+0.5
+C
+0.4
+0.3
+0.2
+0.1
+GPS-onlysystem
+HAPS-aided GPS system
+0
+0
+5
+10
+15
+20
+25
+30
+35
+3Dpositioningerror(m)Denseurbanarea
+1
+0.9
+0.8
+0.7
+0.6
+CDF
+0.5
+0.4
+0.3
+0.2
+0.1
+GPS-only system
+HAPS-aidedGPSsystem
+0
+0
+50
+100
+150
+200
+250
+3Dpositioningerror(m)GPS-only system
+HAPS-aided GPS system
+number of HAPS
+0
+100
+200
+300
+400
+500
+600
+700
+epoch (s)GPS-cnly system
+4
+HAPS-aided GPS system
+300
+Hoe
+number of HAPS
+3
+DOSI
+100
+0
+100
+200
+300
+400
+500
+600
+700
+epoch (s)estimated with higher accuracy in the suburban area than in 
+the dense urban area, hence the HDOP of the HAPS-aided 
+GPS system in the suburban area is better. The cumulative 
+distribution functions of the 3D positioning accuracy in the 
+suburban and dense urban areas are shown in Fig. 7 and Fig. 
+8, respectively. From Fig. 7 and Fig. 8, we can observe that 
+the HAPS-aided GPS system outperforms the GPS-only 
+system, especially in the suburban area. 
+V. CONCLUSION 
+As we are passing 5G and soon entering 6G and beyond, 
+HAPS can be of invisible treasure as it can be used for 
+computation offloading [22], edge computing [23], even base 
+station [24] to meet human needs. HAPS can be another type 
+of ranging source which is quasi-stationary and much closer 
+to the ground of the Earth. Compared to satellite, HAPS 
+exhibits the advantages of lower latency, lower pathloss, 
+lower pseudorange error, and it can provide continuous 
+coverage to reduce the number of handovers for the users in 
+a certain region. Since urban area is the region where GNSS 
+positioning performance degrades severely and where most 
+people live in, deploying several HAPS acting as another type 
+of ranging source on top of a metro city would improve the 
+GNSS positioning performance and maximize the value of 
+the extra payload on HAPS. The HAPS-aided GNSS can also 
+be deployed in the regions with extreme environment such as 
+the Arctic region where the satellite availability is low, and 
+the ionospheric disturbances is severe [25]. From both the 
+simulation and physical experiment results, we observe that 
+HAPS can indeed improve the 3D positioning accuracy, 
+especially in the suburban area. To improve the results of 
+HAPS-aided GPS system in the dense urban area, the receiver 
+clock offset should be estimated with higher accuracy. In 
+future work, the received signal powers of HAPS and satellite 
+will jointly be considered, a satellite selection algorithm will 
+be applied to better emulate the way a modern GNSS receiver 
+processes the raw GNSS data. 
+ACKNOWLEDGMENT 
+This paper is supported in part by Huawei Canada. The 
+Skydel software is a formal donation from Orolia. 
+REFERENCES 
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+ 
+

8dAyT4oBgHgl3EQf2_nF/content/tmp_files/load_file.txt

@@ -0,0 +1,425 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf,len=424
+page_content='High Altitude Platform Station (HAPS)-Aided  GNSS for Urban Areas  Hongzhao Zheng, Mohamed Atia, Halim Yanikomeroglu  Department of Systems and Computer Engineering, Carleton University, Ottawa, Canada  hongzhaozheng@cmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='carleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='ca  Abstract—Today the global averaged civilian positioning  accuracy is still at meter level for all existing Global Navigation  Satellite Systems (GNSSs), and the civilian positioning  performance is even worse in regions such as the Arctic region  and the urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' In this work, we examine the positioning  performance of the High Altitude Platform Station (HAPS)-  aided GPS system in an urban area via both simulation and  physical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' HAPS can support GNSS in many ways,  herein we treat the HAPS as an additional ranging source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' From  both simulation and experiment results, we can observe that  HAPS can improve the horizontal dilution of precision (HDOP)  and the 3D positioning accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The simulated positioning  performance of the HAPS-aided GPS system is subject to the  estimation accuracy of the receiver clock offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' This work also  presents the future work and challenges in modelling the  pseudorange of HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  Keywords—High Altitude Platform Station (HAPS), Global  Navigation Satellite System (GNSS), pseudorange, horizontal  dilution of precision (HDOP)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' INTRODUCTION  The global navigation satellite system (GNSS) has been  around for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Since the first launch of a legacy GNSS  in 1978, the global positioning system (GPS) owned by the  US, the positioning accuracy brought by satellites has been  improving thanks to the ongoing research in the associated  scientific fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Depending on the application, centimeter  level accuracy can be obtained by techniques such as  differential GPS (DGPS), real-time kinematic (RTK), multi-  constellation GNSS and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' For example, the multi-  constellation GNSS (BeiDou + Galileo + GLONASS + GPS)  has been shown to not only shorten the convergence time, but  also to provide centimeter-level positioning accuracy even  with 40° cut-off elevation using the precise positioning  algorithm [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Although numerous techniques have been  developed to achieve centimeter-level positioning accuracy,  many of which are not suitable for civilian applications such  as smartphones, smartwatches, bikes, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Most civilian  applications use single frequency and low cost receivers for  navigation and positioning, hence precise positioning is not  applicable due to reasons such as the incomplete elimination  of the ionospheric delay which appears to be one of the largest  error sources in the pseudorange measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' For similar  reasons, the most common algorithm used in the civilian  applications is therefore the single point positioning algorithm  which only requires a single frequency in localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  However the global averaged positioning accuracy of using  the single point positioning algorithm is still at meter level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  For example, the 95% global averaged horizontal error is less  than or equal to 8 m and the 95% global averaged vertical error  is less than or equal to 13 m for the GPS system [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' the 95%  global averaged horizontal error is less than or equal to 9 m  and the 95 % global averaged vertical error is less than or  equal to 10 m for the BeiDou Navigation Satellite System  (BDS) [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' the 95% global averaged horizontal error is less  than or equal to 5 m and the 95% global averaged   vertical  error is less than or equal to 9 m for GLONASS [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' the 95%  global averaged positioning error is less than or equal to 7 m  for Galileo [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The positioning performance of the GNSS is  even worse in the urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  Today there are many low Earth orbit (LEO) satellites  launched into space, people are also interested in utilizing  LEO satellites to aid positioning service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' For instance, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  prove that the LEO enhanced GNSS can provide centimeter  level Signal-In-Space Ranging Error (SISRE) in real-time  precise point positioning (PPP) application [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Furthermore,  researchers have also been investigating the navigation  performance which relies exclusively on the LEO satellite  signals in case the GNSS signals are unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Khalife et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' have shown that a position root mean squared error  (RMSE) of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='8 m for an unmanned aerial vehicle (UAV) can  be achieved with only two Orbcomm LEO satellites using the  carrier phase differential algorithm [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Compare LEO  satellites with the typical satellites used in the traditional  GNSS which are the medium Earth orbit (MEO) satellites,  LEO exhibits the advantages including but not limited to  shorter propagation delay and lower pathloss due to shorter  distance to the ground user, wider coverage and higher  availability due to the enormous number of satellites  simultaneously visible/available for positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' To further  enhance high bandwidth networking coverage in challenging  areas, High Altitude Platform Stations (HAPS), which resides  in the stratosphere with a typical altitude of about 20 km, can  be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' As urban area is the region where the GNSS  positioning performance degrades most severely, we could  utilize HAPS as another ranging source by equipping it with a  satellite-grade atomic clock on top of a metro city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Since  HAPS is only 20 km above the ground, the pathloss of the  HAPS signal is expected to be much less than that for any  satellites, making the received signal power of HAPS stronger  than that of satellite, which likely renders less estimation error  in the multipath mitigation of the HAPS signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' HAPS is  quasi-stationary as it does not orbit around the globe, this can  reduce the number of handovers during the course of  positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Moreover the HAPS signal does not suffer from  the ionospheric effect since it is transmitted from below the  ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Therefore the pseudorange from HAPS can likely  be estimated with less error compared with that from satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  Similar to the pseudorange from satellites which incorporates  satellite position error, we should also consider the position  error in the pseudorange measurement for HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Fortunately,  there are ongoing research in the literature investigating the  positioning of HAPS and showing HAPS positioning error  can be comparable or even better than the satellite orbit error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  For example, a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='5 m positioning accuracy (circular error  probable [CEP] 68 percent) for HAPS has been shown  achievable using the modified RTK method [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' In fact, there  are a handful of papers in the literature investigating the  HAPS-aided GNSS positioning performance [9]-[12], but  none of which considers utilizing HAPS for the sole mission  of improving the GNSS positioning performance in the urban  area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' In this work we examine the HAPS-aided GNSS  positioning performance in the urban area via both simulation  and physical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' For simplicity, the GNSS signal only  involves the GPS C/A L1 signal, and the single point  positioning algorithm is used to compute the position solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' SYSTEM MODEL  The general system model is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The HAPS  is situated at 20 km above ground in the stratosphere which is  below the ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' There are four satellites shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  1, this is just a reminder that at least four satellites are required  to perform precise 3D localization using GNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Although  only a selection of visible satellites is used in position solution  calculation in reality, in this work all available satellites are  used in position solution calculation for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The  elevation masks for both satellite and HAPS are chosen to be  15 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The pseudorange equation for satellite is given by  𝑝𝑆𝐴𝑇 = 𝜌𝑆𝐴𝑇 + 𝑑𝑆𝐴𝑇 + 𝑐(𝑑𝑡 − 𝑑𝑇𝑆𝐴𝑇) + 𝑑𝑖𝑜𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='𝑆𝐴𝑇  + 𝑑𝑡𝑟𝑜𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='𝑆𝐴𝑇 + 𝜖𝑚𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='𝑆𝐴𝑇 + 𝜖𝑝  (1)  where 𝑝𝑆𝐴𝑇 denotes the satellite pseudorange measurement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  𝜌𝑆𝐴𝑇 is the geometric range between the satellite and receiver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  𝑑𝑆𝐴𝑇 represents the satellite orbit error,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝑐 is the speed of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  𝑑𝑡 is the receiver clock offset from GPS time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝑑𝑇𝑆𝐴𝑇 is the  satellite clock offset from GPS time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝑑𝑖𝑜𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='𝑆𝐴𝑇  denotes the  ionospheric delay for satellite signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝑑𝑡𝑟𝑜𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='𝑆𝐴𝑇 denotes the  tropospheric delay for satellite signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝜖𝑚𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='𝑆𝐴𝑇 is the delay  caused by the multipath for satellite signals and 𝜖𝑝 is the  delay caused by the receiver noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The pseudorange equation  for HAPS is described by  𝑝𝐻𝐴𝑃𝑆 = 𝜌𝐻𝐴𝑃𝑆 + 𝑑𝐻𝐴𝑃𝑆 + 𝑐(𝑑𝑡 − 𝑑𝑇𝐻𝐴𝑃𝑆) + 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆  + 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 + 𝜖𝑝  (2)  where 𝑝𝐻𝐴𝑃𝑆 denotes the HAPS pseudorange measurement,  𝜌𝐻𝐴𝑃𝑆 represents the geometric range between the HAPS and  the receiver, 𝑑𝐻𝐴𝑃𝑆  represents the HAPS position error,  𝑑𝑇𝐻𝐴𝑃𝑆 is the HAPS clock offset from GPS time, 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆  denotes the tropospheric delay for HAPS signals, 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 is  the delay caused by the multipath for HAPS signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' In this  work, the satellite orbit error, the HAPS position error, and  the HAPS clock offset are assumed to be zero for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  The simulated vehicle trajectory originates from Carleton  University in the suburban area and ends at the Rideau Street  of Ottawa in the dense urban area (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' There are four  simulated HAPS where one HAPS is following a circular  trajectory on top of the downtown Ottawa area, and the other  three HAPS are following similar circular trajectories on top  of three populated regions near Ottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Note that HAPS is  quasi-stationary due to factors such as wind, it can move  within a confined space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 3 shows the flowchart of the  single point positioning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Since the HAPS clock  offset in this work is assumed zero, we simply use 𝑑𝑇 to  denote the satellite clock offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' From the data collected by  the GNSS receiver, we shall obtain both the receiver  independent exchange (RINEX) format observation file and  the RINEX navigation file, which contains the satellite  information such as the pseudorange, the ionospheric  parameters, 𝜶, the Keplerian parameters, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' With that  information, we know the pseudo-random noise (𝑷𝑹𝑵) code  which represents the unique number of each satellite, the day  of year (𝐷𝑂𝑌) which represents the day of year at the time of  measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Note that 𝑷𝑹𝑵 is in bold to represent a vector  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 1: System model of the HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 2: Vehicle trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  containing the pseudo-random noise code of all visible  satellites at the current epoch and the current iteration of  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' We can compute the satellite positions, 𝑷𝑺𝑨𝑻, and  satellite clock offset, 𝒅𝑻, using the Keplerian parameters  contained in the navigation file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝑷𝑯𝑨𝑷𝑺  denotes a vector  containing the positions of all HAPS which are generated  using the Skydel GNSS simulator [13], and 𝒑𝑯𝑨𝑷𝑺 denotes a  vector containing the HAPS pseudorange which will be  explained in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' To compute the position solution, 𝒙,  we firstly initialize the receiver position to the center of the  Earth, and the receiver clock offset is initialized to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The  change in estimate, 𝒅𝒙, is initialized to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' For each  epoch of measurement, we will first check if the number of  available ranging sources is more than three as at least four  ranging sources are required to perform precise 3D  localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Since the receiver position is iteratively  estimated, we calculate the elevation angles for both satellites  and HAPS with respect to the recently estimated receiver  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Since both the tropospheric delay and the  ionospheric delay are functions of the receiver position, these  two atmospheric delays are estimated iteratively as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The  Ionosphere  HAPS  HAPS  20km  HAPSfootprint  HAPSfootprint  15°  cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  cellOSM+ relief shading  V  Tracks:  nelByDrivi  Dense  urban Areas  OldOrtowa  Huram  tonbu  neGleb  Suburban  Are  eas  Ottawa  enEza  e45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='40751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='-75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='60857  Googe  Map created at GpSVisualiz  Madutn  con  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 3: Flow chart of the single point positioning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  elevation angle, satellite pseudorange, HAPS pseudorange,  satellite position, satellite clock offset, the tropospheric  delay, 𝒅𝒕𝒓𝒐𝒑, the ionospheric delay, 𝒅𝒊𝒐𝒏, and the pseudo-  random noise (𝑷𝑹𝑵) code are modified iteratively based on  the re-computed elevation angles for both satellites and  HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' To prepare the parameters needed for the least square  methods, the corrected pseudorange needs to be computed as  follows:  𝒑𝑺𝑨𝑻  𝒄  = 𝒑𝑺𝑨𝑻 + 𝑐 ∙ 𝒅𝑻 − 𝒅𝒕𝒓𝒐𝒑,𝑺𝑨𝑻 − 𝒅𝒊𝒐𝒏,𝑺𝑨𝑻         (3)  where 𝒑𝑺𝑨𝑻  𝒄  represents the corrected pseudorange for  satellite, 𝒑𝑺𝑨𝑻  represents the uncorrected pseudorange for  satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Since the pseudorange error of HAPS is modeled as  Gaussian noise representing the estimation residual, the  HAPS pseudorange does not need to be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Due to the  Earth rotation, the positions of satellites and HAPS at the  signal emission time are different from that at the signal  reception time, this is known as the Sagnac effect [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The  coordinates of satellite/HAPS can be transformed from the  signal emission time to the signal reception time by [14]  ∆𝑡𝑅𝑂𝑇 = 𝑡𝑟𝑥 − 𝑡𝑡𝑥                             (4)  𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='𝑟𝑥 = 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='𝑡𝑥                 (5)  where ∆𝑡𝑅𝑂𝑇  denotes the signal propagation time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝑡𝑟𝑥  represents the signal reception time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝑡𝑡𝑥 represents the signal  emission time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='𝑟𝑥 is the 𝑖𝑡ℎ satellite/HAPS coordinates at  the signal reception time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='𝑡𝑥  is the 𝑖𝑡ℎ  satellite/HAPS  coordinates at the signal emission time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 𝜔𝐸  denotes the  Earth’s rotation rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' and 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) is known as the  rotation matrix which is described by  𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)  = [  cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)  sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)  0  − sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)  0  cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)  0  0  1  ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  (6)  The line-of-sight vector, 𝒗, and the geometric range between  ranging sources and receiver, 𝝆 , are then calculated to  compute the a-priori range residual vector 𝒃 and the design  matrix 𝑯, where  𝒃 = 𝒑𝒄 − 𝝆                                 (7)  𝑯 = [𝒗, 𝟏𝑙𝑒𝑛𝑔𝑡ℎ(𝒑𝐜)×1)]                      (8)  where 𝒑𝒄  is the corrected satellite pseudorange combined  with the corrected HAPS pseudorange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' At last, the least  square solution is computed as  𝑸 = (𝑯′𝑯)−1                            (9)  𝒅𝒙 = 𝑸𝑯′𝒃                             (10)  𝑑𝑡 = 𝒅𝒙(4)/𝑐                           (11)  where Q is known as the covariance matrix, 𝒅𝒙(4) means the  fourth element in the vector 𝒅𝒙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The covariance matrix, 𝑸, is  described by  𝑸 =  [  𝜎𝑥  2  𝜎𝑥𝑦  𝜎𝑥𝑧  𝜎𝑥𝑡  𝜎𝑥𝑦  𝜎𝑦  2  𝜎𝑦𝑧  𝜎𝑦𝑡  𝜎𝑥𝑧  𝜎𝑦𝑧  𝜎𝑧  2  𝜎𝑧𝑡  𝜎𝑥𝑡  𝜎𝑦𝑡  𝜎𝑧𝑡  𝜎𝑡  2 ]  (12)  where 𝜎𝑥, 𝜎𝑦, 𝜎𝑧 and 𝜎𝑡 represent the standard deviations of  the receiver coordinates x, y, z in the Earth-centered Earth-  fixed (ECEF) coordinate frame and the receiver clock offset,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The least square solution shall be found when  the norm of the change in receiver position, 𝒅𝒙(1: 3), is  sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' In this work, this threshold is chosen to be  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='01 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' We use the horizontal dilution of precision (HDOP)  and the 3D positioning accuracy as the metrics to examine the  positioning performance of the proposed HAPS-aided GPS  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' To compute the HDOP, we must convert the  covariance matrix into the local north-east-down (NED)  coordinate frame, which can be done with the following  equation [15]:  𝑸𝑵𝑬𝑫 = 𝑹′𝑸̃𝑹                          (13)  where 𝑸̃ and 𝑹 are defined as  𝑸̃ = [  𝜎𝑥  2  𝜎𝑥𝑦  𝜎𝑥𝑧  𝜎𝑥𝑦  𝜎𝑦  2  𝜎𝑦𝑧  𝜎𝑥𝑧  𝜎𝑦𝑧  𝜎𝑧  2  ]                           (14)  Initialization  PsAT,PHAPS,PRN,DOY  x = 04x1  Input  dt = x(4)/c  PHAPs,PsAT,dT,α  dx =x+Inf  stop = 0  No  Exit 4  NsAT + NHAPs ≥ 4  Yes  No  Idx(1:3)I>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content="01  Yes  Finding parameters  For satellites  For HAPS  sAT,dtrop,dion  OHAPS  Applying elevation mask  For satellites  For HAPS  sAT,dtrop,dion,PRN,dT,PsAT  HAPS,PHAPS  Pseudorangecorrection  PSAT,PHAPS  Combining the  corrected  pseudoranges  P'=[PSAT,PHAPS]  Correcting for the Sagnac effect  (i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=', Earth rotation)  PSAT,PHAPS  Combiningthecorrected  ranging source positions  P° = [PSAT, PHAPS]  Finding parameters  V,P,b,H,Q  Output  Computingtheposition solution  using the Least Square method  x,dt  x,dt𝑹 = [  −sin 𝜆  cos 𝜆  0  − cos 𝜆 sin 𝜑  − sin 𝜆 sin 𝜑  cos 𝜑  cos 𝜆 cos 𝜑  sin 𝜆 cos 𝜑  sin 𝜑  ]        (15)  where 𝜆 and 𝜑 represent the longitude and latitude of the  receiver, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Then the HDOP is described by  𝐻𝐷𝑂𝑃 = √𝜎𝑛2 + 𝜎𝑒2                         (16)  where 𝜎𝑛, 𝜎𝑒, and 𝜎𝑑 represent the receiver position errors in  the local north, east and down directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' SIMULATION OF THE HAPS-AIDED GPS SYSTEM  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Simulation Setup  The system model is established using the default Earth  orientation parameters of the Skydel GNSS simulation  software [13] which considers all GPS satellites orbiting  around the Earth and transmitting the L1 C/A code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The  Saastamoinen model is chosen to emulate the tropospheric  effect and the Klobuchar model is chosen to emulate the  ionospheric effect along with the software default Klobuchar  parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=', alpha and beta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The output from Skydel  contains the ECEF coordinates of satellites at the signal  emission time, the ionospheric corrections, the tropospheric  corrections, the satellite clock offsets, the ECEF coordinates  of the receiver, the signal emission time, and so forth, at each  time stamp from the start of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The receiver clock  offset in the simulation is zero by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The correction terms  in the pseudorange equation of satellite including the satellite  orbit error, the multipath and the receiver noise are not  separately considered in the simulation, instead a pseudorange  error is introduced to reflect the presence of those effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The  pseudorange error of satellite is featured using the built-in first  order Gauss-Markov process with the default time constant of  10 s and the standard deviation of 6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The continuous model  for the first order Gauss-Markov process is described by [16]  𝑥̇ = −  1  𝑇𝑐 𝑥 + 𝑤                                   (17)  where 𝑥  represents a random process with zero mean,  correlation time 𝑇𝑐, and noise 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The autocorrelation of the  first order Gauss-Markov process is described by [17]  𝑅(∆𝑡) = 𝜎2𝑒−|∆𝑡|  𝜏                                 (18)  where ∆𝑡 represents the sampling interval, 𝜎 and 𝜏 denote the  standard deviation and the time constant of the first order  Gauss-Markov process, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The pseudorange of  HAPS is simulated by adding Gaussian noise to the geometric  range between HAPS and receiver, where the Gaussian noise  represents the sum of all kinds of estimation residuals  including the HAPS position, the HAPS clock offset, the  tropospheric delay, the multipath and the receiver noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The  pseudorange error for HAPS is modelled using the Gaussian  noise with standard deviations of 2 m and 5 m representing the  suburban and the dense urban scenario, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The  characteristics of the pseudorange errors for the suburban  scenario and the dense urban scenario are set to be the same  for satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Note that by doing this, the positioning  performance of the GPS-only system stays the same in both  suburban scenario and dense urban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The standard  deviation for the HAPS pseudorange error is enforced to be  smaller than that for the satellite pseudorange error in both  suburban scenario and dense urban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' All the available  satellites (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=', satellites with elevation angles greater than the  predefined elevation mask) are simultaneously utilized for  positioning as if all satellites above the elevation mask are in  line of sight (LOS) with the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Under this setting, we  examine the 3D positioning performance for the GPS-only  system, the one-HAPS with GPS system, the four-HAPS with  GPS system and the four-HAPS-only system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' For the one-  HAPS with GPS system, we use the HAPS on top of the  downtown Ottawa area which elevation is above 80°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Simulation Results  The cumulative distribution functions of the 3D  positioning accuracy for different systems with the standard  deviations of the HAPS pseudorange error being 2 m and 5 m  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 4, we  can observe that with much less pseudorange error for HAPS,  the four-HAPS with GPS system achieves the best  positioning performance, the one-HAPS with GPS system  achieves almost the same positioning performance as the  GPS-only system, and the four-HAPS-only system achieves  slightly worse performance than the four-HAPS with GPS  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The reasons why the four-HAPS-only system does  not achieve the best positioning performance is potentially  due to the following reasons 1) it has much fewer ranging  sources in receiver position computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 2) the ranging  source geometry is poor as the elevation angles for all four  HAPS at any given time are above 40° with one even above  80°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 5, we see that with the HAPS pseudorange  error similar but slightly smaller than the satellites’  pseudorange error, the four-HAPS-only system achieves the  worst positioning performance but the four- HAPS with GPS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 4: CDF for 3D position accuracy (suburban scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 5: CDF for 3D position accuracy (dense urban scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  Denseurbanscenario(HAPSprstd=5m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='6  DF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='2  GPS-onlysystem  One-HAPSwithGPSsystem  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='1  Four-HAPSwithGPSsystem  Four-HAPS-only system  0  0  5  10  15  20  25  30  35  3Dpositionalaccuracy(m)inlocalNEDframeSuburbanscenario(HAPSprstd=2m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='6  DF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='5  C  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='2  GPS-onlysystem  One-HAPSwithGPSsystem  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='1  Four-HAPS with GPS system  Four-HAPS-only system  0  0  5  10  15  20  25  30  35  3Dpositionalaccuracy(m)inlocalNEDframesystem still outperforms the other systems considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' FIELD EXPERIMENTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Experiment Setup  To verify and support the simulation results, we also  process the raw GNSS data collected along the vehicle  trajectory which is similar to the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 2 with a  slight difference due to partial road closure on the day of data  collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The raw GNSS data are collected using the Ublox  EVK-M8T GNSS unit and processed using the single point  positioning package developed by Napat Tongkasem [18]  with proper modification so that HAPS can be incorporated  in the single point positioning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Table I gives the  specifications of the EVK-M8T GNSS unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' To reflect  realistic LOS conditions for HAPS, the LOS probability with  respect to the HAPS elevation angle in the urban area is  implemented based on [19] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Note that the LOS  probability for HAPS provided by [19] is generated based on  the city of Chicago and enforcing the LOS probability on  HAPS in the dense urban area in Ottawa might be too harsh  considering the incompatible city scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The pseudorange of  HAPS in the experiment is modeled as the addition of the  geometric range between the satellite and receiver, the  receiver clock offset multiplied by the speed of light and the  pseudorange error representing the sum of all kinds of  estimation residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The pseudorange errors for HAPS in the  suburban area and in the dense urban area are simulated as  Gaussian noise with standard deviations of 2 m and 5 m,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Since the vehicle trajectory involves both  suburban area and dense urban area, the entire route is  divided into two parts where the first part is considered as the  suburban scenario and the second part is considered as the  dense urban scenario (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' By observing the  positioning performance of the GPS-only system using the  real GPS data, the LOS probability for the suburban area is  applied to HAPS for epochs less than 380 s, and the LOS  probability for the dense urban area is applied to HAPS for  epochs greater than or equal to 380 s (refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Since  the GNSS receiver does not provide accurate receiver clock  offset with respect to the GPS time, the receiver clock offset  in each epoch is estimated by making use of the ground truth  receiver position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The ground truth data is provided by Ublox  EVK-M8U  GNSS  unit,  which  is  equipped  with  accelerometer and gyroscope, hence it can perform sensor  fusion to get better positioning performance and dead  reckoning when the signal quality degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  EVK-M8T GNSS UNIT SPECIFICATIONS [21]  Parameter  Specification  Serial Interfaces  1 USB V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='0  1 RS232, max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='baud rate 921,6 kBd  DB9 +/- 12 V level  14 pin – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='3 V logic  1 DDC (I2C compatible) max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 400 kHz  1 SPI-clock signal max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 5,5 MHz – SPI DATA  max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 1 Mbit/s  Timing Interfaces  2 Time-pulse outputs  1 Time-mark input  Dimensions  105 × 64 × 26 mm  Power Supply  5 V via USB or external powered via extra power  supply pin 14 (V5_IN) 13 (GND)  Normal Operating  Temperature  −40℃ to +65℃  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 6: HDOP (top) and 3D position accuracy (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Experiment Results  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 6 shows the HDOP, and the 3D positioning accuracy  overlapped with the number of visible HAPS at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  As we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 6, the HDOP and 3D positioning  accuracy of the HAPS-aided GPS system are better than that  of the GPS-only system in both suburban area and dense  urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Moreover, we can observe that the positioning  performance of the HAPS-aided GPS system is more stable  than the GPS-only system as there are less spikes for the  HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Note that, the pseudorange of  HAPS in the experiment is modeled as a function of the  receiver clock offset, which is estimated with the best effort,  additional error should be expected in the pseudorange of  HAPS with the magnitude depending on the quality of all  visible satellite signals and the ground truth receiver position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  As we would expect the quality of the satellite signals in the  suburban area is better compared to that in the dense urban  area, the receiver clock offset would also be expected to be  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 7: CDF of 3D position accuracy in the suburban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 8:  CDF of 3D position accuracy in the dense urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  Suburbanarea  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='6  CDF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='5  C  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='1  GPS-onlysystem  HAPS-aided GPS system  0  0  5  10  15  20  25  30  35  3Dpositioningerror(m)Denseurbanarea  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='6  CDF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='1  GPS-only system  HAPS-aidedGPSsystem  0  0  50  100  150  200  250  3Dpositioningerror(m)GPS-only system  HAPS-aided GPS system  number of HAPS  0  100  200  300  400  500  600  700  epoch (s)GPS-cnly system  4  HAPS-aided GPS system  300  Hoe  number of HAPS  3  DOSI  100  0  100  200  300  400  500  600  700  epoch (s)estimated with higher accuracy in the suburban area than in  the dense urban area, hence the HDOP of the HAPS-aided  GPS system in the suburban area is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The cumulative  distribution functions of the 3D positioning accuracy in the  suburban and dense urban areas are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 8, we can observe that  the HAPS-aided GPS system outperforms the GPS-only  system, especially in the suburban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' CONCLUSION  As we are passing 5G and soon entering 6G and beyond,  HAPS can be of invisible treasure as it can be used for  computation offloading [22], edge computing [23], even base  station [24] to meet human needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' HAPS can be another type  of ranging source which is quasi-stationary and much closer  to the ground of the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Compared to satellite, HAPS  exhibits the advantages of lower latency, lower pathloss,  lower pseudorange error, and it can provide continuous  coverage to reduce the number of handovers for the users in  a certain region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Since urban area is the region where GNSS  positioning performance degrades severely and where most  people live in, deploying several HAPS acting as another type  of ranging source on top of a metro city would improve the  GNSS positioning performance and maximize the value of  the extra payload on HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The HAPS-aided GNSS can also  be deployed in the regions with extreme environment such as  the Arctic region where the satellite availability is low, and  the ionospheric disturbances is severe [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' From both the  simulation and physical experiment results, we observe that  HAPS can indeed improve the 3D positioning accuracy,  especially in the suburban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' To improve the results of  HAPS-aided GPS system in the dense urban area, the receiver  clock offset should be estimated with higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' In  future work, the received signal powers of HAPS and satellite  will jointly be considered, a satellite selection algorithm will  be applied to better emulate the way a modern GNSS receiver  processes the raw GNSS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  ACKNOWLEDGMENT  This paper is supported in part by Huawei Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' The  Skydel software is a formal donation from Orolia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content='  REFERENCES  [1] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
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+page_content='GOV, USA, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' Accessed on: Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
+page_content=' 9, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'}
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+On the η1(1855), π1(1400) and π1(1600) as dynamically generated states and their
+SU(3) partners
+Mao-Jun Yan,1, ∗ Jorgivan M. Dias,1, † Adolfo Guevara,1, ‡
+Feng-Kun Guo,1, 2, 3, § and Bing-Song Zou1, 2, 4, ¶
+1CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
+Chinese Academy of Sciences, Beijing 100190, China
+2School of Physical Sciences, University of Chinese Academy of Sciences,
+Beijing 100049, China
+3Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China
+4Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
+In this work, we interpret the newly observed η1(1855) resonance with exotic JP C =
+1−+ quantum numbers in the I = 0 sector, reported by the BESIII Collaboration, as a
+dynamically generated state from the interaction between the lightest pseudoscalar mesons
+and axial-vector mesons. The interaction is derived from the lowest order chiral Lagrangian
+from which the Weinberg-Tomozawa term is obtained, describing the transition amplitudes
+among the relevant channels, which are then unitarized using the Bethe-Salpeter equation,
+according to the chiral unitary approach. We evaluate the η1(1855) decays into the ηη′ and
+K ¯K∗π channels and find that the latter has a larger branching fraction. We also investigate
+its SU(3) partners, and according to our findings, the π1(1400) and π1(1600) structures
+may correspond to dynamically generated states, with the former one coupled mostly to
+the b1π component and the latter one coupled to the K1(1270) ¯K channel. In particular,
+our result for the ratio Γ(π1(1600) → f1(1285)π)/Γ(π1(1600) → η′π) is consistent with the
+measured value, which supports our interpretation for the higher π1 state. We also report
+two poles with a mass about 1.7 GeV in the I = 1/2 sector, which may be responsible for
+the K∗(1680). We suggest searching for two additional η1 exotic mesons with masses around
+1.4 and 1.7 GeV. In particular, the predicted η1(1700) is expected to have a width around
+0.1 GeV and can decay easily into K ¯Kππ.
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+§ [email protected]
+¶ [email protected]
+arXiv:2301.04432v1  [hep-ph]  11 Jan 2023
+
+2
+I.
+INTRODUCTION
+Over the last two decades, the experimental observation of many new hadronic states is chal-
+lenging our current understanding of hadrons as conventional mesons and baryons with valence
+contents of quark-antiquark and three quarks, respectively, since most of them do not fit in the
+well-known quark model. This difficulty brought back a long-standing discussion on the exotic
+hadronic structures, i.e., multiquark configurations that might have quantum numbers beyond
+those assigned to the conventional mesons and baryons [1, 2].
+Exotic quark configurations such as tetraquarks [3, 4], hadron-hadron molecules [5], glueballs,
+and hybrids [6, 7], among others, have been suggested to describe suitably most of the properties of
+these new states, such as the JPC quantum numbers, mass, and decay width, especially for those
+lying in the charmonium and bottomonium spectra.
+On the other hand, distinguishing the exotic states from the conventional hadrons is a more
+complicated task in the light quark sector. Many states have their masses close to each other, and
+the possibility of mixing brings additional difficulty to the problem. The situation improves as
+the quantum numbers do not fall into those allowed by the conventional quark model. It seems
+to be the case of the newly discovered state, dubbed η1(1855), by the BESIII Collaboration [8, 9],
+observed in the invariant mass distribution of the η η′ meson pair in the J/ψ → γ η η′ decay
+channel with a significance of 19σ. Its mass and width reported by BESIII are 1855 ± 9+6
+−1 MeV
+and 188 ± 18+3
+−8 MeV, respectively, with likely JPC = 1−+ quantum numbers, which cannot be
+formed by a pair of quark and antiquark. The η1(1855) is not the only state experimentally found
+with that set of quantum numbers. As of today, three other hadronic structures, called π1(1400),
+π1(1600) and π1(2015), with JPC = 1−+, were observed by several collaborations [7, 10].
+From the theoretical point of view, the hybrid model has been used to investigate these exotic
+meson states, in particular the 1−+ ones. Lattice quantum chromodynamics (QCD) calculations
+have pointed out hybrid supermultiplets with exotic JPC quantum numbers, including the 1−+
+one [11–16]. In this picture, however, the mass of the lightest 1−+ state and decay modes are in-
+consistent with the corresponding experimental results, while the π1(1600) and π1(2015) structures
+can fit into the nonets predicted by lattice QCD [7].
+The newly observed η1(1855) state has also been the focus of some studies. In particular, the
+authors in Ref. [17] proposed two hybrid nonet schemes in which the η1(1855) resonance can be
+either the lower or higher mass state with isospin I = 0. In Ref. [18], an effective Lagrangian
+respecting flavor, parity, and charge conjugation symmetries is used to study the hybrid nonet
+
+3
+decays into two-body meson states. The authors have fixed the couplings to those two-body meson
+states by performing a combined fit to the experimental and lattice results available. As a result,
+the decay width value estimated for the isoscalar member of the hybrid nonet agrees with the
+one observed for η1(1855) state. Also addressing the same picture, Ref. [19] applied the approach
+of QCD sum rules to describe the η1(1855) mass. By contrast, within the same approach, the
+η1(1855) resonance is described as a tetraquark state in Ref. [20].
+The η1(1855) resonance also supports a meson-meson molecule interpretation due to its prox-
+imity to the K ¯K1(1400) threshold, as put forward by Refs. [21, 22]. In particular, the authors
+in Ref. [21] have investigated the K ¯K1(1400) interaction through the one-boson exchange model.
+According to their findings, the K ¯K1(1400) system binds for cutoff values above 2 GeV with a
+monopole form factor. In addition, the comparison between their result for the branching fraction
+B(η1 → η η′) to the experimental one led them to conclude that the K ¯K1(1400) molecule can
+explain the η1(1855) structure.
+An important point to be addressed is the meson-meson interaction around the K1(1400) ¯K
+threshold for the JPC = 1−+ quantum numbers. In this sector, many meson-meson pairs may
+contribute to that interaction, so a coupled-channel treatment seems appropriate to take these
+contributions into account. In particular, hadron-hadron interactions in coupled channels have
+been studied in many works to describe the properties of the new hadronic systems experimentally
+observed. In those cases, these hadronic structures are called dynamically generated states.
+Following this approach, in this work, we aim to explore the η1(1855), π1(1400), and π1(1600)
+hadronic systems as dynamically generated states from pseudoscalar-axial vector meson interactions
+in coupled channels. Specifically, the low-energy interactions are given by the Weinberg-Tomozawa
+(WT) term from chiral Lagrangians at the leading order of the chiral expansion by treating the
+axial vector mesons as matter fields and the pseudoscalar mesons as the pseudo-Nambu-Goldstone
+bosons of the spontaneous breaking of chiral symmetry.
+Such Lagrangians have been used to
+study many hadron structures stemming from meson-meson and meson-baryon interactions in
+coupled channels in light and heavy sectors, see, e.g., Refs. [23–27]. In our case, the amplitudes
+obtained from the WT term are unitarized via the Bethe-Salpeter equation from which bound
+states/resonances manifest as poles in the physical/unphysical Riemann sheets of the scattering
+matrices. The existence of a whole family of kaonic bound states has been pointed out in Ref. [28]
+based on unitarizing the WT term for the scattering of the kaon off isospin-1/2 matter fields
+taking heavy mesons and doubly-charmed baryons as examples. As we shall show in this work,
+the newly observed η1(1855) structure may correspond to a dynamically generated state from the
+
+4
+pseudoscalar-axial vector interaction in the isospin I = 0 sector coupling strongly to the K1(1400) ¯K
+channel. Moreover, the π1(1400) and π1(1600), may be assigned as the η1(1855) SU(3) partners
+which are also dynamically generated from the pseudoscalar-axial vector meson interactions in the
+I = 1 sector. The former resonance couples mainly to the b1π channel, and the latter has the
+K1(1270) ¯K as its main coupled channel.
+In addition, we have also found two poles around 1.7 GeV in the I = 1/2 sector. These poles
+are particularly interesting as they could be the origin of the K∗(1680) structure observed experi-
+mentally [10], which is the main component of the 1− contribution to the φK mass distribution in
+the B → J/ψφK decays recently measured by LHCb [29].
+This paper is organized as follows. In Section II, we discuss the relevant channels contributing
+to the pseudoscalar-axial vector meson interactions and the use of the chiral unitary approach
+(ChUA) for the evaluation of the transition amplitudes among those channels. In Sections III
+and IV, we investigate the dynamical generation of poles stemming from those interactions in the
+I = 0 and I = 1 sectors and discuss their possible decay channels. Finally, in Section V, we also
+explore the dynamical generation of poles for I = 1/2 and their connection to the vector K∗(1680)
+structure observed experimentally. Section VI gives a summary.
+II.
+COUPLED CHANNEL SCATTERING IN CHIRAL UNITARY APPROACH
+We investigate the interactions between axial and pseudoscalar mesons in coupled channels in
+the 1300 ∼ 2000 MeV energy range. First, we need to determine the space of states contributing
+to the interaction in this energy range.
+In Tables I, II, III, and IV, we list all the relevant channels for the problem under consideration
+along with their corresponding mass thresholds. The channels are organized from the lower to
+higher mass values and by the isospin, 0, 1 and 1/2, respectively.
+TABLE I. JP C = 1−+ meson-meson channels with I = 0. The threshold masses are in the units of MeV.
+Channel
+a1π
+K1(1270) ¯K
+f1(1285)η
+K1(1400) ¯K
+f1(1420)η
+Threshold
+1368
+1748
+1829
+1898
+1973
+TABLE II. JP C = 1−+ meson-meson channels with I = 1. The threshold masses are in the units of MeV.
+Channel
+b1π
+f1(1285)π
+f1(1420)π
+K1(1270) ¯K
+a1η
+K1(1400) ¯K
+Threshold
+1367
+1419
+1564
+1748
+1777
+1895
+
+5
+TABLE III. JP = 1− meson-meson channels with I = 1/2. The threshold masses are in the units of MeV.
+Here the flavor-neutral axial vector mesons have JP C = 1++.
+Channel
+a1K
+f1(1285)K+
+K1(1270)η
+f1(1420)K
+K1(1400)η
+Threshold
+1725
+1777
+1800
+1921
+1947
+TABLE IV. JP = 1− meson-meson channels with I = 1/2. The threshold masses are in the units of MeV.
+Here the flavor-neutral axial vector mesons have JP C = 1+−.
+Channel
+h1(1170)K
+b1K
+K1(1270)η
+h1(1415)K
+K1(1400)η
+Threshold
+1661
+1725
+1800
+1911
+1947
+In what follows, we shall discuss the relevant scattering amplitudes among all those channels
+above for each isospin sector. These transitions can be written in the form of the WT term which
+then is unitarized. Notice that the channels displayed in Tables III and IV, in principle, should be
+grouped in the same space of states since they share identical isospin and JP quantum numbers.
+However, the relevant transitions among them arise only at the next-to-leading order in the chiral
+expansion; see the discussion around Eq. (17) below. Thus, such transitions are of higher order
+than that of the WT term and will be neglected here.
+A.
+The Weinberg-Tomozawa term
+In order to study the interactions among all the channels listed in the previous tables, we have
+to evaluate the interactions between the pseudoscalar and axial-vector mesons.
+The latter are
+organized in two SU(3) octets according to their JPC quantum numbers.
+A1 =
+�
+�
+�
+�
+�
+a0
+1
+√
+2 + f8
+1
+√
+6
+a+
+1
+K+
+1A
+a−
+1
+− a0
+1
+√
+2 + f8
+1
+√
+6
+K0
+1A
+K−
+1A
+¯K0
+1A
+− 2f8
+1
+√
+6
+�
+�
+�
+�
+�
+(1)
+is the octet of resonances of axial-vector states with JPC = 1++ for the flavor-neutral mesons, and
+B1 =
+�
+�
+�
+�
+�
+b0
+1
+√
+2 + h8
+1
+√
+6
+b+
+1
+K+
+1B
+b−
+1
+− b0
+1
+√
+2 + h8
+1
+√
+6
+K0
+1B
+K−
+1B
+K0
+1B
+− 2
+√
+6h8
+1
+�
+�
+�
+�
+�
+(2)
+describes the octet of axial-vector resonances with JPC = 1+−.
+The singlet and I = 0 octet
+flavor eigenstates are not mass eigenstates; that is, the pairs of f1(1420), h1(1415) (also known as
+
+6
+TABLE V. Two sets of values of the axial-vector meson mixing angles taken from Ref. [30]. Set B is preferred
+in Ref. [30]. The η-η′ mixing angle θP is taken from Ref. [31]. For more discussions about these mixing
+angles, we refer to the review of Quark Model in the Review of Particle Physics [10].
+Angles
+θK1
+θ3P1
+θ1P1
+θP
+Set A
+57◦
+52◦
+−17.5◦ −17◦
+Set B
+34◦
+23.1◦
+28.0◦
+−17◦
+h1(1380)) and f1(1285), h1(1170) mesons are mixtures of the singlet (1) and octet (8) mesons such
+that
+�
+� |f1(1285)⟩
+|f1(1420)⟩
+�
+� =
+�
+� cos θ3P1
+sin θ3P1
+− sin θ3P1 cos θ3P1
+�
+�
+�
+�
+��f1
+1
+�
+��f8
+1
+�
+�
+� ,
+(3)
+and
+�
+� |h1(1170)⟩
+|h1(1415)⟩
+�
+� =
+�
+� cos θ1P1
+sin θ1P1
+− sin θ1P1 cos θ1P1
+�
+�
+�
+�
+��h1
+1
+�
+��h8
+1
+�
+�
+� .
+(4)
+Furthermore, the K1A and K1B members of the multiplets in Eqs. (1) and (2) are the strange
+partners of the a1(1260) and b1(1235), and their mixture contributes to the physical K1(1270) and
+K1(1400) mesons, that is
+�
+� |K1(1270)⟩
+|K1(1400)⟩
+�
+� =
+�
+� sin θK1
+cos θK1
+cos θK1 − sin θK1
+�
+�
+�
+� |K1A⟩
+|K1B⟩
+�
+� .
+(5)
+The corresponding values for the mixing angles in Eqs. (3), (4), and (5) are listed in Table V, where
+they are grouped into two sets, denoted by A and B. Although set B is preferred in Ref. [30], we
+will use both sets to have an estimate of the uncertainties caused by such an angle.
+In order to determine the WT term we start with the Lagrangian (see, e.g., Ref. [32])
+L0 = −1
+4
+�
+VµνV µν − 2M2
+V VµV µ�
+,
+(6)
+where ⟨, ⟩ takes trace in the SU(3) flavor space,
+Vµν = DµVν − DνVµ ,
+(7)
+while Dµ is the chirally covariant derivative, which when acting on SU(3) octet matter fields reads
+as
+Dµ = ∂µ + [Γµ, ] ,
+(8)
+
+7
+with [ , ] the usual commutator. In addition, Γµ stands for the chiral connection, given by
+Γµ = 1
+2
+�
+u†∂µu + u∂µu†�
+,
+(9)
+with
+u = exp
+�
+i
+√
+2Fπ
+φ8
+�
+,
+(10)
+where Fπ = 92.1 MeV is the pion decay constant [10], and φ8 is the pseudoscalar SU(3) octet, that
+is
+φ8 =
+�
+�
+�
+�
+�
+π0
+√
+2 +
+1
+√
+6η8
+π+
+K+
+π−
+− 1
+√
+2π0 +
+1
+√
+6η8
+K0
+K−
+¯K0
+− 2
+���
+6η8
+�
+�
+�
+�
+� .
+(11)
+In addition, the physical η and η′ mesons are the mixtures of η8 and η1
+�
+� |η⟩
+|η′⟩
+�
+� =
+�
+� − sin θP cos θP
+cos θP
+sin θP
+�
+�
+�
+�
+��η1�
+��η8�
+�
+� ,
+(12)
+where η1 becomes the ninth pseudo-Goldstone boson in large Nc QCD [33–36]. The Goldstone
+boson nonet is written as
+φ9 = φ8 + 1
+√
+3η1,
+(13)
+which leads to a relation in the commutator
+�
+φ9, ∂µφ9�
+=
+�
+φ8, ∂µφ8�
+.
+(14)
+Therefore, only the scattering of the octet Goldstone bosons off the axial-vector mesons in
+Weinberg-Tomozawa term contributes to JP(C) = 1−(+) spectrum.
+The covariant derivative Dµ by means of the connection Γµ encodes the leading order interaction
+between the pseudoscalar mesons and the vector field Vµ [32, 37, 38]. Therefore, by replacing the
+Vµ field to the axial-vector field Aµ corresponding to either the A1 or B1 multiplet, the chiral tran-
+sition between φ8 (pseudoscalar) and A (1+) (axial-vector) is described by the following interaction
+Lagrangian
+LI = − 1
+4F 2π
+�
+[Aµ, ∂νAµ]
+�
+φ8, ∂νφ8��
+,
+(15)
+
+8
+which accounts for the WT interaction term for the PA → PA process, with P and A corresponding
+to the pseudoscalar and axial-vector mesons, respectively. From this Lagrangian we obtain the S-
+wave transition amplitude among the channels listed in Tables I, II, III and IV, that is
+Vij(s) = −ϵ · ϵ′
+8F 2π
+Cij
+�
+3s −
+�
+M2 + m2 + M′2 + m′2�
+− 1
+s
+�
+M2 − m2� �
+M′2 − m′2��
+,
+(16)
+where ϵ (ϵ′) stands for the polarization four-vector of the incoming (outgoing) axial-vector me-
+son [25, 39]. The masses M (M′) , m (m′) correspond to the initial (final) axial-vector mesons and
+initial (final) pseudoscalar mesons, respectively. The indices i and j represent the initial and final
+PA states, respectively. The coefficients Cij are given in Tables VI, VII, VIII, and IX.
+TABLE VI. Cij coefficients in Eq. (16) for axial and pseudoscalar pairs coupled to JP C = 1−+ in S-wave
+and I = 0.
+Cij
+a1π
+K1(1270) ¯K
+f1(1285)η
+K1(1400) ¯K
+f1(1420)η
+a1π
+−4
+�
+3
+2 sin θK1
+0
+�
+3
+2 cos θK1
+0
+K1(1270) ¯K
+−3
+− 3
+√
+2 sin θ3P1 sin θK1
+0
+− 3
+√
+2 cos θ3P1 sin θK1
+f1(1285)η
+0
+− 3
+√
+2 cos θK1 sin θ3P1
+0
+K1(1400) ¯K
+−3
+− 3
+√
+2 cos θ3P1 cos θK1
+f1(1420)η
+0
+TABLE VII. Cij coefficients in Eq. (16) for axial and pseudoscalar pairs coupled to JP C = 1−+ in S-wave
+and I = 1.
+Cij
+b1π
+f1(1285)π
+f1(1420)π
+K1(1270) ¯K
+a1η
+K1(1400) ¯K
+b1π
+−2
+0
+0
+cos θK1
+0
+− sin θK1
+f1(1285)π
+0
+0
+�
+3
+2 sin θK1 sin θ3P1
+0
+�
+3
+2 cos θK1 sin θ3P1
+f1(1420)π
+0
+�
+3
+2 cos θ3P1 sin θK1
+0
+�
+3
+2 cos θK1 cos θ3P1
+K1(1270) ¯K
+−1
+−
+�
+3
+2 sin θK1
+0
+a1η
+0
+−
+�
+3
+2 cos θK1
+K1(1400) ¯K
+−1
+Before proceeding, let us discuss the A1φ8 → B1φ8 transitions, with A1 and B1 the two SU(3)
+octets of axial-vector mesons and φ8 the octet of the pseudo-Nambu-Goldstone bosons. Let A1µ
+and B1µ denote the fields for the 1++ and 1+− axial-vector meson multiplets, respectively. Under
+parity transformation, we have A1µ → −Aµ
+1 and B1µ → −Bµ
+1 ; under charge conjugation, we have
+A1µ → AT
+1µ and B1µ → −BT
+1µ. Then the A1φ8 → B1φ8 transitions can only arise at O
+�
+p2�
+with p
+
+9
+TABLE VIII. Cij coefficients in Eq. (16) for axial and pseudoscalar pairs coupled to JP = 1− in S-wave
+and I = 1/2. Here the flavor-neutral axial mesons have JP C = 1++.
+Cij
+a1K
+f1(1285)K
+K1(1270)η
+f1(1420)K
+K1(1400)η
+a1K
+−2
+0
+− 3
+2 sin θK1
+0
+− 3
+2 cos θK1
+f1(1285)K
+0
+3
+2 sin θK1 sin θ3P1
+0
+3
+2 sin θK1 cos θK1
+K1(1270)η
+0
+3
+2 cos θ3P1 sin θK1
+0
+f1(1420)K
+0
+3
+2 cos θ3P1 cos θK1
+K1(1400)η
+0
+TABLE IX. Cij coefficients in Eq. (16) for axial and pseudoscalar pairs coupled to JP = 1− in S-wave and
+I = 1/2. Here the flavor-neutral axial mesons have JP C = 1+−.
+Cij
+h1(1170)K
+b1K
+K1(1270)η
+h1(1415)K
+K1(1400)η
+h1(1170)K
+0
+0
+3
+2 cos θK1 sin θ1P1
+0
+3
+2 sin θK1 sin θ1P1
+b1K
+−2
+− 3
+2 cos θK1
+0
+− 3
+2 sin θK1
+K1(1270)η
+0
+3
+2 cos θK1 cos θ1P1
+0
+h1(1415)K
+0
+3
+2 sin θK1 cos θ1P1
+K1(1400)η
+0
+the momentum scale in the chiral power counting. They are given by operators such as
+⟨A1µ[B1ν, [uµ, uν]]⟩ ,
+(17)
+with
+uµ = i
+�
+u†∂µu − u∂µu†�
+.
+(18)
+Such terms are one order higher in the chiral power counting than the WT terms describing the
+A1φ8 → A1φ8 and B1φ8 → B1φ8 transitions, and thus will be neglected.
+B.
+Unitarization procedure
+The unitarization procedure we adopt follows ChUA in which the scattering amplitudes in
+Eq. (16) are the elements of a matrix, denoted by V , such that it enters as an input to solve the
+Bethe-Salpeter equation, which in its on-shell factorization form, reads [23]
+T = (1 − V G)−1 V .
+(19)
+
+10
+The V matrix describes the transition between the channels listed in Tables I, II, III, and IV. In
+addition, G is the diagonal loop function matrix whose diagonal matrix elements are given by
+Gl = i
+�
+d4q
+(2π)4
+1
+q2 − m2
+l + iϵ
+1
+(q − P)2 − M2
+l + iϵ ,
+(20)
+with ml and Ml the masses of the pseudoscalar and axial-vector mesons, respectively, involved in
+the loop in the channel l, and P the total four-momentum of those mesons (P 2 = s). After the
+integration over the temporal component q0, Eq. (20) becomes
+Gl(s) =
+�
+d3q
+(2π)3
+ω1 + ω2
+2ω1ω2
+1
+(P 0)2 − (ω1 + ω2)2 + iϵ
+,
+(21)
+with ω1 =
+�
+Ml2 + |⃗q|2 and ω2 =
+�
+ml2 + |⃗q|2, and can be regularized by means of a cutoff in
+the three-momentum qmax. On the other hand, the function Gl can also be regularized using a
+subtraction constant as [40]
+GDR
+l
+(s) =
+1
+16π2
+�
+αl(µ) + log M2
+l
+µ2 + m2
+l − M2
+l + s
+2s
+log m2
+l
+M2
+l
++ pl
+√s
+�
+log s − m2
+l + M2
+l + 2pl
+√s
+−s + m2
+l − M2
+l + 2pl
+√s
++ log s + m2
+l − M2
+l + 2pl
+√s
+−s − m2
+l + M2
+l + 2pl
+√s
+��
+,
+(22)
+where pl is the three-momentum of the mesons in the center-of-mass (c.m.) frame
+pl =
+��
+s − (Ml + ml)2� �
+s − (Ml − ml)2�
+2√s
+,
+(23)
+while µ is an arbitrary scale of the regularization. Any changes in the µ scale can be absorbed by the
+subtraction constant αl(µ) such that the result is independent of the scale. We may determine the
+subtraction constant for each intermediate state of the scattering problem by comparing Eqs. (21),
+regularized using qmax, and (22) at the threshold. The equivalence between the two prescriptions
+for the loop-function is discussed in, e.g., Refs. [41–43]. In this work, we follow Ref. [44] and set
+µ = 1 GeV and α = −1.35, which is obtained by matching to hard cutoff regularization with
+qmax ≃ 0.7 GeV in the f1(1285)η channel. This set of parameters are used for all channels, and
+a variation of the cutoff within qmax = (0.7 ± 0.1) GeV, and correspondingly α(µ = 1 GeV) =
+−1.35 ± 0.17, will be used to show the dependence of the results on this parameter.
+C.
+Searching for poles
+We move on to the complex energy plane to search for poles in the T-matrix. Specifically, for a
+single-channel problem, there are two Riemann sheets for the complex energy plane. Bound states
+
+11
+show up as poles, below the threshold, in the transition matrix on the real energy axis on the first
+Riemann sheet, while virtual states manifest themselves below the threshold on the real axis on the
+second Riemann sheet, and resonances correspond to poles off the real axis on the second Riemann
+sheet. The Riemann sheets come about because the G loop function has a cut extending from
+the threshold to infinity which is usually chosen to be along the positive real axis. For n coupled
+channels, there are n cuts and thus 2n Riemann sheets. From unitarity and the Schwarz reflection
+principle, the discontinuity of the Gl function can be read off from its imaginary part,
+Im Gl(s) = −
+pl
+8π√s ,
+(24)
+which we can use to perform an analytic continuation to the entire complex plane. In this case,
+the Gl loop function on the “second” Riemann sheet with respect to the lth channel reads
+GII
+l (s) = GI
+l(s) + i
+pl
+4π√s ;
+(25)
+the lower half plane of this Riemann sheet is directly connected to the physical region when the lth
+channel is open, i.e., Re(√s) ≥ m + M. We will label the Riemann sheets according to the sign of
+the imaginary part of the corresponding c.m. momentum for each channel (see the next section).
+Furthermore, it is also possible to determine the pole couplings to the lth channel. Note that
+close to the pole singularity the T-matrix elements Tij(s) admit a Laurent expansion,
+Tij(s) = gi gj
+s − zp
++ regular terms,
+(26)
+where zp = (Mp −iΓ/2)2 is the pole location on the complex energy plane, with Mp and Γ standing
+for the pole mass and width, respectively. Therefore, the product of couplings gigj is the residue
+at the pole in Tij(s) which takes values on the Riemann sheet where the pole is located. In this
+way, the couplings can be evaluated straightforwardly. For instance, for a diagonal transition it is
+given by
+g2
+i = r
+2π
+� 2π
+0
+Tii(z(θ))eiθdθ
+= lim
+s→zp(s − zp)Tii(s) =
+� d
+ds
+1
+Tii(s)
+�−1
+s=zp
+,
+(27)
+where z(θ) = zp + i reiθ with r the radius of contour for the integral, and the two lines give two
+equivalent ways of computing residues.
+
+12
+TABLE X. The poles (in GeV) and their corresponding couplings (in GeV) to the channels contributing to the
+PA interaction with I = 0 and exotic quantum numbers JP C = 1−+. The corresponding Riemann sheet for
+each pole is listed below the pole position. The dominantly coupled channel is emphasized in boldface for each
+pole. The errors of the poles are from varying the subtraction constant within α(µ = 1 GeV) = −1.35±0.17,
+and only the central values of the couplings are given.
+Poles (Set A)
+Channels
+1.39 ± 0.01 − i(0.04 ± 0.01)
+a1π
+K1(1270) ¯
+K
+f1(1285)η
+K1(1400) ¯
+K
+f1(1420)η
+(− + + + +)
+gl
+5.21 + i3.01
+1.22 + i0.78
+0.01 + i0.02
+0.36 + i0.35
+0.00
+1.69 ± 0.03
+a1π
+K1(1270) ¯
+K
+f1(1285)η
+K1(1400) ¯
+K
+f1(1420)η
+(− + + + +)
+gl
+0.36 + i0.98
+8.16 − i0.17
+3.64 + i0.01
+0.09 − i0.15
+2.46 + i0.01
+1.84 ± 0.03
+a1π
+K1(1270) ¯
+K
+f1(1285)η
+K1(1400) ¯
+K
+f1(1420)η
+(− − − + +)
+gl
+0.07 + i0.28
+0.69 + i0.55
+1.68 + i0.08 9.33 + i0.15 1.16 + i0.06
+Poles (Set B)
+Channels
+1.39 ± 0.01 − i(0.04 ± 0.01)
+a1π
+K1(1270) ¯
+K
+f1(1285)η
+K1(1400) ¯
+K
+f1(1420)η
+(− + + + +)
+gl
+5.21 + i3.03
+0.81 + i0.53
+0.00
+0.55 + i0.54
+0.00
+1.70 ± 0.02
+a1π
+K1(1270) ¯
+K
+f1(1285)η
+K1(1400) ¯
+K
+f1(1420)η
+(− + + + +)
+gl
+0.25 + i0.67
+8.34 − i0.08
+1.27 − i0.01
+0.37 + i0.17
+2.58 − i0.01
+1.84 ± 0.03
+a1π
+K1(1270) ¯
+K
+f1(1285)η
+K1(1400) ¯
+K
+f1(1420)η
+(− − − + +)
+gl
+0.15 + i0.62
+0.33 − i0.27
+1.83 + i0.09 9.05 + i0.17 3.81 − i0.20
+III.
+η1(1855) AND ITS DECAYS
+A.
+Dynamical generation of the η1(1855)
+Following the unitarization procedure described previously, we seek dynamically generated
+states stemming from the S-wave interactions between pseudoscalar and axial-vector mesons. For
+the I = 0 case, the transition amplitudes among the channels listed in Table I are determined using
+Eq. (16) with the Cij coefficients given in Table VI. In Table X, we show the isoscalar poles with
+exotic quantum numbers JPC = 1−+ obtained by solving Eq. (19) using those coefficients as well
+as each set of mixing angles listed in Table V. We also show the couplings of these poles to the
+channels spanning the space of states in Table I.
+Furthermore, in Table X we also highlight the Riemann sheets, the first and the second one
+for each channel, denoted by the + and − signs, respectively. We get three poles such that their
+
+13
+locations are barely affected by the change of the mixing angles from set A to set B listed in
+Table V. The lower pole is at 1.39 GeV with a width of about 0.04 GeV, which is above the a1π
+threshold. In particular, this channel is open for decay, and the fact that it is this channel the
+one for which the pole couples mostly, as pointed out in Table X, explains why that pole has such
+a value for its width. By contrast, although the a1π channel is also open for decay, the pole at
+1.69 GeV has a much smaller width because its coupling to this channel is small compared to
+the one for K1(1400) ¯K, which is the dominant channel for that pole. Similarly, the highest pole,
+located at 1.84 GeV, couples mostly to the K1(1400) ¯K channel, and has a small imaginary part.
+In addition, we can also understand why the highest pole couples more to the K1(1400) ¯K than
+to the f1(1285)η. The latter channel is closer to the pole than the former, but from Table VI,
+the diagonal f1(1285)η transition is not allowed since its WT term is zero. Nevertheless, the pole
+couples to f1(1285)η through the nondiagonal K1(1400) ¯K–f1(1285)η transition, which leads to a
+small coupling.
+B.
+Effects of the widths of the axial-vector mesons
+So far we have neglected the nonzero widths of the axial-vector mesons. In order to investigate
+their effects on the results, we use complex masses for the intermediate resonances, that is, Mi →
+Mi − iΓi/2. However, by doing that, the analytic properties are lost such that the poles of the
+T matrix do not correspond to the masses and widths of the obtained resonances any more. On
+the other hand, we can see the impact of such nonzero widths on the lineshapes of the transition
+matrix elements.
+In Fig. 1 we show a comparison between the lineshape for the T-matrix element corresponding
+to the elastic transition TK1(1400) ¯
+K→K1(1400) ¯
+K with and without including the widths for the inter-
+mediate particles. This channel has the strongest coupling to the pole at 1.84 GeV; therefore, we
+expect that any nontrivial structure should manifest most in its associated T-matrix element. The
+dashed and solid lines are the TK1(1400) ¯
+K→K1(1400) ¯
+K with zero and nonzero width, respectively, for
+both sets A and B of mixing angles in Table 1. Notice that, for the case of zero width approxima-
+tion, the TK1(1400) ¯
+K→K1(1400) ¯
+K lineshape has narrow peaks around 1845 MeV, right at the range
+of energy where we expect the η1(1855) manifests in our model. The inclusion of finite widths for
+the axial-vector mesons changes the sharp peak to a broad bump with a width of about 0.2 GeV,
+which is around the width of the K1(1400) [10]. Notice that the width matches nicely that of the
+η1(1855) measured by BESIII,
+�
+188 ± 18+3
+−8
+�
+MeV [8]. In the following, we will continue to present
+
+14
+w/o Γ
+w/o Γ
+w/ Γ
+w/ Γ
+1600
+1700
+1800
+1900
+2000
+0
+10
+20
+30
+40
+50
+s [MeV]
+|T44
+2
+Set A
+Set B
+FIG. 1. The blue dashed and solid lines are, respectively, the modulus squared of the T-matrix element, cor-
+responding to the diagonal K1(1400) ¯K → K1(1400) ¯K transition, evaluated with and without the inclusion
+of the widths associated with the axial-vector mesons taking part in the loop function Gl (Eq. (20)).
+predictions neglecting the width effects of the axial-vector mesons.
+Let us briefly discuss the other two predicted isoscalar exotic η1 mesons in Table X. The one
+with a mass of about 1.39 GeV, denoted as η1(1400), is expected to be rather broad due to the
+large width of the a1(1260) as it couples most strongly to the a1π channel. It can be searched for
+in final states such as ρππ and K ¯Kππ. The one with a mass around 1.7 GeV, denoted as η1(1700),
+couples most strongly to the K1(1270) ¯K and is expected to have a width similar to that of the
+K1(1270), i.e., around 0.1 GeV. It can also be searched for in final states of K ¯Kππ.
+C.
+The η1(1855) → η′η and K∗ ¯Kπ decays
+Let us first discuss the η1 → ηη′ decay, whose Feynman diagram is shown in Fig. 2. Within
+our approach the η1(1855) structure decays via its K1(1400) ¯K component, with the corresponding
+coupling constant listed in Table X. We also need to evaluate the K1(1400) ¯K → ηη′ transition, for
+which we use the resonance chiral theory (RχT) operators given in Ref. [45].
+The RχT operators can be divided regarding the intrinsic-parity sector to which they contribute.
+Due to its nature, the odd-intrinsic parity sector will contain a Levi-Civita tensor [46–48]; for the
+η1 → ηη′ decay one cannot saturate the Lorentz indices in such tensor to get a nonzero contribution.
+Thus, only the even-intrinsic parity operators must give a nonvanishing contribution. Since the
+chiral O(p2) Lagrangian does not contribute to such processes [49], we will use the O(p4) Lagrangian
+given in Ref. [45]. From these operators, only three will contribute to this decay. To get the largest
+possible contribution from such operators, we use the upper bounds imposed from chiral counting
+
+15
+as done in Ref. [50]. This amounts to making equal the three coupling constants and setting them
+to λA
+1 = λA
+2 = λA
+3 = g = 0.025 GeV−1, which gives a Lagrangian
+L = g
+�
+⟨Aµν (uµuαhνα + hναuαuµ)⟩ + ⟨Aµν (uαuµhνα + hναuµuα)⟩ + ⟨Aµν (uµhναuα + uαhναuµ)⟩
+�
+,
+(28)
+where uµ has been given in Eq. (18), hµν = D{µuν} is the symmetrized covariant derivative of uµ
+and the spin-1 resonance field is given in the antisymmetric tensor formalism [37]. However, since
+the η1 → K1 ¯K transition is given in terms of Proca fields, we need to express the K1 as a Proca
+field. Following Ref. [49], the antisymmetric tensor field can be expressed in terms of the Proca
+one as follows,
+Rµ =
+1
+MR
+∂νRνµ,
+(29)
+where MR is the mass of the resonance. Using the Lagrangian of Eq.(28) and expressing the axial
+resonance in the Proca representation, we get the η1 → ηη′ decay amplitude
+Mη1→ ηη′ = −
+4m2
+η1
+3F 3πmK1
+ggK1(1400) ¯
+KGK1 ¯
+K
+��
+αp2
+η′ + 1
+√
+2βp2
+η
+�
+εη1 · pη +
+�
+pη ↔ pη′��
+,
+(30)
+where Fπ is the pion decay constant, gK1 ¯
+K is the coupling constant of the pole to the K1(1400) ¯K
+channel, GK1 ¯
+K is the loop function for the K1 and ¯K mesons , εη1 is the η1 vector polarization,
+and pη(′) is the momentum of the η(′). Here, α and β are given in terms of the η-η′ mixing angle
+α = cos 2θP + 2
+√
+2 sin 2θP ,
+(31a)
+β = 2
+√
+2 cos 2θP − sin 2θP .
+(31b)
+K1
+¯K
+η
+η′
+η1 (1855)
+FIG. 2. Diagram corresponding to the η1 → ηη′ decay through the K1 ¯K loop.
+Although one might try to rely in a much simpler way to describe the direct coupling of one axial-
+vector and three pseudoscalar fields by means of the Hidden Local Symmetry (HLS) Lagrangian
+
+16
+[51–53], it is worth to notice that nonetheless, the total amplitude for this process given by the
+HLS Lagrangian vanishes, which coincides with Eq.(30) in the chiral limit.
+The decay of η1 state into ηη′ is given by
+Γ2B =
+1
+2J + 1
+1
+8πM2η1
+|Mη1→ ηη′|2 q ,
+(32)
+with the amplitude Mη1→ ηη′ in Eq. (30), while J stands for the η1 spin. Besides that, q reads
+q =
+1
+2Mη1
+λ1/2 �
+M2
+η1, m2
+η′, m2
+η
+�
+,
+(33)
+with Mη1, mη′, and mη the masses for the η1(1855), η′, and η mesons, respectively, where
+λ (x, y, z) = x2 + y2 + z2 − 2xy − 2yz − 2zx is the K¨all´en triangle function.
+Therefore, we
+get the following results for the decay width in this channel
+Γ2B =
+�
+�
+�
+(19 ± 4) MeV (set A) ,
+(7 ± 2) MeV (set B) ,
+(34)
+where the error is from choosing subtraction constant to be in the range α(µ = 1GeV) = −1.35 ±
+0.17, corresponding to the hard cutoff qmax = (0.7±0.1) GeV as discussed at the end of Section II B.
+For set A, our result agrees with that of Ref. [21], where the η1(1855) was assumed to be a K1 ¯K
+molecule and the same θK1 mixing angle was used for accounting for the K1A and K1B mixture
+contributing to the physical K1(1270) and K1(1400) states.
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+FIG. 3. Feynman Diagram associated with the three-body decay of the pole through its main component
+K1 ¯K.
+As for the η1 → ¯KK∗π three-body decay, Fig. 3 shows the Feynman diagrams contributing
+to this process. In particular, the η1(1855) decays through its molecular components, that in our
+approach are the K1(1270) ¯K and K1(1400) ¯K. In this case, the contribution from the K1(1270) ¯K
+component can be ignored for the following reasons: 1) from Table X, we see that the relative
+coupling strength for the K1(1270) ¯K channel is much smaller than that for the K1(1400) ¯K one;
+
+117
+2) the branching ratio B[K1(1270) → K∗π] is only 16%, while 96% of the K1(1400) decays is
+dominated by the K∗π. Therefore, from Fig. 3 the η1(1855) → ¯KK∗π amplitude is written as
+M3B = gK1(1400) ¯
+K
+�
+−gµν + pµpν
+M2
+K1
+�
+1
+p2 − M2
+K1 + i MK1ΓK1
+gK∗π εµ
+η1εν
+K∗ ,
+(35)
+where gK1(1400) ¯
+K is the coupling of the pole associated with the η1 state to the K1(1400) ¯K channel,
+gK∗π is the K1(1400)K∗π coupling extracted from the K1(1400) → K∗π reaction in the Review of
+Particle Physics (RPP) [10], and εµ
+η1 and εν
+K∗ are the polarization vectors of the η1 and K∗ mesons,
+respectively.
+The differential decay width for the η1 → ¯KK∗π process is given by
+dΓ
+dMK1 ¯
+K
+=
+1
+(2π)3
+pK ˜pπ
+4M2η1
+|M3B|2
+1
+2J + 1 ,
+(36)
+where
+˜pπ =
+1
+2MK1
+λ1/2 �
+M2
+K1, m2
+K∗, m2
+π
+�
+,
+(37)
+and
+pK =
+1
+2Mη1
+λ1/2 �
+M2
+η1, m2
+K, M2
+K1
+�
+,
+(38)
+with MK1, mK∗, mπ being the masses of the K1(1400), K∗ and π mesons.
+From Eq. (36) we obtain the following results for the η1 → ¯KK∗π decay width
+Γ3B =
+�
+81+11
+−24 MeV
+�A ,
+Γ3B =
+�
+74+12
+−24 MeV
+�B ,
+(39)
+where the uncertainties come from the subtraction constant (cutoff) used to regularize the loops
+in Eq. (22) (Eq. (21)). As can be seen from Eq. (39), we obtain similar results whether we use the
+sets A or B. For the sake of comparison to other works, we evaluate the ratio Γ2B/Γ3B, and get
+Γ2B
+Γ3B
+=
+�
+0.23−0.08
++0.16
+�A or
+�
+0.10−0.03
++0.08
+�B ,
+(40)
+which is consistent to the results in Ref. [21], where the η1 is also assumed to be a K1(1400) ¯K
+molecular state. On the other hand, adopting the same multiquark configuration than the present
+work and Ref. [21], the authors of Ref. [22] have found a different result for the ratio, Γ2B/Γ3B ≈
+0.03. Nevertheless, in all the cases the results point out that the ¯KK∗π three-body channel is more
+likely than the ηη′ one.
+
+18
+IV.
+THE π1(1400/1600) DYNAMICAL GENERATION
+The WT amplitudes for the pseudoscalar-axial vector meson interactions with I = 1 are given
+by Eq. (16), with the corresponding Cij coefficients listed in Table VII. In this case, from Eq. (19),
+we get two π1 poles shown in Table XI.
+TABLE XI. Poles and their corresponding couplings to the channels contributing to the PA interaction
+with JP C = 1−+ and I = 1. The errors of the poles are from varying the subtraction constant within
+α(µ = 1 GeV) = −1.35 ± 0.17, and only the central values of the couplings are given.
+Poles (Set A)
+Channels
+1.47 ± 0.01 − i(0.12 ± 0.02)
+b1π
+f1(1285)π
+f1(1420)π
+K1(1270) ¯
+K
+a1η
+K1(1400) ¯
+K
+(− − + + ++)
+gl
+5.22 + i4.40 0.02 − i0.09
+0.03 − i0.05
+1.25 + i1.27
+0.02 − i0.12 1.33 + i1.63
+1.75 ± 0.02 − i(0.02 ± 0.01)
+b1π
+f1(1285)π
+f1(1420)π
+K1(1270) ¯
+K
+a1η
+K1(1400) ¯
+K
+(− − − + ++)
+gl
+0.10 + i0.95
+2.73 − i0.02
+1.89
+5.84 − i1.85 3.49 − i0.03 2.65 − i0.53
+Poles (Set B)
+Channels
+1.47 ± 0.01 − i(0.12 ± 0.02)
+b1π
+f1(1285)π
+f1(1420)π
+K1(1270) ¯
+K
+a1η
+K1(1400) ¯
+K
+(− − + + ++)
+gl
+5.27 + i4.31 0.01 − i0.03
+0.03 − i0.06
+1.97 − i1.81
+0.02 − i0.08 0.91 + i1.07
+1.77 ± 0.01 − i(0.01 ± 0.01)
+b1π
+f1(1285)π
+f1(1420)π
+K1(1270) ¯
+K
+a1η
+K1(1400) ¯
+K
+(− − − + ++)
+gl
+0.13 + i1.44
+1.37 − i0.25
+2.86 − i0.50
+4.80 − i2.29 3.53 − i0.64 4.54 − i1.77
+Similar to the previous section, we also provide the couplings of these dynamically generated
+states to the channels listed in Table II. Table XI shows a broad π1 pole at 1.47 GeV, and a width of
+about 0.12 GeV.1 This state is above the b1π and f1(1285)π thresholds. Its large width stems from
+the large coupling to the b1π and the fact that this channel is open for decaying. The f1(1285)π
+channel is also open. However, according to Table VII, the corresponding WT term in Eq. (16) is
+zero for the diagonal f1(1285)π transition. On the other hand, the next π1 pole in Table XI has a
+sizeable dependence on the mixing angles. Using set A, we find that pole at 1.75 GeV. It couples
+most strongly to the K1(1270) ¯K channel, which is closed for decaying. Nonetheless, the state
+can decay into b1π and f1(1285)π, albeit their corresponding couplings are small compared to the
+K1(1270) ¯K one, but still large enough to provide a sizeable width for the pole. In contrast, when
+set B is adopted, the higher π1 pole is now located at 1.77 GeV, above the f1(1420)π threshold,
+1 As discussed in Section III B, the widths of the dynamically generated poles will be significantly increased once
+the width effects of the axial-vector mesons are taken into account; see also the discussions below.
+
+19
+which is now open. One might think that the width should increase since now three channels are
+open for decaying. However, although the coupling to the f1(1420)π has increased in this case, at
+the same time the couplings to the other open channels have decreased. Hence, the overall effect
+leads to a smaller width compared to the previous case.
+π
+f1(1285)
+π1 (1600)
+K1/a1
+¯K/η
+π
+η′
+π1 (1600)
+FIG. 4. a) Diagram corresponding to the π1(1600) → f1(1285)π reaction, and b) the π1(1600) → η′π decay
+also via the AP loop. The filled circles represent the effective couplings of the π1 to the AP meson pairs
+calculated from the residues. The rectangles are the AP → η′π transition amplitudes at tree level.
+The lower pole mass is slightly higher than the mass of the π1(1400) state listed in RPP,
+(1354 ± 25) MeV [10].
+Notice that we use the same subtraction constant for all channels.
+In
+principle, it can take different values and lead to a shift of the poles. In addition, we did not
+include in the loops the b1 width, that is relatively large and whose effects could influence the pole
+position. However, it is expected to affect more the imaginary part of the pole than the real one
+(see Fig. 5(a) below). We can get a rough estimate of this change by adding the b1 width to the
+previous result for Im(z1), with z1 the lower π1 pole, i.e.,
+Γb1 + 2Im(z1) ≈ 0.4 GeV ,
+(41)
+which is close to the π1(1400) width reported in RPP, (330 ± 35) MeV [10]. From these results,
+we are led to claim that the lower π1 pole may explain the π1(1400) resonance; in other words, the
+π1(1400) is suitably described in our approach as a dynamically generated state with the b1π as
+its main component.
+Alternatively, following the prescription used in Section III, we can also study the changes in
+the results caused by the inclusion of the finite widths for the axial-vector mesons by looking at
+the line shape for the relevant T-matrix elements. In Fig. 5(a) we show the line shapes for the
+T-matrix element corresponding to the elastic b1π → b1π transition, which is the one we would
+expect the lower pole in Table XI manifests most due to its large coupling to the b1π channel. It
+becomes clear that the bumps become broader when the widths of axial-vector mesons are taken
+into account. A similar behavior can be seen in Fig. 5(b) for the T-matrix element associated
+
+20
+with the scattering of K1 (1270) ¯K, which is the channel to which the higher π1 pole couples most
+strongly.
+w/o Γ
+w/o Γ
+w/ Γ
+w/ Γ
+1300
+1400
+1500
+1600
+1700
+0.0
+0.5
+1.0
+1.5
+2.0
+s [MeV]
+|T11
+2
+Set A
+Set B
+(a) Modulus square of elastic b1π scattering
+w/o Γ
+w/o Γ
+w/ Γ
+w/Γ
+1400
+1500
+1600
+1700
+1800
+1900
+0
+1
+2
+3
+4
+s [MeV]
+|T44
+2
+Set A
+Set B
+(b) Modulus square of elastic K1 (1270) ¯K
+scattering
+FIG. 5. The dashed and solid lines correspond to zero and full widths of the axial-vector mesons in G.
+The higher π1 pole, denoted now by z2, has a mass consistent with that of the π1(1600), whose
+pole mass has been reported to be
+�
+1623 ± 47+24
+−75
+�
+MeV in Ref. [54] and (1564 ± 24 ± 86) MeV in
+Ref. [55]. It can decay into the η′π and f1(1285)π channels. The corresponding diagrams for both
+amplitudes are illustrated in Fig. 4, from which we have
+Mf1(1285)π = gf1(1285)πεη1 · εf1 ,
+(42)
+and
+Mη′π = gK1 ¯
+KGK1 ¯
+KVK1 ¯
+K,η′π · εη1 + ga1ηGa1ηVa1η,η′π · εη1 ,
+(43)
+with εη1 and εf1 the polarization vectors of the η1 and f1 (1285) mesons. Here gf1(1285)π, gK1 ¯
+K and
+ga1η are the effective coupling of the z2 pole to the corresponding couplings, and GK1 ¯
+K and Ga1η
+are the loops involving the K1 ¯K and a1η mesons, respectively. Notice that the effective couplings
+are computed from the residues of the T matrix elements; thus they contain contributions from all
+coupled channels.
+In order to compare our findings with the experimental information, we evaluate the ratio
+R1 = |Mf1(1285)π|2 q
+|Mη′π|2 ˜q
+,
+(44)
+where q and ˜q are the momentum in the c.m. frame of the f1(1285)π and η′π pairs, respectively.
+Numerically, Eq. (44) gives
+R1 =
+�
+�
+�
+�
+2.4+0.8
+−0.6
+�A ,
+�
+2.1+0.4
+−0.3
+�B .
+(45)
+
+21
+The ratio is slightly bigger for the mixing angles in the set A. Nevertheless, the result in Eq. (45)
+is consistent to the corresponding ratio 3.80 ± 0.78 reported by the E852 Collaboration [56]. This
+good agreement with the experimental data supports the molecular picture for the π1(1600) state.
+V.
+DYNAMICAL GENERATION IN I = 1/2 SECTOR
+In the I = 1/2 sector, the corresponding WT amplitudes are given by Eq. (16) with the Cij
+coefficients given in Tables VIII and IX. For each case, we have found two poles for parameter sets
+A and B, as shown in Table XII and XIII.
+TABLE XII. Poles and their corresponding couplings to the channels contributing to the PA interaction
+with JP = 1−. Here the flavor-neutral axial mesons have JP C = 1++. The errors of the poles are from
+varying the subtraction constant within α(µ = 1 GeV) = −1.35 ± 0.17, and only the central values of the
+couplings are given.
+Poles (Set A)
+Channels
+1.69 ± 0.02
+a1K
+f1(1285)K
+K1(1270)η f1(1420)K
+K1(1400)η
+(+ + + + +)
+gl
+6.89
+0.89
+3.75
+0.54
+2.10
+Poles (Set B)
+Channels
+1.70 ± 0.02
+a1K
+f1(1285)K
+K1(1270)η f1(1420)K
+K1(1400)η
+(+ + + + +)
+gl
+6.58
+0.25
+2.45
+0.27
+3.15
+TABLE XIII. Poles and their corresponding couplings to the channels contributing to the PA interaction
+with JP = 1−. Here the flavor-neutral axial mesons have JP C = 1+−. The errors of the poles are from
+varying the subtraction constant within α(µ = 1 GeV) = −1.35 ± 0.17, and only the central values of the
+couplings are given.
+Poles (Set A)
+Channels
+1.70 ± 0.02
+h1(1170)K
+b1K
+K1(1270)η
+h1(1415)K
+K1(1400)η
+(− + + + +)
+gl
+0.20
+6.46
+2.38 − i0.01
+0.50
+3.21 − i0.02
+Poles (Set B)
+Channels
+1.69 ± 0.02
+h1(1170)K
+b1K
+K1(1270)η
+h1(1415)K
+K1(1400)η
+(− + + + +)
+gl
+0.55 − i0.01 6.78 + i0.02 3.69 − i0.06 0.83 − i0.01 2.17 − i0.04
+Similarly to the previous cases, the poles are located on the same Riemann sheets in both sets
+of mixing angles. The interactions in the a1K and b1K channels are strong to generate a bound
+
+22
+state in each of them. The existence of a lower h1 (1170) K channel below the b1K threshold moves
+the pole in Table XIII to Riemann sheet (− + + + +). It has a nonzero imaginary part of a few
+MeV, which is not shown in the table due to precision.
+As discussed before, the I = 1/2 poles in Tables
+XII and XIII will receive sizeable widths
+once the width effects of the axial-vector mesons are taken into account, and it is expected that
+the widths are of the order of a few hundred MeV, like those of the b1 and a1 mesons. Although
+we neglected the transitions between the A1P and B1P sectors as discussed around Eq. (17) in
+Section II, strange mesons are not C-parity eigenstates and the two dynamically generated I = 1/2
+1− states will inevitably mix. The two mixed states together could correspond to the 1− K∗ (1680)
+structure [10].
+VI.
+CONCLUSIONS
+We have studied the interactions between the pseudoscalar and axial-vector mesons in coupled
+channels with JPC = 1−(+) quantum numbers for the isospin 0, 1, and 1/2 sectors. Using the
+chiral unitary approach, we describe the interaction with the Weinberg-Tomozawa term derived
+from chiral Lagrangians. The transition amplitudes among all the relevant channels are unitarized
+using the Bethe-Salpeter equation from which resonances (bound states) manifest themselves as
+poles on the (un)physical Riemann sheets of the complex energy plane.
+We consider the physical isoscalar axial-vector states as mixtures of the corresponding SU(3)
+singlets and octets. In addition, the K1(1270) and K1(1400) physical states are also mixtures of
+the K1A and K1B mesons, which are the strange partners of the a1 and b1 resonances, respectively.
+We group into two sets, called A and B, the mixing angles accounting for such mechanisms and
+investigate their influence on the pole positions.
+According to our findings, we obtain poles with JP(C) = 1−(+) quantum numbers in the energy
+range from 1.30 to 2.00 GeV, in each isospin sector studied (I = 0, 1, 1/2). The 1−+ quantum
+numbers are exotic in the sense that they cannot be formed from a pair of quark and antiquark.
+In particular, we have found an isoscalar state that may correspond to the η1(1855) state, newly
+observed by the BESIII Collaboration [8]. In addition, we have also found two dynamically gener-
+ated isovector states that we assign to be the π1(1400) and π1(1600) resonances. Hence, within our
+formalism, they are dynamically generated through the pseudoscalar-axial vector meson interac-
+tions, with the η1(1855) state coupling mostly to K1(1400) ¯K channel, while the π1(1400) couples
+strongly to the b1π, and π1(1600) structure couples most strongly to the K1(1270) ¯K. We also
+
+23
+find two I = 1/2 JP = 1− states with a mass around 1.7 GeV. They combined together could be
+responsible to the observed K∗(1680) structure.
+In addition, we also evaluate the decays of the η1(1855) and the π1(1600). We find that the
+three-body decay channel ¯KK∗π has a significantly larger branching fraction than the η′η, which
+is the channel where the observation of the η1(1855) was made. The obtained ratio between the
+π1(1600) → f1(1285)π and π1(1600) → η′π decays, given by Eq. (45), is consistent with the
+corresponding experimental value.
+We suggest searching for two additional η1 exotic mesons with masses of about 1.4 and 1.7 GeV,
+respectively. In particular, the latter should be relatively narrow with a width around 0.1 GeV
+and one of its main decay channels is K ¯Kππ.
+ACKNOWLEDGMENTS
+M. J. Y is grateful to Shuang-Shi Fang and M. P. Valderrama for valuable discussions. This
+project is supported in part by the National Natural Science Foundation of China (NSFC) under
+Grants No. 12125507, No. 11835015, and No. 12047503; by the China Postdoctoral Science Foun-
+dation under Grant No. 2022M713229; by the NSFC and the Deutsche Forschungsgemeinschaft
+(DFG) through the funds provided to the Sino-German Collaborative Research Center TRR110
+“Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG
+Project-ID 196253076); and by the Chinese Academy of Sciences under Grant No. XDB34030000.
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+Byte Pair Encoding for Symbolic Music
+Nathan Fradet 1 2 Jean-Pierre Briot 1 Fabien Chhel 3 Amal El Fallah Seghrouchni 1 Nicolas Gutowski 4
+Abstract
+The symbolic music modality is nowadays mostly
+represented as discrete and used with sequential
+models such as Transformers, for deep learning
+tasks. Recent research put efforts on the tokeniza-
+tion, i.e. the conversion of data into sequences
+of integers intelligible to such models. This can
+be achieved by many ways as music can be com-
+posed of simultaneous tracks, of simultaneous
+notes with several attributes. Until now, the pro-
+posed tokenizations are based on small vocabular-
+ies describing the note attributes and time events,
+resulting in fairly long token sequences. In this
+paper, we show how Byte Pair Encoding (BPE)
+can improve the results of deep learning models
+while improving its performances. We experiment
+on music generation and composer classification,
+and study the impact of BPE on how models learn
+the embeddings, and show that it can help to in-
+crease their isotropy, i.e., the uniformity of the
+variance of their positions in the space.
+1. Introduction
+Deep learning tasks on symbolic music are nowadays mostly
+tackled by sequential models1, such as the Transformers
+(Vaswani et al., 2017). These models receive sequences of
+tokens as input, and convert them to learned embedding
+vectors. A token is an integer associated to a high level
+element, such as a word or sub-word in natural language,
+and both are linked in a vocabulary that acts as a look-up
+table. An embedding represents the semantic information of
+a token as a vector of fixed-size, and is learning contextually
+by the model. To use such models for symbolic music, one
+needs to tokenize the data, i.e., convert it to sequences of
+tokens that can be decoded back. This can be achieved by
+several ways, as music can be composed of simultaneous
+tracks, of simultaneous notes with several attributes such as
+1LIP6, Sorbonne University - CNRS, Paris, France 2Aubay,
+Boulogne-Billancourt, France 3 ESEO-TECH / ERIS, Angers,
+France 4University of Angers, Angers, France. Correspondence to:
+Nathan Fradet <[email protected]>.
+1Commonly referred as Language Models (LM)
+their pitch and duration.
+Recently, the token representation of symbolic music has
+been extensively studied, with the goal to improve 1) the
+results, e.g. the quality of generated results or the accuracy
+of a certain Music Information Retrieval (MIR) task, and;
+2) the efficiency of the models. The former is tackled with
+more expressive representations (Huang & Yang, 2020; Ker-
+marec et al., 2022), and the latter by representations based
+on either token combinations (Payne, 2019; Donahue et al.,
+2019), or embedding pooling (Hsiao et al., 2021; Zeng et al.,
+2021; Ren et al., 2020), which reduce the overall sequence
+length. Still, current tokenizations only use tokens represent-
+ing the values of time and note attributes, such as Pitch or
+Duration. This comes with a big limitation: these tokens do
+not carry much information by themselves, and neither their
+associated embeddings. By analogy to natural language,
+these tokens are closer to the characters than words. Yet, the
+expressive information carried by music is deduced by the
+combinations of its notes and their attributes. Considering
+the infinite possible arrangements, deep learning models
+may struggle to implicitly learn their common features.
+In this paper, we study the application of Byte Pair En-
+coding (BPE, described in Section 3) for symbolic music
+generation, aiming to improve the two objectives mentioned
+above, while making the models learn more isotropic em-
+bedding representations in some cases. To the best of our
+knowledge, BPE has yet not been studied for the symbolic
+music modality, although it can be applied on top of any
+music tokenization that do not perform embedding pooling.
+This work aims at closing this gap by shedding light on the
+results and performance gains of using BPE:
+• We experiment on two public datasets (Wang et al.,
+2020b; Kong et al., 2021), with two base tokenizations,
+on which BPE is learned with several vocabulary sizes,
+on the generation and composer classification tasks,
+and show that it improves the results;
+• We compare BPE with other sequence reduction tech-
+niques introduced in recent research;
+• We study the geometry of the learned embeddings, and
+show that BPE can improve their isotropy;
+• We show some limits of BPE, such as on the proportion
+arXiv:2301.11975v1  [cs.LG]  27 Jan 2023
+
+Byte Pair Encoding for Symbolic Music
+of sampled tokens, and that the vocabulary size has to
+be carefully chosen.
+The source code is provided for reproducibility: https:
+//github.com/Natooz/BPE-Symbolic-Music
+The paper is organised as follows: Section 2 reviews the
+related work while Section 3 sheds light on the BPE tech-
+nique. Section 4 describes our experimental settings and
+Section 5 describes the evaluation metrics that we use for
+the experimental evaluation. Section 6 presents the results
+and analysis. Furthermore, Section 7 provides an additional
+study on the impact of BPE on how the models learn the
+embeddings. Finally, Section 8 presents our conclusion and
+perspectives.
+2. Related work
+In this section we start by reminding research of specific
+music representation of symbolic music generation. Then,
+we present how recent works put efforts on different strate-
+gies to reduce the sequence length. Finally, we explain their
+limitations which conduce us to propose our novel approach
+that is to apply Byte Pair Encoding in the field of symbolic
+music for reducing sequence length.
+2.1. Representation of symbolic music
+Most works on symbolic music generation from deep learn-
+ing use a specific music representation. Early research in-
+troduced representations specifically tied to the training
+data being used, such as DeepBach (Hadjeres et al., 2017),
+FolkRNN (Sturm et al., 2015) or BachBot (Liang et al.,
+2017). Non-sequential models such as MuseGAN (Dong
+et al., 2018) often represent music as pianoroll matrices.
+Since, more universal representations have been studied, al-
+lowing to convert any sequence of (simultaneous) notes into
+tokens (Oore et al., 2018; Huang & Yang, 2020; Hadjeres
+& Crestel, 2021; Fradet et al., 2021). Some of them are
+depicted in Figure 1.
+2.2. Sequence reduction strategies
+In more recent works, efforts have been put towards the
+efficiency. Indeed, most recent models are based on the
+Transformer architecture (Vaswani et al., 2017). The atten-
+tion mechanism, at the heart of Transformers, has however
+a time and space complexity that grows quadratically with
+the input sequence length. This is a well known bottleneck,
+that led researchers to work on more efficient attention esti-
+mations (Tay et al., 2021), down to linear complexity. In the
+field of symbolic music specifically, researchers worked on
+strategies to reduce the sequence length in order to increase
+1) the efficiency of the models; 2) the scope of the attention
+mechanism; 3) the quality of the generated results. These
+Bar Pos. 0
+Pitch D3 
+Vel. 22Dur. 7Pos. 7
+Pitch A3 
+Vel. 24Dur. 7Pos. 15
+Pitch E4
+Vel. 24Dur. 7Pos. 27
+Pitch G3
+Vel. 16Dur. 3 Bar Pos. 0
+Pitch A3
+Vel. 20Dur. 31
+Ti.-Sh. 0
+Pitch D3 
+Vel. 22Dur. 7
+Pitch A3 
+Vel. 24Dur. 7
+Pitch E4
+Vel. 24Dur. 7
+Pitch G3
+Vel. 16Dur. 3
+Pitch A3
+Vel. 20Dur. 31
+Ti.-Sh. 8
+Ti.-Sh. 8
+Ti.-Sh. 12
+Ti.-Sh. 4
+N.-On D3 
+Vel. 22
+N.-On A3 
+Vel. 24
+N.-Off A3
+N.-On E4
+Vel. 24
+N.-Off E4 N.-On G3
+Vel. 16
+N.-On A3
+Vel. 20
+Ti.-Sh. 7
+Ti.-Sh. 7
+Ti.-Sh. 7
+N.-Off D3
+Ti.-Sh. 3
+Ti.-Sh. 3
+N.-Off G3
+Ti.-Sh. 31
+N.-Off A3
+Music score
+MIDI-Like
+REMI
+Structured
+Figure 1. A sheet music and several token representations.
+strategies can be split in two categories: 1) embedding pool-
+ing strategies such as Compound Word (Hsiao et al., 2021)
+(CPWord), Octuple (Zeng et al., 2021) or PopMag (Ren
+et al., 2020); 2) token combination strategies such as in
+MuseNet (Payne, 2019) or LakhNES (Donahue et al., 2019).
+Embedding pooling consists in merging the embeddings of
+several distinct tokens with a pooling operation. This is
+often done by concatenating the embeddings and projecting
+the sequence, resulting in an aggregated embedding of fixed
+size. Token combinations is simply the use of a vocabu-
+lary containing tokens that represent several values, e.g.,
+Pitch-x Duration-y that represent both the pitch and
+velocity information.
+2.3. Limitations
+However, these strategies show the following limitations.
+Embedding pooling: 1) requires a more complex training
+procedure; 2) for generation, inferring from such model
+can be seen as sampling from a multivariate distribution,
+which can be a delicate operation; 3) the results can easily
+degenerate if the pooling does not yield semantically rich
+embeddings that represent the underlying tokens. On the
+other hand, token combinations of entire types of tokens can
+lead to large vocabularies with unused tokens and potentially
+non-optimized or unbalanced token distributions.
+To the best of our knowledge, no work has been conducted
+on applying BPE, introduced in Section 3, to symbolic mu-
+sic generation. A similar technique is used with Sympho-
+nyNet (Liu et al., 2022), which does not rely on token adja-
+cency but rather on the concurrence of multiple notes, and
+they only experimented with a vocabulary size of 1k tokens.
+The following section describes the Byte Pair Encoding
+technique, its algorithm and depicts how it can be relevant
+to use in the field of symbolic music.
+
+1
++Byte Pair Encoding for Symbolic Music
+3. Byte Pair Encoding
+Byte Pair Encoding (BPE) (Gage, 1994) is a data com-
+pression technique. It converts the most recurrent succes-
+sive bytes (or in our case tokens) in a corpus into newly
+created ones.
+For instance, in the character sequence
+aabaabaacaa, the sub-sequence aa occurs three times
+and is the most recurrent. Learning and applying BPE on
+this sequence would replace aa with a new symbol, e.g., d,
+resulting in a reduced sequence dbdbdcd. The latter can
+be reduced again by replacing the db subsequence, giving
+eedcd. In the context of deep learning, BPE naturally in-
+creases the size of the vocabulary, while reducing the overall
+sequence lengths. In practice BPE is learned on a corpus
+until the vocabulary reaches a target size. BPE learning is
+described by the pseudo-code of Algorithm 1.
+Algorithm 1 Learning of BPE pseudo-code
+Require: Base vocabulary V, target vocabulary size N,
+dataset X
+1: while |V|< N do
+2:
+Find s = {t1, t2} ∈ V2, from X, the most recurrent
+token succession
+3:
+Add a new token t in V, mapping to s
+4:
+Substitute every occurrence of s in X with t
+5: end while
+6: return V
+BPE is nowadays largely used in the NLP field as it allows
+to encode rare words and segmenting unknown or com-
+posed words as sequences of sub-word units (Sennrich et al.,
+2016).
+In symbolic music, notes are represented by successions
+of tokens that represent the values of their attributes. In
+this context, BPE can allow to represent a note, or even a
+succession of notes, that is very recurrent in the dataset, as
+a single token. For instance, a note that would be coded
+as the succession of tokens Pitch D3, Velocity 60,
+Duration 2.0 could be replaced by a single new one.
+Rare note (and attributes) would still be encoded as non-
+BPE tokens. The same logic applies to time tokens, that can
+also be associated to note tokens.
+4. Experimental settings
+This section details the experimental protocol by describing
+the models, the training and the datasets used along with the
+specific tokenization processes.
+4.1. Model and training
+As we specifically focus on sequential models, we exper-
+iment with the state of the art deep learning architecture
+for most NLP tasks at the time of writing, the Transformer
+(Vaswani et al., 2017) architecture. The generator uses a
+causal attention mask and is trained with teacher forcing,
+while the classifier does not use attention mask and is first
+pre-trained to retrieve randomized tokens then finetuned to
+classify the input sequences. They are respectively similar
+to GPT2 (Radford et al., 2019) and BERT (Devlin et al.,
+2019). The details of implementation, such as their sizes
+and training, can be found in Appendix A
+All models receive sequences between 384 and 460 tokens,
+beginning with special BOS (Beginning of Sequence) and
+ending EOS (End of Sequence) tokens. We split datasets
+in two subsets: one only used for training and updating
+the models, one for validation to monitor trainings, that is
+also used to test the models after training. These subsets
+represent respectively 65% and 35% of the original datasets.
+4.2. Datasets
+We experiment with two datasets: POP909 (Wang et al.,
+2020b) and GiantMIDI (Kong et al., 2021).
+The POP909 dataset (Wang et al., 2020b) is composed of
+909 piano tracks of Pop musics, with aligned MIDI and
+audio versions. Each MIDI file contains three tracks: the
+first is the lead melody, the second is secondary melodies
+and bridges, the third is the arrangements with chords and
+arpeggios. For our experiments we merge all three tracks
+into a single one.
+The GiantMIDI dataset (Kong et al., 2021) is composed
+of 10k piano MIDI files, transcribed from audio to MIDI
+without downbeat and tempo estimation. Each file contains
+a single track of non-interrupted piano music, often with
+complex melodies and harmonies. Considering the com-
+plexity of its content, we make the assumption that it is a
+difficult dataset for a model to learn from.
+We perform data augmentation on the pitch dimension on
+both datasets. Each MIDI file is augmented up and down to
+two octaves.
+4.3. Music tokenization
+We experiment with Remi (Huang & Yang, 2020) and
+TSD (for Time Shift Duration) as base tokenizations, on
+which BPE will be applied on top. Both tokenizations de-
+scribe notes as a succession of the Pitch, Velocity and
+Duration tokens. Remi represents time with Bar and
+Position tokens, which respectively indicates when a
+new bar is beginning and at which position within the time
+is. TSD represents time with TimeShift tokens, indicat-
+ing explicitly time movements.
+When tokenizing symbolic music, it is common to down-
+sample continuous features to discrete sets of values. For
+instance, velocities can be downsampled from 128 to 32
+
+Byte Pair Encoding for Symbolic Music
+values. These sets should be sufficiently precise so that
+the global information remains coherent (Huang & Yang,
+2020; Oore et al., 2018; Hadjeres & Crestel, 2021). Down-
+sampling features helps models to learn more easily, as it
+allows to reduce the perplexity of the predictions, especially
+for values which are less commons in the training set. The
+details of our downsamplings can be found in Appendix B.
+BPE is learned from tokenized corpuses, up to a maximum
+of 1500 randomly picked files, to reduce the learning time.
+We choose to experiment with six vocabulary sizes. One
+without BPE, and five where the original vocabulary size is
+multiplied by 4, 10, 20, 50 and 100.
+To extend our analysis, we also experiment with a version
+of TSD and Remi where Pitch and Velocity tokens
+are merged (PVm), and one where Pitch, Velocity and
+Duration are merged (PVDm). PVm is similar to the
+strategy used with MuseNet (Payne, 2019). We finally ex-
+periment with the CPWord (Hsiao et al., 2021) and Octuple
+(Zeng et al., 2021) embedding pooling strategies, that we
+group with Remi in our experiments as they represent time
+similarly. We use the same pooling strategy, and sample
+independently from the logits of each output modules. For
+implementation simplicity reasons, all embeddings have the
+same size than the model dimension.
+5. Evaluation metrics
+Generative models are often evaluated with automatic met-
+rics on the generated results. Image and audio models are
+assessed with the Fr´echet Inception Distance (FID) (Heusel
+et al., 2017) and Fr´echet Audio Distance (FAD) (Kilgour
+et al., 2019), both comparing the distribution of original data
+and generated results. Language models are often assessed
+with BLEU (Papineni et al., 2002), ROUGE (Lin, 2004) or
+other metrics that compare generated results with reference
+sentences.
+Automatic evaluation of symbolic music remains however
+an open issue. It exists no reference-free metric measuring
+its quality or fidelity. Metrics with reference such as BLEU
+may be suited for machine translation tasks, but remains
+irrelevant for open-ended generation, such as in our case.
+We then perform both human and automatic evaluations, as
+commonly done for symbolic music (Huang & Yang, 2020;
+Huang et al., 2018; Hsiao et al., 2021). Our automatic met-
+rics aim to measure the errors of prediction of the models,
+and the similarity of some features.
+5.1. Tokenization syntax error
+Every tokenization has an underlying syntax of token type
+and value successions, that can normally be made. For
+instance, if the last token of an input sequence is of type
+Pitch, a tokenization could require that the next token to
+predict must be of type Velocity. We could also expect
+a model to not predict more than once the same note at a
+same moment, or to not go back in time.
+Successions of incorrect token types can be interpreted as
+errors of prediction. These errors can help us to measure
+if a model has efficiently learned the music representation
+and if it can yield coherent results. With this motivation,
+we introduce a new metric we called Tokenization Syntax
+Errors (TSE).
+Velocity
+Pitch
+Duration
+Position
+Bar
+(a) REMI.
+Velocity
+Note-On
+Note-Off
+Time-Shift
+(b) MIDI-Like
+Figure 2. Directed graphs of the token types succession (without
+additional tokens) for a) REMI (Huang & Yang, 2020) and b)
+MIDI-Like (Oore et al., 2018).
+We distinguish five categories of errors:
+• TSEtype: the predicted token does not have a type
+that should follow the previous one. For any tokeniza-
+tion, we can draw a directed graph representing the
+possible token types successions, such as in Figure 2.
+• TSEtime: when using Position tokens, the pre-
+dicted Position value is inferior or equal to the
+current one, making the time goes backward.
+• TSEdupn (duplicated note): when the model predicts
+a note that has already been played at the current mo-
+ment (by the same instrument).
+• TSEnnof (no NoteOff): when using NoteOn and
+NoteOff, and that a NoteOn token has been pre-
+dicted with no NoteOff later to end it, or too distant
+in time.
+• TSEnnon (no NoteOn): when a NoteOff token is
+predicted but the corresponding note has not been
+played.
+For a given sequence of tokens, TSE measures the ratio,
+scaled between 0 and 1, of errors for these five categories.
+A TSE of 0 means that there is no error in the sequence,
+while a ratio of 1 means only errors were predicted. Our
+experiments are not concerned by the last two categories as
+we do not use NoteOff tokens.
+Finally, we should mention that most of these errors can
+be avoided by a ruled-based sampling. When predicting a
+token, one can easily keep track of the time, notes played
+and token types to automatically exclude invalid predictions.
+
+Byte Pair Encoding for Symbolic Music
+In practice, this can be achieved by setting the invalid indices
+of the predicted logits to −∞ before applying softmax.
+5.2. Feature similarity
+We expect models to generate continuations that keep the
+features of the input prompt consistent. For instance, it
+should predict first notes within the same scale and with
+the same velocity range. We measure this similarity by
+calculating the overlapping area of distributions of features,
+for the prompt and the first 16 generated beats.
+Previous works (Yang & Lerch, 2020; Choi et al., 2020;
+Mittal et al., 2021; von R¨utte et al., 2022) use the proba-
+bility density function of the distributions, estimated with
+kernel density estimations, and emphasizes that it smooths
+and transforms the distributions into more general repre-
+sentations. While this method can be suited for continuous
+modalities, it can lead to inaccuracies with categorical ones.
+Here, pitch and duration features can be considered as dis-
+crete. Their distributions are both sparse, containing for
+instance many white keys and fewer black keys, yet adja-
+cent and corresponding to close integer values in the MIDI
+format. In order to be more accurate, we measure this simi-
+larity with the histogram intersection of these features, as
+described in Equation (1).
+Similarity (D1, D2) = HI (Hist (D1) , Hist (D2))
+HI(x, y) = �
+i min(xi, yi), xi ≥ 0, yi ≥ 0
+(1)
+Hist : R|D| �→ Ne returns the normalized histogram of
+a distribution of a feature with e elements, HI stands for
+Histogram Intersection.
+5.3. Human evaluations
+For each experiment, we select 40 prompts of 8 beats. For
+each prompts, we generate continuations of 1k tokens with
+the benchmarked models. Three musicians open the con-
+tinuations as a MIDI file, allowing them to listen the tracks
+and also visualize them as piano rolls. Among the tracks,
+they are asked to select the one: 1) with the highest fidelity
+on pitch scale, velocity, note density and rhythm, with the
+prompt; 2) they subjectively prefer overall, considering its
+correctness, structure and richness.
+6. Results and analysis
+We focus on how BPE is learned on the corpuses, then on its
+benefits for music generation and composer classification.
+6.1. BPE learning
+Figure 3 shows the distribution of token types combina-
+tions of the learned BPE tokens.
+We observe that the
+majority of the combinations learned on the Remi tok-
+enization represent notes, by their Pitch, Velocity and
+4
+10
+20
+50
+100
+BPE Factor
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Proportion
+Pch-Vel-Dur
+Pch-Vel-Dur-TimeShift
+Vel-Dur-TimeShift
+Vel-Dur
+Pch-Vel-Dur-Pch-Vel-Dur
+TimeShift-Pch
+Other
+(a) TSD
+4
+10
+20
+50
+100
+BPE Factor
+0.0
+0.2
+0.4
+0.6
+0.8
+Proportion
+Pch-Vel-Dur
+Pch-Vel-Dur-Pos
+Vel-Dur
+Pos-Pch-Vel-Dur
+Pch-Vel-Dur-Pch-Vel-Dur
+Pos-Pch
+Other
+(b) Remi
+Figure 3. Normalized distributions of the token types of the BPE
+tokens, per BPE factor for the POP909 dataset.
+0
+2k
+4k
+6k
+8k
+10k
+12k
+14k
+Vocabulary size
+2.0
+2.5
+3.0
+3.5
+4.0
+Avg. token combinations
+POP909 TSD
+POP909 REMI
+GiantMIDI TSD
+GiantMIDI REMI
+0
+2k
+4k
+6k
+8k
+10k
+12k
+14k
+Vocabulary size
+5
+10
+15
+20
+25
+30
+35
+Max. token combinations
+Figure 4. Average (left) and maximum (right) number of token
+combinations represented by BPE tokens in function of the vocab-
+ulary size.
+Duration attributes. For TSD, the combinations also in-
+clude TimeShift tokens early in the learning. This dif-
+ference mostly comes from common TimeShift tokens
+following notes, whereas for Remi the notes are distributed
+at different Position(s). As the vocabulary grows, the
+combinations tend to be more diverse. The distribution for
+the GiantMIDI dataset are showned in Appendix C.
+Figure 4 shows the evolution of the average number of non-
+BPE token combinations represented by the BPE tokens.
+At the beginning of the learning, the mean number of com-
+binations grows more quickly as the most recurrent token
+successions are often made of more than two tokens. The
+POP909 dataset being smaller than GiantMIDI, it naturally
+leads to a higher maximum number of combinations as the
+latter is more diverse. When the vocabulary begins to con-
+
+Byte Pair Encoding for Symbolic Music
+Table 1. Metrics of generated results. TSE numbers are all scaled at e-3 for better readability. Sim stands for similarity, the best results are
+the closest to the datasets. Hum. Fidelity and Overall are the human evaluations.
+Data / Strategy
+TSEtype (↓)
+TSEdupn (↓)
+TSEtime (↓)
+Sim. pit.
+Sim. vel.
+Sim. dur.
+Hum. Fidelity (↑)
+Hum. Overall (↑)
+POP909 TSD
+0.66 ± 0.13
+0.84 ± 0.12
+0.69 ± 0.14
+No BPE
+1.0 ± 1.8
+13.6 ± 8.0
+-
+0.59 ± 0.08
+0.82 ± 0.10
+0.64 ± 0.09
+0.00
+0.00
+BPE×4
+0.2 ± 0.9
+21.9 ± 19.9
+-
+0.65 ± 0.07
+0.82 ± 0.10
+0.74 ± 0.08
+0.24
+0.19
+BPE×10
+0.5 ± 2.2
+13.4 ± 14.6
+-
+0.64 ± 0.07
+0.78 ± 0.12
+0.74 ± 0.07
+0.53
+0.42
+BPE×20
+0.8 ± 2.1
+12.8 ± 11.0
+-
+0.62 ± 0.07
+0.79 ± 0.11
+0.70 ± 0.09
+0.20
+0.31
+BPE×50
+22.4 ± 24.0
+4.4 ± 5.3
+-
+0.56 ± 0.07
+0.70 ± 0.12
+0.62 ± 0.11
+0.02
+0.02
+BPE×100
+21.5 ± 40.2
+35.6 ± 56.0
+-
+0.54 ± 0.08
+0.66 ± 0.14
+0.63 ± 0.10
+0.00
+0.00
+PVm
+6.1 ± 6.6
+6.9 ± 9.3
+-
+0.59 ± 0.08
+0.78 ± 0.12
+0.73 ± 0.08
+0.01
+0.06
+PVDm
+23.6 ± 19.3
+0.2 ± 0.7
+-
+0.43 ± 0.09
+0.57 ± 0.19
+0.54 ± 0.12
+0.00
+0.00
+POP909 REMI
+0.66 ± 0.13
+0.84 ± 0.12
+0.69 ± 0.14
+No BPE
+0.0 ± 0.1
+115.4 ± 33.8
+74.7 ± 26.7
+0.61 ± 0.08
+0.85 ± 0.09
+0.72 ± 0.07
+0.02
+0.03
+BPE×4
+0.1 ± 0.4
+65.7 ± 21.7
+154.9 ± 27.6
+0.55 ± 0.09
+0.77 ± 0.12
+0.70 ± 0.09
+0.27
+0.34
+BPE×10
+0.3 ± 1.1
+52.3 ± 18.3
+167.1 ± 30.5
+0.49 ± 0.08
+0.77 ± 0.10
+0.63 ± 0.09
+0.52
+0.44
+BPE×20
+0.8 ± 2.2
+81.8 ± 37.3
+242.6 ± 46.5
+0.46 ± 0.08
+0.71 ± 0.13
+0.61 ± 0.10
+0.12
+0.12
+BPE×50
+37.8 ± 35.5
+128.2 ± 22.2
+324.1 ± 21.5
+0.30 ± 0.12
+0.56 ± 0.20
+0.55 ± 0.12
+0.00
+0.00
+BPE×100
+83.9 ± 78.0
+136.3 ± 32.4
+324.6 ± 28.8
+0.28 ± 0.11
+0.54 ± 0.22
+0.55 ± 0.12
+0.00
+0.00
+PVm
+2.3 ± 7.1
+160.0 ± 75.3
+102.7 ± 48.2
+0.60 ± 0.08
+0.77 ± 0.12
+0.69 ± 0.09
+0.05
+0.04
+PVDm
+49.3 ± 46.2
+99.8 ± 25.1
+301.9 ± 26.5
+0.32 ± 0.13
+0.50 ± 0.24
+0.45 ± 0.12
+0.02
+0.02
+CPWord
+331.9 ± 33.8
+144.5 ± 46.8
+99.3 ± 16.6
+0.57 ± 0.08
+0.85 ± 0.07
+0.73 ± 0.09
+0.00
+0.00
+Octuple
+-
+789.3 ± 111.1
+891.9 ± 76.1
+0.05 ± 0.15
+0.07 ± 0.21
+0.06 ± 0.17
+0.00
+0.00
+GiantMIDI TSD
+0.49 ± 0.17
+0.74 ± 0.18
+0.52 ± 0.23
+No BPE
+0.2 ± 1.1
+3.9 ± 4.6
+-
+0.50 ± 0.10
+0.77 ± 0.12
+0.63 ± 0.13
+0.24
+0.19
+BPE×4
+0.5 ± 1.4
+15.2 ± 18.1
+-
+0.51 ± 0.10
+0.75 ± 0.13
+0.62 ± 0.14
+0.33
+0.27
+BPE×10
+1.5 ± 3.3
+35.2 ± 45.6
+-
+0.51 ± 0.11
+0.68 ± 0.17
+0.65 ± 0.13
+0.29
+0.37
+BPE×20
+0.0 ± 0.0
+17.5 ± 29.3
+-
+0.52 ± 0.09
+0.73 ± 0.15
+0.65 ± 0.12
+0.11
+0.08
+BPE×50
+0.0 ± 0.3
+6.8 ± 8.5
+-
+0.50 ± 0.09
+0.70 ± 0.13
+0.64 ± 0.11
+0.00
+0.00
+BPE×100
+1.5 ± 3.7
+1.1 ± 1.5
+-
+0.46 ± 0.09
+0.63 ± 0.17
+0.53 ± 0.13
+0.01
+0.01
+PVm
+3.0 ± 3.7
+0.7 ± 1.3
+-
+0.46 ± 0.11
+0.69 ± 0.15
+0.67 ± 0.11
+0.02
+0.09
+PVDm
+35.6 ± 56.1
+0.5 ± 1.2
+-
+0.39 ± 0.13
+0.61 ± 0.18
+0.25 ± 0.18
+0.00
+0.00
+GiantMIDI REMI
+0.49 ± 0.17
+0.74 ± 0.18
+0.52 ± 0.23
+No BPE
+0.2 ± 0.9
+57.8 ± 40.2
+95.1 ± 42.8
+0.53 ± 0.10
+0.75 ± 0.14
+0.63 ± 0.13
+0.00
+0.01
+BPE×4
+0.2 ± 0.8
+44.3 ± 23.5
+82.3 ± 36.4
+0.46 ± 0.11
+0.71 ± 0.15
+0.62 ± 0.12
+0.41
+0.43
+BPE×10
+2.5 ± 3.5
+31.7 ± 20.2
+175.6 ± 60.3
+0.43 ± 0.10
+0.63 ± 0.21
+0.54 ± 0.15
+0.53
+0.52
+BPE×20
+0.7 ± 2.4
+36.6 ± 29.3
+221.9 ± 66.4
+0.33 ± 0.12
+0.65 ± 0.16
+0.46 ± 0.15
+0.02
+0.01
+BPE×50
+34.8 ± 11.1
+80.5 ± 53.1
+316.4 ± 54.1
+0.36 ± 0.11
+0.58 ± 0.18
+0.30 ± 0.23
+0.00
+0.00
+BPE×100
+476.1 ± 148.3
+159.8 ± 60.1
+285.3 ± 31.5
+0.19 ± 0.10
+0.59 ± 0.20
+0.20 ± 0.19
+0.00
+0.00
+PVm
+0.7 ± 2.4
+53.8 ± 47.4
+181.5 ± 56.9
+0.46 ± 0.11
+0.70 ± 0.15
+0.60 ± 0.14
+0.00
+0.01
+PVDm
+31.9 ± 63.9
+65.6 ± 28.8
+285.6 ± 32.6
+0.33 ± 0.14
+0.58 ± 0.19
+0.29 ± 0.17
+0.02
+0.02
+CPWord
+408.9 ± 28.3
+160.1 ± 54.4
+69.3 ± 16.7
+0.51 ± 0.11
+0.81 ± 0.09
+0.69 ± 0.12
+0.00
+0.00
+Octuple
+-
+763.8 ± 134.4
+894.3 ± 62.1
+0.03 ± 0.11
+0.06 ± 0.19
+0.04 ± 0.15
+0.00
+0.00
+tain BPE tokens with a large number of combinations, it
+starts to specialize on very specific note successions that
+may appear in few data samples. In particular, big jumps
+of maximum number of combinations, e.g. from 14 to 27
+for POP909 Remi, indicate that two already big BPE tokens
+represent the most recurrent succession. These numbers,
+correlated with the model, dataset sizes and overall token
+distribution of the dataset, might help to choose an optimal
+vocabulary size.
+Further analysis in Appendix C shows that BPE consider-
+ably reduces the sequence length, and so the training and
+generation time, at the cost of an increased tokenization
+time. Tokenization of data is however often performed once,
+and the training time gain is very likely to be larger than the
+tokenization time loss.
+6.2. Generated results
+For the generation task, we generate continuations of input
+prompt from the validation subset. The continuations are
+autoregressively generated with 1024 steps, with nucleus
+sampling (Holtzman et al., 2020), with p = 0.9.
+The results of all metrics are reported in Table 1. For TSD,
+BPE allows to reduce both the token type and note dupli-
+cation errors in most cases, while the time errors slightly
+increase for Remi baselines. These results show that models
+can easily scale to bigger vocabularies, up to a certain limit.
+Here, starting from a BPE factor of 50, the TSE seems to
+increase, as do the other results. BPE tends to however
+produce results with features slightly less similar, especially
+with big vocabulary sizes.
+We gathered a total of 400 human evaluations. They show
+that BPE with factors of 4 and 10 significantly outperform
+other baselines, in all experiments. BPE helps models to
+
+Byte Pair Encoding for Symbolic Music
+Table 2. Number of tokens sampled and not sampled by generative
+models, respectively right and left separated by |.
+Strategy
+POP909 TSD
+POP909 Remi
+GiantMIDI TSD
+GiantMIDI Remi
+No BPE
+116 | 23 (16%)
+141 | 11 (7%)
+136 | 3 (2%)
+151 | 1 (0%)
+BPE×4
+454 | 102 (18%)
+487 | 121 (19%)
+456 | 100 (17%)
+386 | 222 (36%)
+BPE×10
+479 | 911 (65%)
+514 | 1006 (66%)
+456 | 934 (67%)
+618 | 902 (59%)
+BPE×20
+592 | 2188 (78%)
+552 | 2488 (81%)
+478 | 2302 (82%)
+504 | 2536 (83%)
+BPE×50
+521 | 6429 (92%)
+249 | 7351 (96%)
+401 | 6549 (94%)
+155 | 7445 (97%)
+BPE×100
+521 | 13379 (96%)
+244 | 14956 (98%)
+281 | 13619 (97%)
+89 | 15111 (99%)
+PVm
+321 | 426 (57%)
+338 | 422 (55%)
+342 | 405 (54%)
+369 | 391 (51%)
+PVDm
+391 | 13712 (97%)
+144 | 13972 (98%)
+252 | 13851 (98%)
+166 | 13950 (98%)
+Table 3. Average accuracy of classification models.
+Strategy
+TSD (↑)
+Remi (↑)
+TSD Large (↑)
+Remi Large (↑)
+No BPE
+0.196 ± 0.031
+0.169 ± 0.021
+0.208 ± 0.033
+0.175 ± 0.022
+BPE×4
+0.218 ± 0.033
+0.168 ± 0.021
+0.226 ± 0.034
+0.171 ± 0.022
+BPE×10
+0.226 ± 0.038
+0.190 ± 0.030
+0.228 ± 0.037
+0.201 ± 0.034
+BPE×20
+0.236 ± 0.038
+0.195 ± 0.026
+0.240 ± 0.039
+0.210 ± 0.029
+BPE×50
+0.199 ± 0.027
+0.207 ± 0.032
+0.247 ± 0.041
+0.216 ± 0.035
+BPE×100
+0.122 ± 0.009
+0.119 ± 0.008
+0.243 ± 0.037
+0.126 ± 0.010
+PVm
+0.199 ± 0.027
+0.150 ± 0.016
+0.213 ± 0.029
+0.188 ± 0.025
+PVDm
+0.226 ± 0.035
+0.192 ± 0.028
+0.228 ± 0.036
+0.194 ± 0.029
+CPWord
+-
+0.204 ± 0.28
+-
+0.214 ± 0.024
+Octuple
+-
+0.274 ± 0.041
+-
+0.283 ± 0.043
+generate more correct and pleasant music. We make the
+assumption that having a larger set of learned embeddings
+help the model to capture more easily the global melody, har-
+mony and music structure, and in turn improve the generated
+results. These embedding, when well learned contextually,
+may represent richer and more explicit information.
+Table 2 shows that while models give high probabilities to
+more unique tokens with BPE in absolute number, the pro-
+portion of sampled tokens decreases. Models tend to focus
+on the sets of more recurrent tokens and omitting more rare
+ones. Beyond a BPE factor of 20 (or vocabulary size be-
+tween 2k and 2.5k tokens), the models are even focusing on
+a more restricting sets of tokens. These numbers highlight
+the limitations of using a too large vocabulary size, as the
+extra effort is unlikely to result in better results.
+6.3. Composer classification
+Composer classification is performed with the top-10 most
+present composers of the GiantMIDI dataset. The results,
+reported in Table 3, show that BPE outperforms other base-
+lines. Here, the model seems to benefit from larger vocabu-
+lary sizes. We also remark that the model size plays in its
+capacity to handle large vocabularies. While the results of
+BPE100 for the small model indicate it was unable to learn
+anything, the larger one performed almost as good as the top
+baseline. A second observation is the good performances
+of embedding pooling strategies (CPWord and OCtuple).
+While they performed poorly for generative tasks, they are
+among the best for this classification task. They seem to be
+better for MIR tasks than generation. As stated in Section 1,
+generation implies sampling, and sampling from several
+distributions is delicate, as for training a model with an
+autoregressive objective on several output distributions.
+Table 4. IsoScore results.
+Generator
+POP909 TSD
+POP909 Remi
+GiantMIDI TSD
+GiantMIDI Remi
+No BPE
+0.09
+0.14
+0.08
+0.09
+BPE×4
+0.02
+0.04
+0.02
+0.02
+BPE×10
+0.12
+0.11
+0.02
+0.07
+BPE×20
+0.13
+0.05
+0.02
+0.02
+BPE×50
+0.02
+0.01
+0.01
+0.01
+BPE×100
+0.01
+0.01
+0.01
+0.00
+PVm
+0.02
+0.02
+0.01
+0.02
+PVDm
+0.00
+0.00
+0.00
+0.00
+CPWord
+-
+0.04
+-
+0.08
+Octuple
+-
+0.04
+-
+0.02
+Classifier
+TSD (↑)
+Remi (↑)
+TSD Large (↑)
+Remi Large (↑)
+No BPE
+0.74
+0.71
+0.80
+0.77
+BPE×4
+0.35
+0.33
+0.54
+0.37
+BPE×10
+0.36
+0.31
+0.48
+0.50
+BPE×20
+0.54
+0.57
+0.64
+0.53
+BPE×50
+0.77
+0.80
+0.75
+0.82
+BPE×100
+0.82
+0.90
+0.87
+0.89
+PVm
+0.27
+0.27
+0.32
+0.32
+PVDm
+0.69
+0.88
+0.88
+0.88
+CPWord
+-
+0.08
+-
+0.05
+Octuple
+-
+0.08
+-
+0.06
+7. Learned embedding spaces
+Results presented in this section rely on Table 4, and Fig-
+ures 5 and 6. Isotropy is a measure of the uniformity of the
+space occupied by a distribution, across all dimensions. In
+our case, the distribution is a manifold X ∈ RN×d where
+N = |V | and d is the model/embedding dimension. It
+has been associated with improved performances with lan-
+guage models (Bi´s et al., 2021; Liang et al., 2021), mostly
+because embeddings are more discriminative and enable
+models to capture and distinguish more easily subtle seman-
+tic information. It has been observed that representations
+from Transformers often exhibit anisotropy, i.e., they tend
+to occupy only a small subspace of the embedding space,
+and often not uniformly (Gao et al., 2019; Ethayarajh, 2019;
+Wang et al., 2020a; Gong et al., 2018; Reif et al., 2019),
+especially causal generative models (Ethayarajh, 2019).
+Isotropy is often estimated by different ways: singular value
+decomposition (Bi´s et al., 2021; Gao et al., 2019; Liang
+et al., 2021; Wang et al., 2020a), intrinsic dimension (Cai
+et al., 2021), partition function (Arora et al., 2016; Mu &
+Viswanath, 2018), average cosine similarity (Ethayarajh,
+2019). Although these methods are correlated with isotropy,
+recent research shed light on some of their limits (Rudman
+et al., 2022). We choose to estimate it with intrinsic value,
+IsoScore (Rudman et al., 2022), singular value and cosine
+similarity, to have results that corroborate and complement
+themselves. The results of the two latter can be found in
+Appendix D. For tokenizations with embedding pooling, we
+used 50k randomly sampled embeddings of combinations
+of tokens representing notes, as using all the embedding
+combinations would be intractable and would not reflect the
+ones actually learned by the models. Results for tokeniza-
+tions where N ≲ d (no BPE) have to be interpreted loosely.
+Isotropy cannot be reliably measured with less samples than
+the number of dimensions they occupy. The estimations are
+more accurate when N ≫ d, as more samples populate all
+
+Byte Pair Encoding for Symbolic Music
+Gen. POP909 TSD
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+0
+20
+40
+60
+80
+Dimension
+lPCA
+MLE
+MOM
+TLE
+TwoNN
+FisherS
+Gen. POP909 Remi
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+CPWord
+Octuple
+0
+20
+40
+60
+lPCA
+MLE
+MOM
+TLE
+TwoNN
+FisherS
+Gen. GiantMIDI TSD
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+0
+5
+10
+15
+20
+25
+lPCA
+MLE
+MOM
+TLE
+TwoNN
+FisherS
+Gen. GiantMIDI Remi
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+CPWord
+Octuple
+0
+20
+40
+60
+80
+lPCA
+MLE
+MOM
+TLE
+TwoNN
+FisherS
+Clasmall TSD
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+0
+20
+40
+60
+80
+Dimension
+200
+300
+400
+500
+600
+700
+lPCA
+MLE
+MOM
+TLE
+TwoNN
+FisherS
+Clasmall Remi
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+CPWord
+Octuple
+0
+20
+40
+60
+80
+100
+200
+300
+400
+500
+600
+700
+800
+lPCA
+MLE
+MOM
+TLE
+TwoNN
+FisherS
+Clalarge TSD
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+0
+20
+40
+60
+80
+100
+200
+400
+600
+800
+1000
+lPCA
+MLE
+MOM
+TLE
+TwoNN
+FisherS
+Clalarge Remi
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+CPWord
+Octuple
+0
+20
+40
+60
+80
+100
+200
+400
+600
+800
+1000
+lPCA
+MLE
+MOM
+TLE
+TwoNN
+FisherS
+Figure 5. Intrinsic dimension estimations. A second x axis has been added on the right for lPCA on classifier plots for better readability.
+Gen. POP909 no BPE
+Gen. POP909 BPE×20
+Clasmall GiantMIDI no BPE
+Clasmall GiantMIDI BPE×20
+Figure 6. 3d UMAP representations of learning embedding spaces,
+with TSD tokenization. Abbreviations in legend stand for: Pi:
+Pitch; V: Velocity; D: Duration; Po: Position: TS: TimeShift.
+dimensions of Rd.
+The IsoScore results (See Table 4), show that BPE does
+not increase the score for generative models. It seems that
+big vocabularies with BPE yield lower IsoScore results,
+that corroborate with the intrinsic dimension results (Fig-
+ure 5). Causal generative models have been shown to learn
+anisotropic embedding representations (Cai et al., 2021;
+Ethayarajh, 2019). Embeddings form cones and clusters,
+that can be observed in Figure 6. As we estimated isotropy
+on all embeddings altogether, the presence of clusters nat-
+urally correlate with anisotropy, as the variance is mostly
+pronounced on their distances. The cluster themselves might
+be more isotropic (Cai et al., 2021).
+On the other hand, BPE can help bi-directional models
+to learn more isotropic embedding representations. The
+IsoScore grows with the vocabulary size, as do the intrinsic
+dimension. In Figure 6 we observe that the embeddings
+have no preferred direction in space, forming a sphere (See
+more figures in Appendix D).
+8. Conclusion
+We showed that BPE can increase the quality of results
+for symbolic music generation, and composer classifica-
+tion, while improving the performances, with a well chosen
+vocabulary size. BPE can be applied on top of any tokeniza-
+tion, and we advice the reader to do so for projects involving
+symbolic music. The drawbacks are a time-consuming vo-
+cabulary learning, and a slower tokenization of data. BPE
+can also helps models to learn more isotropic embedding
+representations. Future work will explore more in depth
+the isotropy of clusters of embeddings of generative models.
+We also plan to experiment with larger model, dataset and
+vocabulary sizes, hoping to find guidelines for choosing an
+optimum vocabulary size.
+
+Special
+Pitch
+Velocity
+Duration
+TimeShift
+8
+7
+6
+5
+4
+3
+8
+7
+6
+-3
+5
+-2
+-1
+4
+0Special
+Pitch
+Velocity
+Duration
+Time-Shift
+Pi-V-D
+7
+Pi-V-D-TS
+V-D-TS
+6
+V-D
+TS-Pi
+5
+Other BPE
+4
+5
+4
+3
+5
+6
+7
+2
+8
+9
+10
+1
+11
+12Special
+Pitch
+Velocity
+Duration
+TimeShift
+4.5
+4.0
+3.5
+3.0
+2.5
+2.0
+3.5
+3.0
+2.5
+-4.0
+2.0
+-3.5
+-3.0
+1.5
+-2.5
+-2.0
+1.0
+-1.5Special
+Pitch
+Velocity
+Duration
+Time-shift
+8.0
+Pi-V-D
+7.5
+V-D-Pi
+7.0
+V-D-TS
+6.5
+Pi-V-D-TS
+6.0
+V-D
+5.5
+Other BPE
+5.0
+8.5
+8.0
+7.5
+7.0
+3.0
+3.5
+6.5
+4.0
+6.0
+4.5
+5.0
+5.5
+5.5
+6.0
+5.0Byte Pair Encoding for Symbolic Music
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+Byte Pair Encoding for Symbolic Music
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+
+Byte Pair Encoding for Symbolic Music
+A. Model and training
+Table 5. Model configurations. The number of parameters is based on the baseline with no BPE, and may vary depending on the baseline
+with the size of the first and last layers. Gen stands for generator and Cla for classifier.
+Gen
+Clasmall
+Clalarge
+Dimension
+512
+768
+1024
+Nb attention heads
+8
+12
+16
+Nb layers
+10
+10
+18
+Feedforward size
+2048
+2048
+3078
+Parameters
+32.6M
+58.0M
+193.3M
+The sizes of the models are reported in Table 5. The generator is trained with a teacher forcing objective on 100k steps. The
+classifier pre-trained on 60k steps to retrieve the value of randomized positions. Between 1 to 15% of each input sequences
+is randomized during pre-training. It is then fine-tuned on 100k steps to predict the composer of the input sequence, from
+the first output hidden state, i.e., the BOS position, which is projected through an output classification layer. The input
+embedding and output pre-training module weights are tied to improve the performances (Press & Wolf, 2017).
+The batch size is set to 16 for the generator, and 24 for the classifier. All trainings are done with automatic mixed-precision
+(Micikevicius et al., 2018), the Adam optimizer (Kingma & Ba, 2015) with β1 = 0.9, β2 = 0.999 and ϵ = 10−8, and
+dropout, weight decay and a gradient clip norm of respectively 10−1, 10−2 and 3. We use a one cycle learning rate scheduler:
+the initial learning rate is close to 0 and gradually grows for the 30% first steps to 5e−6, 1e−6 and 5e−7 for the generators,
+classifier pre-training and classifier fine-tuning respectively, then slowly decreases down to 0. We perform 5 validations steps
+every 30 training steps, and compute their average accuracy and loss. The model parameters are saved when the validation
+loss is the lowest ever observed, and after training the last version saved is used for testing. The training is stopped early if
+the validation losses did not decrease for 15k steps and 25k steps for respectively the generator and classifier.
+B. Data downsampling
+0
+1
+2
+3
+4
+5
+6
+7
+duration
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+density
+Dataset
+POP909
+GiantMIDI
+0
+20
+40
+60
+80
+100
+120
+velocity
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+density
+Dataset
+POP909
+GiantMIDI
+Figure 7. Distributions of the note durations and velocities of the POP909 and GiantMIDI datasets. The duration axis is limited to 7 beats.
+Figure 7 shows the distributions of velocity and duration values of the notes from the two datasets we use. As there is a
+larger proportion of low note durations (below two beats), we decided to downsample the Duration and TimeShift
+tokens with different resolutions: those up to one beat are downsampled to 8 samples per beat (spb), those from one to
+two beats to 4 spb, those from two to four beats to 2 spb, and those from four to eight beats to 1 spb. This way, short
+notes are represented more precisely than longer ones, reducing the vocabulary size. For Remi, Position tokens are
+downsampled to 8 spb, resulting in 32 different tokens as we only consider the 4/* time signature. This allows to represent
+the 16th note. We only consider pitches within the recommended range for piano (program 0) specified in the General MIDI
+2 specifications2: 21 to 108. We then deduplicate all duplicated notes. Velocities are downsampled to 8 distinct values. No
+additional token (e.g., Chord, Tempo) is used.
+2Available on the MIDI Manufacturers Association website
+
+Byte Pair Encoding for Symbolic Music
+C. BPE Learning
+Table 6. Vocabulary sizes, mean tokens per beat (tpb), and variation of tpb from without BPE, average tokenizing time and detokenizing
+time. A maximum of 1000k randomly sampled MIDI files were used for each row. Vocabulary sizes for CPWord and Octuple are the
+product of the sizes of their ”sub-vocabularies”, or in other words the number of possible token combinations, and are rounded for better
+readability. Tokenizing and detokenizing times were run on an Intel Xeon Gold 5128 CPU.
+Data
+Vocab. size
+tpb
+tpb variation (%)
+Tok. time (sec)
+Detok. time (sec)
+POP909 TSD
+No BPE
+139
+17.81 ± 4.12
+-
+0.04 ± 0.02
+0.01 ± 0.02
+BPE×4
+556
+9.71 ± 2.12
+-45.50
+0.20 ± 0.05
+0.02 ± 0.02
+BPE×10
+1390
+8.05 ± 1.75
+-54.80
+0.44 ± 0.10
+0.02 ± 0.02
+BPE×20
+2780
+6.95 ± 1.53
+-60.99
+0.77 ± 0.18
+0.02 ± 0.02
+BPE×50
+6950
+5.84 ± 1.28
+-67.20
+1.59 ± 0.37
+0.02 ± 0.02
+BPE×100
+13.9k
+5.33 ± 1.16
+-70.10
+2.72 ± 0.63
+0.02 ± 0.02
+PVm
+747
+12.72 ± 2.92
+-28.59
+0.03 ± 0.01
+0.01 ± 0.01
+PVDm
+14.1k
+7.63 ± 1.73
+-57.17
+0.02 ± 0.01
+0.01 ± 0.01
+POP909 Remi
+No BPE
+152
+18.06 ± 4.12
+-
+0.03 ± 0.02
+0.01 ± 0.01
+BPE×4
+608
+10.55 ± 2.26
+-41.61
+0.21 ± 0.05
+0.02 ± 0.02
+BPE×10
+1520
+8.85 ± 1.90
+-51.00
+0.47 ± 0.11
+0.02 ± 0.02
+BPE×20
+3040
+8.01 ± 1.74
+-55.64
+0.86 ± 0.19
+0.02 ± 0.02
+BPE×50
+7600
+7.32 ± 1.58
+-59.46
+1.97 ± 0.43
+0.02 ± 0.02
+BPE×100
+15.2k
+6.70 ± 1.43
+-62.92
+3.64 ± 0.79
+0.02 ± 0.02
+PVm
+760
+12.97 ± 2.92
+-28.19
+0.02 ± 0.01
+0.01 ± 0.01
+PVDm
+14k
+7.88 ± 1.73
+-56.38
+0.02 ± 0.01
+0.01 ± 0.01
+CPWord
+49k
+7.88 ± 1.73
+-56.38
+0.03 ± 0.01
+0.03 ± 0.02
+Octuple
+161k
+5.09 ± 1.21
+-71.81
+0.02 ± 0.01
+0.02 ± 0.02
+GiantMIDI TSD
+No BPE
+139
+15.64 ± 6.29
+-
+0.08 ± 0.10
+0.03 ± 0.05
+BPE×4
+556
+8.87 ± 3.30
+-43.26
+0.45 ± 0.57
+0.08 ± 0.16
+BPE×10
+1390
+7.88 ± 2.86
+-49.64
+1.04 ± 1.29
+0.07 ± 0.16
+BPE×20
+2780
+7.04 ± 2.40
+-54.98
+1.90 ± 2.34
+0.07 ± 0.15
+BPE×50
+6950
+5.94 ± 2.20
+-62.03
+4.11 ± 5.03
+0.07 ± 0.14
+BPE×100
+13.9k
+5.45 ± 2.04
+-65.15
+7.49 ± 9.16
+0.07 ± 0.14
+PVm
+747
+11.26 ± 4.46
+-28.03
+0.06 ± 0.08
+0.03 ± 0.04
+PVDm
+14.1k
+6.57 ± 2.59
+-57.98
+0.06 ± 0.07
+0.02 ± 0.03
+GiantMIDI Remi
+no BPE
+152
+15.89 ± 6.42
+-
+0.08 ± 0.10
+0.04 ± 0.05
+BPE×4
+608
+9.58 ± 3.39
+-39.70
+0.53 ± 0.67
+0.08 ± 0.18
+BPE×10
+1520
+8.18 ± 2.96
+-48.51
+1.22 ± 1.51
+0.08 ± 0.17
+BPE×20
+3040
+7.22 ± 2.78
+-54.56
+2.18 ± 2.70
+0.08 ± 0.17
+BPE×50
+7600
+6.41 ± 2.42
+-59.67
+4.87 ± 5.98
+0.08 ± 0.16
+BPE×100
+15.2k
+5.96 ± 2.20
+-62.51
+9.08 ± 11.12
+0.07 ± 0.15
+PVm
+760
+11.30 ± 4.66
+-28.90
+0.06 ± 0.08
+0.03 ± 0.04
+PVDm
+14k
+6.94 ± 2.63
+-56.29
+0.05 ± 0.06
+0.02 ± 0.03
+CPWord
+49k
+6.90 ± 2.55
+-56.60
+0.07 ± 0.09
+0.07 ± 0.10
+Octuple
+161k
+4.37 ± 1.88
+-72.51
+0.05 ± 0.06
+0.05 ± 0.07
+4
+10
+20
+50
+100
+BPE Factor
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Proportion
+Vel-Dur-TimeShift
+Pch-Vel-Dur
+Pch-Vel-Dur-TimeShift
+Vel-Dur-Pch
+Vel-Dur
+Pch-Vel-Dur-Pch
+Other
+(a) TSD
+4
+10
+20
+50
+100
+BPE Factor
+0.0
+0.2
+0.4
+0.6
+0.8
+Proportion
+Pch-Vel-Dur
+Pch-Vel-Dur-Pos
+Vel-Dur
+Pch-Vel-Dur-Pch-Vel-Dur
+Pos-Pch-Vel-Dur
+Pos-Pch
+Other
+(b) Remi
+Figure 8. Normalized distributions of token types per BPE factor for the GiantMIDI dataset.
+Table 6 shows the vocabulary sizes, sequence length variation and tokenization times of all baselines. When learning BPE,
+the average number of tokens per beat (tpb) quickly decreases, so the sequence length. As the vocabulary grows, the tpb
+decreases more slowly, as the most recurrent token successions have already be learned and replaced. A lower tpb allows to
+generate faster.
+
+Byte Pair Encoding for Symbolic Music
+The tradeoff of BPE, besides the vocabulary learning time, is the tokenization time, as a MIDI file is first tokenized without
+BPE, then BPE is applied by finding the token subsequences to be replaced by the BPE tokens. The decoding step time, i.e.,
+the time of the conversion of tokens to a MIDI file, is almost not impacted by BPE. The tokenization and detokenization
+times have been gotten with MidiTok (Fradet et al., 2021) which is implemented in Python. The tokenization time could be
+decreased if performed by a faster compiled language such as Rust or C. The Figure 8 complements the Figure 3, with the
+GiantMIDI dataset.
+D. Learned embedding space
+Figure 9 shows the singular values for the generative and classification models. As the different tokenizations features
+vocabularies with very different sizes, the values are normalized for better readability. Note that the NoBPE tokenizations
+feature vocabularies with a size inferior to the embedding dimension. NoBPE adj. corresponds to the NoBPE results adjusted
+to cover the x-axis on the whole embedding size.
+Figure 10 shows the pairwise cosine similarity of the learned embedding vectors, for the TSD and Remi representation on
+the POP909 dataset. The first tokens up to 90 are Pitches, followed by Velocities up to 125, Durations up to 160
+and then Time-Shift or Position. Without BPE, we can clearly distinguish patterns in the cosine similarity matrices.
+These high similarities shows that embeddings are close to each other. With BPE and larger vocabulary sizes, the average
+cosine similarity tend to decrease, especially between BPE tokens. Embeddings are less similar and more discriminative.
+UMAP (McInnes et al., 2018) representations shown in Figure 6, Figure 12 and Figure 11 have been calculated with the
+default parameters of the official Python package. We clearly see that generative models learn clusters of embeddings of the
+same type, distant from each other. The embeddings do not occupy the space uniformly. On the other hand, pre-trained
+bi-directional models learn more isotropic embedding representations. The embeddings are spread uniformly across all
+directions, for all token types.
+
+Byte Pair Encoding for Symbolic Music
+Gen POP909
+100
+101
+102
+Dimension
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Singular value
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+noBPE adj.
+100
+101
+102
+Dimension
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Singular value
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+CPWord
+Octuple
+noBPE adj.
+Gen GiantMIDI
+100
+101
+102
+Dimension
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Singular value
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+noBPE adj.
+100
+101
+102
+Dimension
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Singular value
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+CPWord
+Octuple
+noBPE adj.
+Clasmall
+100
+101
+102
+Dimension
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Singular value
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+noBPE adj.
+100
+101
+102
+Dimension
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Singular value
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+CPWord
+Octuple
+noBPE adj.
+Clalarge
+100
+101
+102
+103
+Dimension
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Singular value
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+noBPE adj.
+TSD
+100
+101
+102
+103
+Dimension
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Singular value
+noBPE
+bpe4
+bpe10
+bpe20
+bpe50
+bpe100
+PVm
+PVDm
+CPWord
+Octuple
+noBPE adj.
+Remi
+Figure 9. Normalized singular values of embedding matrices of classifier models.
+
+Byte Pair Encoding for Symbolic Music
+TSD
+0
+20
+40
+60
+80
+100
+120
+0
+20
+40
+60
+80
+100
+120
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+200
+400
+600
+800
+1000
+1200
+0
+200
+400
+600
+800
+1000
+1200
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+500
+1000
+1500
+2000
+2500
+0
+500
+1000
+1500
+2000
+2500
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Remi
+0
+20
+40
+60
+80
+100
+120
+140
+0
+20
+40
+60
+80
+100
+120
+140
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+No BPE
+0
+100
+200
+300
+400
+500
+600
+0
+100
+200
+300
+400
+500
+600
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+BPE x4
+0
+200
+400
+600
+800
+1000
+1200
+1400
+0
+200
+400
+600
+800
+1000
+1200
+1400
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+BPE x10
+0
+500
+1000
+1500
+2000
+2500
+3000
+0
+500
+1000
+1500
+2000
+2500
+3000
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+BPE x20
+Figure 10. Pairwise cosine similarity matrix of learned embedding of the generative models, on the POP909 dataset.
+No BPE
+BPE x4
+BPE x10
+BPE x20
+BPE x50
+BPE x100
+PVm
+PVDm
+Figure 11. UMAP 2d representations of the embeddings of classifier models pre-trained with the GiantMIDI dataset and TSD tokenization.
+Abbreviations in legend stand for: Pi: Pitch; V: Velocity; D: Duration; Po: Position; TS: TimeShift.
+
+4.0
+Special
+Pitch
+3.5
+Velocity
+Duration
+3.0
+TimeShift
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+6.5
+7.0
+7.5
+8.0
+8.5
+9.0
+9.5
+10.06
+5
+Special
+Pitch
+Velocity
+！
+C
+4
+Duration
+Time-Shift
+V-D-TS
+V-D
+3 
+Pi-V-D
+V-D-Pi
+Other BPE
+5
+6
+7
+8
+9
+10Special
+Pitch
+Velocity
+8
+Duration
+Time-Shift
+V-D-TS
+7
+Pi-V-D
+V-D-Pi
+Pi-V-D-TS
+V-D
+6
+Other BPE
+5
+4 
+3
+4
+5
+6
+7Special
+9
+Pitch
+Velocity
+Duration
+Time-Shift
+8
+Pi-V-D
+V-D-Pi
+V-D-TS
+7
+Pi-V-D-TS
+V-D
+Other BPE
+6
+5
+2
+3
+4
+5
+6Special
+Pitch
+Velocity
+9
+Duration
+Time-Shift
+Pi-V-D-TS
+8
+Pi-V-D
+V-D-Pi
+V-D-TS
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+1284009 0400344Byte Pair Encoding for Symbolic Music
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+Figure 12. UMAP 3d representations of the embeddings of generative models with the POP909 dataset. Abbreviations in legend stand for:
+Pi: Pitch; V: Velocity; D: Duration; Po: Position: TS: TimeShift.
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9NE1T4oBgHgl3EQfnwSt/content/tmp_files/2301.03313v1.pdf.txt

@@ -0,0 +1,1837 @@
+Published as a conference paper at ICLR 2023
+BQ-NCO: BISIMULATION QUOTIENTING FOR GENER-
+ALIZABLE NEURAL COMBINATORIAL OPTIMIZATION
+Darko Drakulic1
+Sofia Michel1
+Florian Mai2,*
+Arnaud Sors1
+Jean-Marc Andreoli1
+1 NAVER Labs Europe {firstname.lastname}@naverlabs.com
+2 Idiap Research Institute and EPFL [email protected]
+* Work was done as part of an internship at NAVER Labs Europe.
+ABSTRACT
+Despite the success of Neural Combinatorial Optimization methods for end-to-
+end heuristic learning, out-of-distribution generalization remains a challenge. In
+this paper, we present a novel formulation of combinatorial optimization (CO)
+problems as Markov Decision Processes (MDPs) that effectively leverages sym-
+metries of the CO problems to improve out-of-distribution robustness. Starting
+from the standard MDP formulation of constructive heuristics, we introduce a
+generic transformation based on bisimulation quotienting (BQ) in MDPs. This
+transformation allows to reduce the state space by accounting for the intrinsic
+symmetries of the CO problem and facilitates the MDP solving. We illustrate our
+approach on the Traveling Salesman, Capacitated Vehicle Routing and Knapsack
+Problems. We present a BQ reformulation of these problems and introduce a sim-
+ple attention-based policy network that we train by imitation of (near) optimal
+solutions for small instances from a single distribution. We obtain new state-of-
+the-art generalization results for instances with up to 1000 nodes from synthetic
+and realistic benchmarks that vary both in size and node distributions.
+1
+INTRODUCTION
+Combinatorial Optimization problems are crucial in many application domains such as transporta-
+tion, energy, logistics, etc. Because they are generally NP-hard (Cook et al., 1997), their resolution
+at real-life scales is mainly done by heuristics, which are efficient algorithms that generally produce
+good quality solutions (Boussa¨ıd et al., 2013). However, strong heuristics are generally problem-
+specific and designed by domain experts. Neural Combinatorial Optimization (NCO) is a relatively
+recent line of research that focuses on using deep neural networks to learn such heuristics from data,
+possibly exploiting information on the specific distribution of problem instances of interest (Bengio
+et al., 2021; Cappart et al., 2021). Despite the impressive progress in this field over the last few
+years, their out-of-distribution generalization, especially to larger instances, remains a major hurdle
+(Joshi et al., 2022; Manchanda et al., 2022).
+In this paper, we are interested in constructive NCO methods, which build a solution incrementally,
+by applying a sequence of elementary steps. These methods are often quite generic, see e.g. the
+seminal papers by Khalil et al. (2017); Kool et al. (2019). Most CO problems can indeed be rep-
+resented in this way, although the representation is not unique as the nature of the steps is, to a
+large extent, a matter of choice. Given a choice of step space, solving the CO problem amounts to
+computing an optimal policy for sequentially selecting the steps in the construction. This task can
+typically be performed in the framework of Markov Decision Processes (MDP), through imitation
+or reinforcement learning. The exponential size of the state space, inherent to the NP-hardness of
+combinatorial problems, usually precludes other methods such as (tabular) dynamic programming.
+Whatever the learning method used to solve the MDP, its efficiency, and in particular its out-of-
+distribution generalization capabilities, greatly depends on the state representation. The state space
+is often characterized by deep symmetries, which, if they are not adequately identified and lever-
+aged, hinders the training process by forcing it to independently learn the policy at states which in
+fact are essentially the same (modulo some symmetry).
+1
+arXiv:2301.03313v1  [cs.LG]  9 Jan 2023
+
+Published as a conference paper at ICLR 2023
+In this work, we investigate a type of symmetries which often occurs in MDP formulations of con-
+structive CO heuristics. We first introduce a generic framework to systematically derive a naive CO
+problem-specific MDP. We formally demonstrate the equivalence between solving the MDP and
+solving the CO problem and highlight the flexibility of the MDP formulation, by defining a mini-
+mal set of conditions for the equivalence to hold. Our framework is general and easy to specialize
+to encompass previously proposed learning-based construction heuristics. We then show that the
+state space of this naive MDP is inefficient because it fails to capture deep symmetries of the CO
+problem, even though such symmetries are easy to identify. Therefore, we propose a method to
+transform the naive MDP, based on the concept of bisimulation quotienting (BQ), in order to get
+a reduced state space, which is easier to process by the usual (approximate) MDP solvers. We il-
+lustrate our approach on three well-known CO problems, the Traveling Salesman Problem (TSP),
+the Capacitated Vehicle Routing Problem (CVRP) and Knapsack Problem (KP). Furthermore, we
+propose a simple transformer-based architecture for these problems, that we train by imitation of
+expert trajectories derived from (near) optimal solutions. In particular, we show that our model is
+well-suited for our BQ formulation: it spends a monotonically increasing amount of computation as
+a function of the subproblem size (and therefore complexity), in contrast to most previous models.
+Finally, extensive experiments confirm the validity of our approach, and in particular its state-of-the-
+art out-of-distribution generalization capacity. In summary, our contributions are as follows: 1) We
+present a generic and flexible framework to define a construction heuristic MDP for arbitrary CO
+problems; 2) We propose a method to simplify commonly used “naive” MDPs for constructive NCO
+via symmetry-focused bisimulation quotienting; 3) We design an adequate transformer-based archi-
+tecture for the new MDP, for the TSP, CVRP and KP; 4) We achieve state-of-the-art generalization
+performance on these three problems.
+2
+COMBINATORIAL OPTIMIZATION AS A MARKOV DECISION PROBLEM
+In this section, we present a generic formalization of constructive heuristics which underlies their
+MDP formulation. A deterministic CO problem is denoted by a pair (F, X), where F is its ob-
+jective function space and X its (discrete) solution space. A problem instance f∈F is a mapping
+f:X→R∪{∞}, with the convention that f(x)=∞ if x is infeasible for instance f. A solver for
+problem (F, X) is a functional:
+SOLVE : F → X
+satisfying
+SOLVE(f) = arg min
+x∈X f(x).
+(1)
+Incremental solution construction
+Constructive heuristics for CO problems build a solution se-
+quentially, starting at an empty partial solution and expanding it at each step until a finalized solution
+is reached. Many NCO approaches are based on a formalization of that process as an MDP, e.g.
+Khalil et al. (2017); Kool et al. (2019); Zhang et al. (2020). Such an MDP can be obtained, for an
+arbitrary CO problem (F, X), using the following ingredients:
+• Steps: T is a set of available steps to construct solutions. A partial solution is a pair (f, t1:n) of
+a problem instance f∈F and a sequence of steps t1:n∈T ∗ (the set of sequences of elements of T ).
+Observe that a partial solution (in F×T ∗) is not a solution (in X), but may represent one.
+• Representation: SOL:F×T ∗→X∪{⊥} is a mapping that takes a partial solution and returns ei-
+ther a feasible solution (in which case the partial solution is said to be finalized), or ⊥ otherwise.
+• Evaluation: VAL:F×T ∗→R∪{∞} is a mapping that takes a partial solution and returns an esti-
+mate of the minimum value of its expansions into finalized solutions. If the returned value is finite,
+the partial solution is said to be admissible.
+In order to define an MDP using these ingredients, we assume they satisfy the following axioms:
+∀f∈F, x∈X,
+f(x) < ∞
+⇔
+∃t1:n ∈ T ∗ such that SOL(f, t1:n) = x,
+(2a)
+∀f∈F, t1:n∈T ∗,
+SOL(f, t1:n) ̸= ⊥
+⇒
+∀m∈{1:n−1}, SOL(f, t1:m) = ⊥,
+(2b)
+∀f∈F, t1:n∈T ∗, x∈X,
+SOL(f, t1:n) = x
+⇒
+�
+VAL(f, t1:n) = f(x),
+∀m ∈ {1:n−1}, VAL(f, t1:m) < ∞.
+(2c)
+Equation 2a states that the feasible solutions are exactly those represented by a finalized partial
+solution; Equation 2b states that if a partial solution is finalized then none of its preceding partial
+solutions in the construction can also be finalized; Equation 2c states that the evaluation of a finalized
+2
+
+Published as a conference paper at ICLR 2023
+partial solution is the value of the solution it represents, and all its preceding partial solutions are
+admissible.
+We call a triplet ⟨T , SOL, VAL⟩ satisfying the above axioms a specification of problem (F, X).
+Note that a specification is not intrinsic to the problem. The step space T results from a choice
+of how to construct a solution sequentially. Once T is chosen, SOL is determined, and must satisfy
+Equation 2a and 2b. Then VAL is only loosely constrained by Equation 2c, and can be chosen among
+a wide range of alternatives, including the following straightforward, uninformed one and the ideal,
+but intractable one (and, more likely, somewhere in between these two extremes):
+VALuninformed(f, t1:n) =def f(x) if [SOL(f, t1:n) = x ̸= ⊥] else 0,
+VALideal(f, t1:n) =def min{f(x)|x ∈ X, ∃u1:m ∈ T ∗ s.t. SOL(f, t1:nu1:m) = x},
+with the convention min ∅=∞. Value 0 in the uninformed case can be replaced by any constant.
+Solution construction as an MDP
+Using a specification ⟨T , SOL, VAL⟩ of problem (F, X), one
+can derive a “naive” MDP as follows. States are partial solutions (in F×T ∗); actions are steps
+(in T ); a state is terminal if it is a finalized partial solution; transitions: action u∈T applied to
+a non-terminal state (f, t1:n) leads to state (f, t1:nu) where u is appended to the sequence so far,
+with reward VAL(f, t1:n)−VAL(f, t1:nu), conditioned on VAL(f, t1:nu) being finite. Note that VAL
+has the double role of providing a reward and specifying the set of allowed actions. The number of
+these is expected to be linear, or at worst polynomial, in the size of the instance, since picking a step
+should not be as complex as solving the whole problem.
+Now, assume we have access to a generic solver SOLVEMDP, which, given an MDP M and one of
+its states so, returns an optimal trajectory starting at that state, i.e. arg maxτ R(τ) where τ ranges
+over the M-trajectories starting at so and ending in a terminal state, and R(τ) denotes its cumulated
+reward. Note that because we are dealing with deterministic MDPs, looking for an optimal policy is
+the same as looking for an optimal trajectory for a given set of initial states. That is why SOLVEMDP is
+defined here directly in terms of trajectories rather than policies. SOLVEMDP can then be specialized
+into a solver for the specific CO problem (F, X):
+Proposition 1. Let Mo be the naive MDP obtained from specification ⟨T , SOL, VAL⟩. The proce-
+dure defined as follows (where ϵ denotes the empty sequence) satisfies the requirement of equation 1:
+SOLVE(f ∈ F) =def {SOL(s)|s is the last state of the trajectory SOLVEMDP(Mo, (f, ϵ))}.
+In other words, solving the naive MDP is equivalent to solving the CO problem. The detailed proof
+of Proposition 1 is in Appendix F. Of course, procedure SOLVEMDP may be approximate, in which
+case so is procedure SOLVE. Moreover, its performance depends on that of SOLVEMDP, esp. its out-
+of-distribution generalization capacity, but also on the choice of specification, esp. of action space.
+It is a distinguishing feature of CO from an MDP perspective that the action space is not prescribed
+by the problem.
+The impact of the choice of the VAL mapping depends on the type of learning used by SOLVEMDP.
+When SOLVEMDP learns by reinforcement, VAL is essential, as it provides the rewards which guide
+the resolution. For example, VALuninformed leads to the notoriously hard case of sparse rewards,
+while VALideal (were it tractable) would lead to the trivial case where a myopic policy (greedy in
+its immediate reward) is optimal. Although we do not provide a generic method to design VAL, we
+argue that there are natural candidates, typically based on extending the objective function to partial
+solutions (not just finalized ones). When SOLVEMDP learns by imitation instead, the choice of VAL
+has a much more limited impact: it only serves to define the allowed actions. The critical factor in
+that case is the construction of the training dataset of expert trajectories to imitate.
+Example on TSP
+Consider the widespread CO problem known as the Traveling Salesman Prob-
+lem (TSP) in a Euclidian space V . A TSP solution (in X) is a path, i.e. a finite sequence of pairwise
+distinct nodes. A TSP instance (in F) is given by a finite set D of nodes as points in V , and maps
+any solution (path) to the length of that path (closed at its ends) if it visits exactly all the nodes of
+D, and ∞ otherwise (infeasible solutions).
+A simple specification ⟨T , SOL, VAL⟩ for the TSP is given by: the step space T is the set of nodes;
+for a given instance f and sequence t1:n of steps, SOL(f, t1:n) is either the sequence t1:n if it forms
+3
+
+Published as a conference paper at ICLR 2023
+u
+a
+v
+u
+a
+v
+u
+a
+v
+step|u−(a+v)
+step|u−(a+v)
+≡Φ
+≡Φ
+step|u−(a+v)
+Figure 1: An example of bisimulation commutation in TSP-MDP, and the corresponding path-TSP-
+MDP transition. The step is the same in all three transitions: it is the end node of the dashed arrow.
+And the reward is also the same: it depends only on the distances a, u, v, and not on any of the
+previously visited nodes.
+a path which visits exactly all the nodes of f, or ⊥ otherwise; and VAL(f, t1:n) is either the length
+of path t1:n (closed at its ends) if it forms a path which visits only nodes of f (maybe not all), or ∞
+otherwise. It is easy to show that we thus obtain a specification (as defined by the axioms above)
+of the TSP. In TSP-MDP, the naive MDP obtained from it, the reward of taking action u (a node)
+at state (f, t1:n) is δf(tn, t1)−(δf(tn, u)+δf(u, t1)) where δf is the node distance measured on the
+corresponding points of V in f, conditioned on t1:nu being pairwise distinct nodes of f. Observe
+that when allowed, the reward depends only on the start and end nodes t1, tn of the step sequence.
+3
+BISIMULATION QUOTIENTING FOR COMBINATORIAL OPTIMIZATION
+In our context of deterministic CO problems and therefore deterministic MDPs, the general notion
+of bisimilarity is simplified (Givan et al., 2003): two states are said to be bisimilar if they spawn
+exactly the same action-reward sequences. Likewise, the notion of a binary relation R on states
+being a bisimulation reduces to a commutation between the (deterministic) transitions of the MDP
+and that relation: if s1Rs2 and action a applied to state s1 leads to state s′
+1 with reward r, then
+action a applied to state s2 leads to a state s′
+2 with the same reward r, and s′
+1Rs′
+2. An illustration is
+given in Fig 1. Bisimilarity is equivalently defined as the largest bisimulation (see Appendix H.1).
+Bisimilarity-induced symmetries
+In the naive MDP obtained from a specification of a given CO
+problem, a state is a partial solution and carries the whole information about the “past” decisions
+(steps) leading to it, which may not all be useful for the “future” decisions, i.e. the completion of that
+partial solution. Consider for example the following two states in TSP-MDP, in which the sequence
+of steps of the partial solution is represented as a directed path in red among some of the problem
+instance nodes:
+s1
+s2
+Observe that s1, s2 differ only in the inner nodes of the red path (black diamond-shaped nodes).
+Now, it is easy to see that the successful completions of these two partial solutions are identical:
+they each consist of a path visiting the (same) unvisited (blue) nodes, starting at the end node of the
+red path and ending at its start node, with the same rewards defined by VAL. Consequently, in the
+MDP, the two states s1, s2 spawn exactly the same action-reward sequences and form a bisimilar
+pair. This is the kind of deep symmetries of the problem which we want the MDP to leverage. Of
+course, there exist other kinds of symmetries, e.g. rotational symmetries: if s2 is obtained from s1
+by applying an isometric transformation to all the points in the problem instance, then s1, s2 also
+form a bisimilar pair. However, the latter symmetry is specific to the Euclidian version of the TSP.
+We focus here on the former kind of symmetry as it is more general. Although it has previously been
+noted for routing problems (Peng et al., 2020; Xin et al., 2021b), we show here that it is an inherent
+characteristic of constructive CO approaches.
+4
+
+Published as a conference paper at ICLR 2023
+Bisimulation quotienting
+A classical result on MDPs (Givan et al., 2003) states that all such
+symmetries in any MDP can be leveraged by quotienting it by its bisimilarity relation, i.e. the set of
+all bisimilar pairs. Of course, there is no free lunch: constructing the bisimilarity of an MDP is in
+general intractable. Still, the result remains valuable because it holds for any bisimulation, not just
+the bisimilarity. Therefore one can control the amount of symmetries captured in the quotienting by
+carefully choosing the bisimulation, trading off its closeness to full bisimilarity for tractability.
+We now assume that, for a given CO problem (F, X) we have access not only to a specification
+⟨T , SOL, VAL⟩ with its associated naive MDP, but also to a mapping Φ:F×T ∗→ ˆS from partial
+solutions to some new space ˆS. Typically, Φ(f, t1:n) should capture, within the partial solution
+(f, t1:n), a piece of information as small as possible but sufficient to determine the set of action-
+reward sequences it spawns in the MDP – in other words, a summary of its “past” which is sufficient
+to determine its “future”. We can then define an equivalence relation ≡Φ where two partial solutions
+are equivalent if they have the same image by Φ. For it to be a bisimulation, Φ must satisfy:
+∀s1, s2∈F×T ∗, Φ(s1)=Φ(s2) ⇒
+�
+∀u∈T , Φ(s1u)=Φ(s2u)
+and VAL(s1)−VAL(s1u)=VAL(s2)−VAL(s2u).
+(3)
+Under that assumption, we can construct a new MDP (the quotient of the original one by the bisim-
+ulation) which is equivalent, as far as policy optimization is concerned, to the original one, but
+captures more symmetries of the problem. This allows to reduce the state space and should lead to
+a better performance, whatever the generic MDP solver used afterwards. Furthermore, by construc-
+tion, the equivalence classes are in one-to-one correspondence with the states in ˆS, so that the new
+MDP can be formulated on that space directly.
+Application to the TSP, CVRP and KP
+Let Φ be the mappings from TSP-MDP states (TSP
+states for short) into new objects called “path-TSP” states, informally described by the following
+diagram:
+TSP state
+path-TSP state
+Φ
+The inner nodes (black diamonds) on the red path of visited nodes in the TSP state are removed,
+leaving only the two ends of the red path which constitute two distinguished nodes in the path-TSP
+state. Mapping Φ has been designed to satisfy equation 3, so it induces a bisimulation on TSP-MDP
+(see Figure 1), and TSP-MDP can be turned into an equivalent “path-TSP-MDP” on path-TSP states.
+This path-TSP-MDP can be viewed as solving a variant of the TSP known as path-TSP, hence its
+name. However it is not the naive MDP for that variant since it forgets as it progresses, while naive
+MDPs always accumulate.
+With the CVRP, we define a step as the pair of a node and a binary flag specifying whether that
+node is reached via the depot or directly. We can define a mapping Φ similarly to the TSP case,
+except it is not sufficient to summarize the “past” (the visited nodes) by just the two ends of their
+path: to guarantee equation 3 and the bisimulation property, an additional piece of information must
+be preserved from the past, namely the remaining capacity at the end of the current path. For the
+KP, intuitively, the summary of the “past” is captured by the remaining items and the remaining
+knapsack capacity. This idea can be leveraged to design a bisimulation. Formal descriptions of the
+specifications and bisimulation quotienting for the CVRP and KP are provided in Appendices A
+and B, respectively.
+4
+NEURAL ARCHITECTURE FOR PATH-TSP
+We now describe our proposed policy network for the path-TSP-MDP above. Figure 4 (Appendix)
+provides a quick overview.
+The models for path-CVRP and BQ-KP differ only slightly and
+are presented in Appendix A and
+B. Most neural models for TSP utilize an encoder-decoder
+architecture, in which the encoder computes a representation of the entire graph once, and the
+decoder constructs a solution by taking into consideration the representation of the whole graph
+5
+
+Published as a conference paper at ICLR 2023
+and the partial solution, e.g. Attention Model (Kool et al., 2019), or PointerNetworks (Vinyals
+et al., 2015). In our case, the path-TSP formulation allows us to forget the nodes in the graph that
+have already been visited, except the distinguished origin and destination nodes. As a corollary,
+it also requires re-encoding the remaining nodes at each prediction step – hence removing the
+need for a separate auto-regressive decoder. To encode a path-TSP state, we use a Transformer
+model (Vaswani et al., 2017). Each node is represented by its (x, y) coordinates, so that the input
+feature matrix for an N-node state is an N×2 matrix. We embed these features via a linear layer.
+The remainder of the encoder is based on Vaswani et al. (2017) with the following differences.
+First, we do not use positional encoding since the input nodes have no order. Instead, we learn an
+origin (resp. destination) embedding that is added to the feature embedding of the origin (resp.
+destination) node. Second, we use ReZero (Bachlechner et al., 2021) normalization, which leads to
+more stable training and better performance in our experiments (see ablation study in Appendix D).
+Finally, a last linear layer projects the encoder’s output into a vector of size N, from which unfea-
+sible actions, corresponding to the origin and destination nodes, are masked out, before applying a
+softmax operator so as to interpret the scalar node values for all allowed nodes as action probabilities.
+Training We train our model by imitation of expert trajectories, using a plain cross-entropy loss
+(behaviour cloning). Such trajectories are extracted from pre-computed optimal (or near optimal)
+solutions for instances of a (relatively small) fixed size. Note that (optimal) solutions are not directly
+in the form of trajectories. Equation 2a guarantees that a trajectory exists for any solution, but it is
+usually far from unique. Besides, optimal solutions are costly, so we seek to make the most out of
+each of them. In the TSP case, we observe that given an optimal tour, any sub-path of that tour is
+also an optimal solution to the associated path-TSP sub-problem, hence amenable to our path-TSP
+model. We therefore form minibatches by first sampling a number n between 4 and N (path-TSP
+problems with less than 4 nodes are trivial), then sampling sub-paths of length n – the same n for
+all the minibatch entries so as to simplify batching – from the initial solution set. For the CVRP, the
+procedure is similar, except that, first, the extracted sub-paths must end at the depot, and, second,
+they can follow the sub-tours of the full solution in any order. We observed experimentally that the
+way that order is sampled has an impact on the performance (see Appendix E).
+Complexity Because of the quadratic complexity of self-attention, and the fact that we call our
+model at each construction step, the total complexity is O(N 3)1 where N is the instance size.
+Note that closely related Transformer-based models such as the TransformerTSP (Bresson &
+Laurent, 2021) and the Attention Model (Kool et al., 2019) have a total complexity of O(N 2)2
+At each decision step, for t remaining nodes, our model has a budget of O(t2) compute whereas
+previous models only spend O(t). We believe that this is a useful inductive bias, which enables
+better generalization in particular for larger problem sizes. This hypothesis is supported by the
+fact that replacing the self-attention component with a linear time alternative (i.e., spending O(t)
+operations per step) drastically degrades the generalization ability to larger instances, as we show in
+Appendix D,
+Summary By reformulating TSP-MDP into path-TSP-MDP, the state is made to contain only a very
+concise summary of the “past” of a partial solution (how it was formed) as two distinguished nodes,
+but sufficient to determine its “future” (how it can be completed). Furthermore, at train time, we
+sample optimal solutions and associated path-TSP states amongst all the possible infixes of solutions
+of full problems. These proposed modifications go hand-in-hand. Thanks to the transformation
+to path-TSP-MDP, our model enables better generalization in two important ways: (i) Due to re-
+encoding at each step, the encoder produces a graph representation that is specific to the current
+path-TSP-MDP state. Graphs in these states vary in size and distribution, implicitly encouraging the
+model to work well across sizes and node distributions, and generalize better than if such variations
+were not seen during the training. In this regard, our model is similar to the SW-AM model (Xin
+et al., 2021b), except that they only approximate the re-embedding process in practice. (ii) By
+sampling subsequences from our training instances, we automatically get an augmented dataset,
+which some previous models had to explicitly design their model for (Kwon et al., 2021).
+1More precisely, the complexity is proportional to �N
+t=1 t2 = N(N + 1)(2N + 1)/6 hence the O(N 3).
+2After an encoder of complexity O(N 2), the decoder has linear complexity O(N − t) at step t.
+6
+
+Published as a conference paper at ICLR 2023
+5
+RELATED WORK
+NCO approaches
+Many NCO approaches construct solutions sequentially, via auto-regressive
+models.
+Starting with the seminal work by Vinyals et al. (2015), which proposed the Pointer
+network that was based on RNNs and trained in a supervised way, Bello et al. (2017) trained the
+same model by RL for the TSP and Nazari et al. (2018) adapted it for the CVRP. Kool et al. (2019)
+introduced an attention-based encoder-decoder architecture (AM) trained with RL to solve several
+variants of routing problems – which is reused by Kwon et al. (2021) along with a few extensions
+(POMO). TransformerTSP Bresson & Laurent (2021) use a similar architecture with a different
+decoder on TSP problems. Another line of works is concerned with directly producing a heat-map
+of solution segments: Nowak et al. (2018) trained a Graph Neural Network in a supervised manner
+to output an adjacency matrix, which is converted into a feasible solution using beam search.
+Joshi et al. (2019) followed a similar framework and trained a deep Graph Convolutional Network
+instead, that was used by (Fu et al., 2020) as well.
+Step-wise methods Peng et al. (2020) first pointed out the limitation of the original AM (Kool et al.,
+2019) approach in representing the dynamic nature of routing problems. They proposed to update
+the encoding after each subtour completion for the CVRP. Xin et al. (2021b) proposed a similar
+step-wise strategy but the encodings recomputed after each decision. In practice, their architecture
+is the most similar to ours for the TSP. However, thanks to our principled MDP transformations
+based on bisimulation quotienting, we obtain a superior representation for CVRP: In contrast to our
+approach, their CVRP architecture only provides censored information by omitting the remaining
+vehicle capacity and simply restricting the state to the nodes whose demand is below the remaining
+capacity. Xin et al. (2020) extended on this idea by proposing the Multi-Decoder Attention Model
+(MDAM) that in particular contains a special layer to efficiently approximate the re-embedding
+process. As MDAM constitutes the most advanced version, we employ it as a baseline in our
+experiments.
+Generalizable NCO Generalization to different instances distributions, and esp. larger instances, is
+regarded as one of the major limitations of current NCO approaches (Joshi et al., 2022; Mazyavkina
+et al., 2020). Fu et al. (2020) trained a Graph Convolution model in a supervised manner on small
+graphs and used it to solve large TSP instances, by applying the model on sampled subgraphs and us-
+ing an expensive MCTS search to improve the final solution (Att-GCN+MCTS). While this method
+achieves excellent generalization on TSP instances, MCTS requires a lot of computing resources
+and is essentially a post-learning search strategy. Geisler et al. (2022) investigate the robustness of
+NCO solvers through adversarial attacks and find that existing neural solvers are highly non-robust
+to out-of-distribution examples. They conclude that one way to address this issue is through adver-
+sarial training. In particular, Xin et al. (2021a) trains a GAN to generate instances that are difficult
+to solve for the current model. Manchanda et al. (2022) take a different approach and leverage meta-
+learning to learn a model in such a way that it is easily adaptable to new distributions. Accounting
+for symmetries in a given CO problem is a powerful idea to boost the generalization performance of
+neural solvers. Both Kwon et al. (2021) and Kim et al. (2022) make use of solution symmetricity as
+part of their loss function during training. Problem instance symmetry can be used at training time
+to augment the dataset (Kwon et al., 2021) or enforce robust representations (Kim et al., 2022), or it
+can be used at inference time to augment the set of solutions (Kwon et al., 2021).
+Please note that all of the above are orthogonal to our approach: rather than augmenting data or
+changing the training paradigm, our approach simplifies the state space by transforming the MDP,
+which has beneficial effects irrespective of the method of training.
+6
+EXPERIMENTS
+To verify the effectiveness of our method, we test it on TSP, CVRP and KP. This section presents
+experimental results for TSP and CVRP, while results for KP are presented in Appendix B.
+We train our model and all baselines on synthetic TSP and CVRP instances of size 100, generated
+as in Kool et al. (2019). We choose graphs of size 100 because it is the largest size for which (near)
+optimal solutions are still reasonably fast to obtain, and such training datasets are commonly used
+in the literature. Then, we evaluate trained models on synthetic instances of size 100, 200, 500 and
+1K generated from the same distribution, as well as the standard TSPLib and CVRPLib datasets.
+7
+
+Published as a conference paper at ICLR 2023
+Hyperparameters and training procedure We use the same hyperparameters for all problems.
+The model has 9 layers, each built with 8 attention heads with embedding size of 128 and dimension
+of feed-forward layer of 512. Our model is trained on 1 million instances with 100 nodes split
+into batches of size 1024, for 1000 epochs. Solutions of these problems are obtained by using the
+Concorde solver (Applegate et al., 2015) for TSP and LKH heuristic (Helsgaun, 2017) for CVRP.
+We use Adam (Kingma & Ba, 2017) as optimizer with an initial learning rate of 7.5e−4 and decay
+of 0.98 every 20 epochs.
+Evaluation We compare our model with existing state-of-the-art methods: OR-Tools (Perron &
+Furnon, 2022), LKH (Helsgaun, 2017), and Hybrid Genetic Search (HGS) for the CVRP (Vidal,
+2022) as traditional non-neural methods; Att-GCN+MCTS and NeuralRewriter (Chen & Tian,
+2019) as hybrid methods for TSP and CVRP respectively; and deep learning-based constructive
+methods: AM, TransformerTSP, MDAM and POMO, which were discussed in Section 5. For all
+deep learning baselines we use the model trained on graphs of size 100 and the best decoding
+strategy. Following the same procedure as in Fu et al. (2020), we generate four test datasets with
+graphs of sizes 100, 200, 500 and 1000. For CVRP, we use capacities of 50, 80, 100 and 250,
+respectively. In addition, we report the results on TSPLib instances with up to 4461 nodes and
+all CVRPLib instances with node coordinates in the Euclidian space. For all models, we report
+the optimality gap and the inference time. The optimality gap for TSP is based on the optimal
+solutions obtained with Concorde. For CVRP, although HGS gives better results than LKH, we use
+the LKH solution as a reference to compute the ”optimality” gap, in order to be consistent (and
+easily comparable) with previous works. While the optimality gap is easy to compute and compare,
+measurements of running times are much harder:
+they may vary due to the implementation
+platforms (Python, C++), hardware (GPU, CPU), parallelization, batch size, etc. Therefore, we also
+report the number of solutions generated by each of the constructive deep learning models. In our
+experiments, we run all deep learning models on a single Nvidia Tesla V100-S GPU with 24GB
+memory, and other solvers on Intel(R) Xeon(R) CPU E5-2670 with 256GB memory, in one thread.
+Results Tables 1a and 1b summarize our results on TSP and CVRP, respectively. For both problems,
+our model shows superior generalization on larger graphs, even with the greedy decoding strategy,
+which generates only a single solution while all others generate several hundreds (and select the
+best among them). In terms of running time with greedy decoding, our model is competitive with
+the POMO baseline, and significantly faster than other models. Beam search decoding with beam
+size 16 further improves the quality of solutions, but as expected, it takes approximately 16 times
+longer. Figure 2 shows optimality gap versus running time for our model and other baseline models.
+Our model clearly outperforms other models in terms of generalization to larger instances. The
+only model that is competitive with ours is Att-GCN+MCTS, but it is 2-15 times slower and is
+designed for TSP only. In addition to synthetic datasets, we test our model on TSPLib and VRPLib
+instances, which are of varying graph sizes, node distributions, demand distributions and vehicle
+capacities. Table 1c shows our model’s strength over MDAM and POMO, even with the greedy
+decoding strategy. The effectiveness of our MDP transformation method and the resulting neural
+architecture is confirmed by the results. Thanks to our more principled approach that leads to better
+state representations and a simpler architecture without a decoder, by generating a single solution,
+it is able to outperform MDAM (with 250 solutions), which is closest to our model conceptually.
+Moreover, an ablation study in Appendix D suggests that spending appropriate amounts of compute
+for each subproblem is a crucial factor in our model.
+7
+CONCLUSION
+We have presented a flexible framework to derive MDPs that sequentially construct solutions to
+CO problems. Starting from a naive MDP, we introduced a generic transformation using bisim-
+ulation quotienting, which reduces the state space by leveraging its symmetries. We applied this
+transformation on the TSP and CVRP, for which we also designed a simple attention-based model,
+well-suited to the transformed state representation. We show experimentally that this combination of
+state representation, simple model, and training procedure yields state-of-the-art generalization re-
+sults on diverse benchmarks. While training on relatively small instances allowed us to use imitation
+learning, our approach and model could be similarly used with reinforcement learning. Finally, we
+have focused on deterministic CO problems, leaving the adaptation of our framework to stochastic
+problems as future work.
+8
+
+Published as a conference paper at ICLR 2023
+Table 1: Summary of the experimental results. The bold values represent the best optimality gap
+(lower is better) and fastest inference time. The underlined cells represent the best ratio between the
+quality of the solution and the inference time. #s refers to number of generated solutions.
+#s
+TSP100
+TSP200
+TSP500
+TSP1000
+Concorde
+-
+0.000%
+38m
+0.000%
+2m
+0.000%
+40m
+0.000%
+2.5h
+OR-Tools
+-
+3.765%
+1.1h
+4.516%
+4m
+4.891%
+31m
+5.021%
+2.4h
+Att-GCN+MCTS∗
+-
+0.037%
+15m
+0.884%
+2m
+2.536%
+6m
+3.223%
+13m
+AM bs1024
+1024
+2.510%
+20m
+6.176%
+1m
+17.978%
+8m
+29.750%
+31m
+TransTSP bs1024
+1024
+0.456%
+51m
+5.121%
+1m
+36.142%
+9m
+76.215%
+37m
+MDAM bs50
+250
+0.395%
+45m
+2.044%
+3m
+9.878%
+13m
+19.965%
+1.1h
+POMO augx8
+8N
+0.134%
+1m
+1.572%
+5s
+20.182%
+1m
+40.603%
+10m
+BQ (ours) greedy
+1
+0.540%
+1m
+0.793%
+5s
+1.425%
+1m
+2.335%
+7m
+BQ (ours) bs16
+16
+0.032%
+18m
+0.166%
+1m
+0.682%
+15m
+1.311%
+1.8h
+(a) Experimental results on TSP. ∗We could not run Att-GCN+MCTS code on our architecture so we report
+results from the original paper.
+#s
+CVRP100
+CVRP200
+CVRP500
+CVRP1000
+LKH
+-
+0.000%
+15.3h
+0.000%
+30m
+0.000%
+1.3h
+0.000%
+2.8h
+HGS
+-
+-0.510%
+15.3h
+-1.024%
+30m
+-1.252%
+1.3h
+-1.104%
+2.8h
+OR-Tools
+-
+9.617%
+15.3h
+10.700%
+30m
+11.403%
+1.3h
+13.559%
+2.8h
+NeuRewriter ∗
+-
+3.456%
+1.1h
+29.460%
+9m
+25.051%
+32m
+29.542%
+1.8h
+AM bs1024
+1024
+4.180%
+24m
+7.786%
+1m
+16.964%
+8m
+86.410%
+31m
+MDAM bs50
+250
+2.206%
+56m
+4.332%
+3m
+9.994%
+14m
+28.015%
+1.4h
+POMO augx8
+8N
+0.689%
+1m
+4.767%
+5s
+20.575%
+1m
+141.058%
+10m
+BQ (ours) greedy
+1
+4.832%
+1m
+3.723%
+5s
+3.429%
+1m
+6.809%
+7m
+BQ (ours) bs16
+16
+1.798%
+18m
+1.375%
+1m
+0.817%
+15m
+2.048%
+1.8h
+(b) Experimental results on CVRP. ∗We could not reproduce the reported results for NeuRewriter, so for
+CVRP100 we report results from the original paper and for other sizes we report the best result we got.
+MDAM
+POMO
+BQ (ours)
+Size
+bs50
+x8
+greedy
+bs16
+<100
+3.06%
+0.42%
+0.38%
+0.06%
+100-200
+5.14%
+2.31%
+2.82%
+1.61%
+200-500
+11.32%
+13.32%
+3.31%
+2.07%
+500-1K
+20.40%
+31.58%
+10.08%
+3.04%
+>1K
+40.81%
+62.61%
+11.87%
+8.61%
+All
+19.01%
+26.30%
+6.22%
+3.94%
+MDAM
+POMO
+BQ (ours)
+Set (size)
+bs50
+augx8
+greedy
+bs16
+A (32-80)
+6.17%
+4.86%
+5.85%
+1.96%
+B (30-77)
+8.77%
+5.13%
+7.04%
+3.50%
+F (44-134)
+16.96%
+15.49%
+7.20%
+3.04%
+M (100-200)
+5.92%
+4.99%
+6.69%
+1.85%
+P (15-100)
+8.44%
+14.69%
+4.71%
+1.32%
+X (100-1K)
+34.17%
+21.62%
+10.74%
+8.35%
+All (15-1K)
+22.36%
+15.58%
+8.58%
+5.60%
+(c) Experimental results on TSPLib (left) and CVRPLib (right).
+10
+−4
+10
+−3
+10
+−2
+10
+−1
+10
+0
+Inference time (per instance, in seconds)
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Optimality gap
+AM bs1024
+MDAM bs50
+POMO augx8
+Att- GCN+MCTS
+BQ (ours) greedy
+BQ (ours) bs16
+TSP100
+TSP200
+TSP500
+TSP1000
+10
+−4
+10
+−3
+10
+−2
+10
+−1
+10
+0
+Inference time (per instance, in seconds)
+0
+20
+40
+60
+80
+100
+120
+140
+Optimality gap
+AM bs1024
+MDAM bs50
+POMO augx8
+NeuralRewriter
+BQ (ours) greedy
+BQ (ours) bs16
+CVRP100
+CVRP200
+CVRP500
+CVRP1000
+Figure 2: Generalization results on different graph sizes for TSP (left) and CVRP (right). Lower
+and further left is better.
+9
+
+Published as a conference paper at ICLR 2023
+REPRODUCIBILITY STATEMENT
+In order to ensure the reproducibility of our approach, we have:
+• described precisely our generic theoretical framework (Section ??) and provided a detailed
+proof of Proposition 1 in Appendix F. This should in particular serve to adapt the frame-
+work to other CO problems;
+• explained in detail our proposed model (Section 4 for TSP and Appendix A for CVRP),
+described precisely the training procedure and listed the hyperparameters (Section 6);
+• used public datasets referenced in Section 6.
+Furthermore, we plan to make our code public upon acceptance.
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+Wouter Kool, Herke van Hoof, and Max Welling. Attention, Learn to Solve Routing Problems! In
+International Conference on Learning Representations, 2019.
+Yeong-Dae Kwon, Jinho Choo, Byoungjip Kim, Iljoo Yoon, Youngjune Gwon, and Seung-
+jai Min.
+POMO: Policy Optimization with Multiple Optima for Reinforcement Learning.
+arXiv:2010.16011 [cs], July 2021.
+Florian Mai, Arnaud Pannatier, Fabio Fehr, Haolin Chen, Francois Marelli, Francois Fleuret, and
+James Henderson. HyperMixer: An MLP-based Green AI Alternative to Transformers, March
+2022.
+Sahil Manchanda, Sofia Michel, Darko Drakulic, and Jean-Marc Andreoli. On the Generalization
+of Neural Combinatorial Optimization Heuristics. In European Conference on Machine Learning
+and Principles and Practice of Knowledge Discovery in Databases (ECML-PKDD), Grenoble,
+France, September 2022.
+Nina Mazyavkina, Sergey Sviridov, Sergei Ivanov, and Evgeny Burnaev. Reinforcement Learning
+for Combinatorial Optimization: A Survey. arXiv:2003.03600 [cs, math, stat], March 2020.
+MohammadReza Nazari, Afshin Oroojlooy, Lawrence Snyder, and Martin Takac. Reinforcement
+Learning for Solving the Vehicle Routing Problem. In Advances in Neural Information Processing
+Systems, volume 31. Curran Associates, Inc., 2018.
+Alex Nowak, David Folqu´e, and Joan Bruna. Divide and Conquer Networks. In 6-Th International
+Conference on Learning Representations, Vancouver, Canada, April 2018.
+Bo Peng, Jiahai Wang, and Zizhen Zhang. A Deep Reinforcement Learning Algorithm Using Dy-
+namic Attention Model for Vehicle Routing Problems. arXiv:2002.03282 [cs, stat], February
+2020.
+Laurent Perron and Vincent Furnon. OR-tools. Google, 2022.
+Yi Tay, Mostafa Dehghani, Dara Bahri, and Donald Metzler. Efficient Transformers: A Survey.
+ACM Computing Surveys, April 2022. ISSN 0360-0300. doi: 10.1145/3530811.
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez,
+Lukasz Kaiser, and Illia Polosukhin. Attention Is All You Need. arXiv:1706.03762 [cs], June
+2017.
+11
+
+Published as a conference paper at ICLR 2023
+Thibaut Vidal.
+Hybrid genetic search for the CVRP: Open-source implementation and SWAP*
+neighborhood. Computers & Operations Research, 140:105643, April 2022. ISSN 0305-0548.
+doi: 10.1016/j.cor.2021.105643.
+Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly.
+Pointer Networks.
+In C. Cortes, N. D.
+Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett (eds.), Advances in Neural Information Pro-
+cessing Systems 28, pp. 2692–2700. Curran Associates, Inc., 2015.
+Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Multi-Decoder Attention Model with Embed-
+ding Glimpse for Solving Vehicle Routing Problems, December 2020.
+Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Generative Adversarial Training for Neural
+Combinatorial Optimization Models, September 2021a.
+Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Step-Wise Deep Learning Models for Solving
+Routing Problems. IEEE Transactions on Industrial Informatics, 17(7):4861–4871, July 2021b.
+ISSN 1941-0050. doi: 10.1109/TII.2020.3031409.
+Cong Zhang, Wen Song, Zhiguang Cao, Jie Zhang, Puay Siew Tan, and Chi Xu.
+Learning to
+Dispatch for Job Shop Scheduling via Deep Reinforcement Learning. arXiv:2010.12367 [cs,
+stat], October 2020.
+12
+
+Published as a conference paper at ICLR 2023
+A
+APPLICATION TO THE CVRP
+Problem definition and specification
+The Capacitated Vehicle Routing Problem (CVRP) is a
+vehicle routing problem in which a vehicle (here, a single one) with limited capacity must deliver
+items from a depot location to various customer locations. Each customer has an associated demand,
+which represents an amount of items, and the problem is for the vehicle to provide all the customers
+in the least travel distance, returning as many times as needed to the depot to refill, but without ever
+exceeding the vehicle capacity.
+Formally, we assume given a set of customer nodes, each with a demand (positive scalar), plus a
+depot node. A CVRP solution (in X) is a finite sequence of nodes starting at the depot, which are
+pairwise distinct except for the depot, and respecting the capacity constraint: the total demand of
+any contiguous sub-sequence of customer nodes is below the vehicle capacity. A CVRP instance (in
+F) is given by a finite set D of nodes, including the depot, their coordinates in the Euclidian space
+V , and maps any solution to the length of the corresponding path using the distances in V , if the
+path visits exactly all the nodes of D, or ∞ otherwise (unfeasible solutions).
+A possible specification ⟨T , SOL, VAL⟩ for the CVRP is defined as follows. The step space T is the
+set of pairs of a non depot node and a binary flag indicating whether that node is to be reached via
+the depot or directly. The extension ¯t of a step t is either the singleton of its node component if its
+flag is 0 or the pair of the depot node and its node component if its flag is 1. For a given problem
+instance f and sequence t1:n of steps, SOL(f, t1:n) is either the sequence ¯t1:n if it forms a d-path
+which visits exactly all the nodes of f , or ⊥ otherwise. VAL(f, t1:n) is either the total length of ¯t1:n
+(closed at its end) if it forms a d-path which visits only nodes of f (maybe not all), or ∞ otherwise.
+It is easy to show that ⟨T , SOL, VAL⟩ forms a specification for the CVRP (i.e. satisfies the axioms of
+specifications introduced in Section ??). The naive MDP obtained from it is denoted CVRP-MDP.
+Bisimulation quotienting
+Just as with TSP, we can define a mapping Φ from CVRP-MDP states
+to a new “path-CVRP” state space, informally described by the following diagram.
+C=10
+1
+1
+4
+3
+2
+4
+1
+1
+4
+3
+CVRP state
+C=3
+1
+4
+3
+path-CVRP state
+Φ
+Here, the capacity of the vehicle is C=10, shown next to the (colourless) depot node, and the demand
+of each node is shown next to it, in orange. The black dotted line indicates that the action which
+introduced the node with demand 2 was via the depot: its flag was set to 1 (all the other actions had
+their flag set to 0 in this simple example). The green dotted line indicates how the path is closed
+to measure its length. After the node with demand 2, the path of visited nodes (in red) continues
+with nodes with demand 4 and 1, respectively, so that the remaining capacity at the end of the path
+is C−(2+4+1)=3. Compared to TSP, this is the additional piece of information in the summary
+of the “past” (path of visited nodes) which is preserved in the path-CVRP state, together with the
+origin and destination of the path. Mapping Φ thus defined satisfies Equation 3, hence induces a
+bisimulation on CVRP-MDP states, and by quotienting, one obtains an MDP which can be defined
+directly on path-CVRP states.
+Model architecture for CVRP
+The model architecture for CVRP is almost the same as for TSP,
+with a slight difference in the input sequence and in the output layer. In the TSP model, the input to
+the node embedding layer for a N-node state is a 2×N matrix of coordinates. For CVRP, we use two
+additional channels: one for node demands, and one for the current vehicle capacity, repeated across
+all nodes. The demand is set to zero for the origin and destination nodes. We obtain a 4×N matrix of
+features, which is passed through a learned embedding layer. As for TSP, a learned origin-signalling
+(resp. destination-signalling) vector is added to the corresponding embeddings. The rest of the
+architecture, in the form of attention layers, is identical to TSP, until after the action scores projection
+layer. In the case of TSP, the projection layer returns a vector of N scores, where each score, after
+13
+
+Published as a conference paper at ICLR 2023
+a softmax, represents the probability of choosing the node as the next step in the construction. In
+the case of CVRP, the model returns a matrix of scores of dimension N×2, corresponding to each
+possible actions (node-flag pair) and the softmax scopes over this whole matrix. As usual, a mask is
+always applied to unfeasible actions before the softmax operator: those which have higher demand
+than the remaining vehicle capacity, as well as the origin and destination nodes.
+B
+APPLICATION TO THE KNAPSACK PROBLEM
+Problem definition and specification
+The Knapsack Problem (KP) is classical combinatorial op-
+timization problem in which we need to pack items, with given values and weights, into a knapsack
+with a given capacity. The objective is to maximize the total value of packed items. Formally, we
+assume given a set of items, each with a value and weight. A KP solution (in X) is a subset of the
+items which respects a capacity constraint (“c-subset”): total weight of the items of the subset must
+not exceed the knapsack capacity. A KP instance (in F) is given by a finite set of D items and maps
+any c-subset to the sum of values of its items.
+A simple problem specification ⟨T , SOL, VAL⟩ can be defined as follows. The step space T is equal
+to the set of items, . For a partial solution (f, t1:n), if the selected items satisfy the capacity con-
+straints and adding any of the remaining items results in an infeasible solution, then SOL(f, t1:n)
+returns the subset of selected items; otherwise it returns ⊥. Finally, VAL(f, t1:n) is either the sum
+of the values of the items in t1:n if they satisfy the capacity constraint and ∞ otherwise. Similarly
+to the TSP and CVRP cases, it is easy to show that ⟨T , SOL, VAL⟩ forms a specification for the KP.
+The naive MDP obtained from it is denoted MDP-KP.
+Bisimulation quotienting
+As it was the case for TSP and CVRP, we can define a mapping Φ
+from KP-MDP state to a new “BQ-KP” state space, informally described by the following diagram.
+3
+7
+9
+1
+1
+2
+4
+5
+8
+8
+6
+weights
+values
+1
+9
+2
+8
+3
+7
+1
+6
+7
+3
+9
+C = 20
+3
+9
+1
+4
+5
+8
+8
+6
+1
+2
+3
+1
+6
+7
+3
+9
+C = 10
+KP-state
+BQ-KP-state
+Φ
+Here, capacity of the knapsack is C = 20 and each item is defined by its weight (bottom cell) and
+value (top cell). Mapping Φ for KP is straightforward - simply saying, it removes all picked items
+and update the remaining capacity by subtracting total weight of removed items from the previous
+capacity.
+Model architecture for KP
+The model architecture for KP is again very similar to previously
+described models for TSP and CVRP. The input to the model is a 3 × N tensor composed of items
+properties (values, weights) and the additional channel for the remaining knapsack’s capacity. By
+definition, the solution has no order (the result is a set of items), so there is no need to add tokens for
+origin and destination. A part from excluding these tokens and different input dimensions, the rest of
+the model is identical to the TSP model. The output is a vector of N probabilities over all items with
+a mask over the unfeasible ones (with weights larger than remaining knapsack’s capacity). In the
+training, at each construction step, any item of the ground-truth solution is a valid choice. Therefore
+we use a multi-class cross-entropy loss.
+Experimental results for KP
+We generate the training dataset as described in Kwon et al. (2021).
+We train our model on 1M KP instances of size 200 and capacity 25, with values and weights ran-
+domly sampled from the unit interval. We use the dynamic programming algorithm from ORTools
+to compute the ground-truth optinal solutions. As hyperparameters, we use the same as for the TSP.
+Then, we evaluate our model on test datasets with the number of items equal 200, 500 and 1000
+and capacity of 25 and 50, for each problem size. Table B shows the performance of our model
+compared to POMO, one of the best performing NCO models on KP. Although our model does not
+outperform it in every dataset, it achieves better overall performance. It should be noted again that
+POMO builds N solutions per instance and choose the best one, while our model generate a single
+solution per instance but still achieves better results.
+14
+
+Published as a conference paper at ICLR 2023
+Optimal
+POMO (single traj.)
+POMO (all traj.)
+BQ (greedy)
+value
+value
+opt gap
+value
+opt gap
+value
+opt gap
+N=200
+C=25
+58.023
+57.740
+0.476%
+58.007
+0.017%
+57.970
+0.081%
+C=50
+80.756
+79.483
+1.544%
+79.787
+1.170%
+80.710
+0.056%
+N=500
+C=25
+90.986
+85.309
+6.217%
+86.516
+4.897%
+90.150
+0.904%
+C=50
+129.326
+128.950
+0.291%
+129.272
+0.042%
+128.369
+0.739%
+N=1000
+C=25
+128.692
+120.757
+5.386%
+123.572
+3.973%
+121.217
+5.808%
+C=50
+182.898
+170.920
+6.545%
+172.427
+5.724%
+175.093
+4.267%
+All
+-
+3.552%
+2.648%
+1.980%
+Table 2: Experimental results on KP.
+Greedy
+Beam size 16
+Beam size 64
+Full graph
+0.79%
+5s
+0.17%
+1m
+0.08%
+5m
+TSP200
+100KNNs
+1.31%
+3s
+0.23%
+33s
+0.10%
+3m
+Full graph
+1.71%
+1m
+0.68%
+15m
+0.54%
+1h
+TSP500
+100KNNs
+2.58%
+18s
+0.92%
+3m
+0.69%
+12m
+250KNNs
+1.56%
+32s
+0.67%
+9m
+0.53%
+30m
+Full graph
+2.34%
+7m
+1.31%
+1.8h
+1.19%
+7.3h
+TSP1000
+100KNNs
+3.34%
+25s
+1.69%
+6m
+1.45%
+24m
+250KNNs
+2.53%
+1m
+1.43%
+23m
+1.19%
+1.4h
+Full graph
+4.80%
+5s
+2.42%
+1m
+1.82%
+5m
+CVRP200
+100KNNs
+5.18%
+3s
+2.12%
+33s
+1.68%
+3m
+Full graph
+4.74%
+1m
+2.10%
+15m
+1.59%
+1h
+CVRP500
+100KNNs
+5.14%
+18s
+2.02%
+3m
+1.74%
+12m
+250KNNs
+4.58%
+32s
+1.86%
+9m
+1.14%
+30m
+Full graph
+8.00%
+7m
+3.19%
+1.8h
+2.39%
+7.3h
+CVRP1000
+100KNNs
+8.25%
+25s
+4.76%
+6m
+3.58%
+24m
+250KNNs
+7.51%
+1m
+3.08%
+23m
+2.28%
+1.4h
+Table 3: Improving the model performance using a k-nearest-neighbor heuristic.
+C
+IMPROVING THE MODEL PERFORMANCE WITH A k-NEAREST-NEIGHBOR
+HEURISTIC
+Our decoding strategy could be further improved by using a k-nearest-neighbor heuristic to restrict
+the search space and reduce the inference time. For both greedy and beam search strategies, at
+every step, it is possible to reduce the remaining graph by considering only a certain number of
+neighbouring nodes. Table 3 presents the experiments on TSP and CVRP where we apply the model
+just on a certain number on nearest neighbours of the origin. This approach clearly reduces the
+execution time, but also in some cases even improves the performance in terms of optimality gap.
+The same heuristic can be applied on Knapsack problem, where model could be applied just on a
+certain number of items with highest values.
+D
+ABLATION STUDY
+D.1
+TRANSFORMER VS HYPERMIXER AS MODEL
+In Section 6 we have shown that our model has an excellent generalization ability to graphs of
+larger size. In Section ??, we hypothesize that this has to do with the fact that a subproblem of
+size t spends O(t2) computation operations due to the quadratic complexity of the Transformer
+encoder’s self-attention component, which is responsible for mixing node representations. To test
+this hypothesis, we experiment with replacing self-attention with an efficient mixing component (see
+Tay et al. (2022) for an overview), namely the recent linear-time HyperMixer (Mai et al., 2022). We
+chose this model because it does not assume that the input is ordered, unlike e.g. sparse attention
+alternatives.
+15
+
+Published as a conference paper at ICLR 2023
+Seed
+TSP100
+TSP200
+TSP500
+TSP1000
+1
+2.10%
+8.38%
+34.91%
+71.30%
+2
+1.38%
+3.54%
+98.59%
+628.71%
+3
+1.93%
+4.14%
+120.18%
+216.77%
+4
+1.37%
+4.54%
+46.23%
+104.85%
+5
+1.25%
+3.66%
+61.99%
+524.43%
+Table 4: Experimental results on TSP with HyperMixer for five different seeds.
+Experimental Details
+For comparability, we set the model and training parameters to the same as
+for Transformers, so the experiments only differ in token mixing component that is used. The only
+other difference is that we used Layer Normalization Ba et al. (2016) instead of ReZero Bachlechner
+et al. (2021), which leads to more stable training for HyperMixer. Since we observed relatively large
+sensitivity to model initialization, we are reporting the results for 5 different seeds.
+Results
+Table 4 shows the result for HyperMixer with greedy decoding. While the model reaches
+lower but acceptable performance than Transformers on TSP100, it generalizes poorly to instances
+of larger size. Moreover, performance is very sensitive to the seed. These results suggest that the
+computation spent by self-attention is indeed necessary to reach the generalization ability of our
+model, which increases the compute with the size of the (sub)problem.
+D.2
+APPROXIMATED MODEL
+As mentioned in Section 5, existing works have also noted the importance of accounting for the
+change of the state after each action: Xin et al. (2021b; 2020) claimed that models should recompute
+the embeddings after each action. However because of the additional training cost, they proposed
+the following approximation: fixing lower encoder levels and recomputing just the top level with a
+mask of already visited nodes. They hypothesis a kind of hierarchical feature extraction property
+that may make the last layers more important for the fine-grained next decision. In contrast, we call
+our entire model after each construction step; effectively recomputing the embeddings of each state.
+We hypothesis that this property may explain the superior performance (Table 1) w.r.t MDAM model
+Xin et al. (2020). In order to support this hypothesis, we have implemented an approximated version
+of our model as follows. We fixed the bottom layers of our model and recomputed just the top layer,
+by masking already visited nodes and adding the updated information (origin and destination tokens
+for TSP). As expected, inference time is 1.6 times shorter, but performance is severely degraded: we
+obtained optimality gap of 9.833% (vs 0.540% with original model) on TSP100.
+D.3
+REZERO VS BATCHNORM AS NORMALIZATION
+Most NCO works that use transformer networks (Kool et al., 2019)(Kwon et al., 2021)(Xin et al.,
+2020) use batch normalization(Ioffe & Szegedy, 2015) rather than layer normalization (Ba et al.,
+2016) in attention layers. We find ReZero normalization (Bachlechner et al., 2021) to work even
+better. Figure 3 shows the effect of using ReZero compared to batch normalization in our Trans-
+former network. Using it leads to more stable training, better performance, and drastically lower
+variance between seeds.
+E
+ON THE IMPACT OF EXPERT SOLUTIONS
+Our datasets consist of pairs of a problem instance and a solution (tour). On the other hand, in this
+paper, we use imitation learning, which requires instead pairs of a problem instance and (expert)
+trajectory in the MDP. Now, a solution may be obtained from multiple trajectories in the MDP. For
+example, with TSP, a solution is a loop in a graph, and one has to decide at which node its construc-
+tion started and in which direction it proceeded. With CVRP, the order in which the subtours are
+constructed needs also to be decided. Hence, all our datasets are pre-processed to transform solu-
+tions into corresponding construction trajectories (a choice for each or even all possible ones). We
+experimentally observed that this transformation has an impact on the performance. For example,
+with CVRP, choosing, for each solution, the construction in the order in which LKH3 displays it,
+16
+
+Published as a conference paper at ICLR 2023
+0
+200
+400
+600
+800
+1000
+Epochs
+0
+5
+10
+15
+20
+Optimality gap
+BatchNorm, seed 0
+BatchNorm, seed 1
+ReZero, seed 0
+ReZero, seed 1
+Figure 3: Training curves showing the effect of the choice of normalization layer on validation
+performance
+which does not seem arbitrary, yields to 1.3 point better opt-gap performance compared to following
+a random ordering of the sub-tours. We hypothesize that if there is any bias in the display of the
+optimal solution - for example, shorter tour first, or closest node first - it requires slightly less model
+capacity to learn action imitation for this display rather than for all possible displays.
+F
+PROOF OF PROPOSITION 1 (SOUNDNESS OF THE NAIVE MDP)
+We show here that procedure SOLVE satisfies SOLVE(f)= arg minx∈X f(x). We first show the
+following general lemma:
+Let Y
+ψ→X
+f→R∪{∞} be arbitrary mappings, if ψ is surjective then
+arg min
+x∈X f(x) = ψ(arg min
+y∈Y f(ψ(y)))
+Simple application of the definition of arg min (as a set). The subscript ∗ denotes the steps where
+the assumption that ψ is a surjection is used:
+x′ ∈ ψ(arg min
+y
+f(ψ(y)))
+iff
+∃y′ ∈ arg min
+y
+f(ψ(y)) x′ = ψ(y′)
+iff
+∃y′ x′ = ψ(y′) ∀y f(ψ(y′)) ≤ f(ψ(y))
+iff
+∃y′ x′ = ψ(y′) ∀y f(x′) ≤ f(ψ(y))
+iff∗
+∀y f(x′) ≤ f(ψ(y))
+iff∗
+∀x f(x′) ≤ f(x)
+iff
+x′ ∈ arg min
+x f(x)
+Let (F, X) be a CO problem with specification ⟨T , SOL, VAL⟩ and M the naive MDP obtained from
+it. For each f∈F, let vf=VAL(f, ϵ), Xf={x∈X|f(x)<∞} and let Yf be the set of M-trajectories
+which start at (f, ϵ) and end at a stop state.
+17
+
+Published as a conference paper at ICLR 2023
+...
+Transformer encoder 
+activation = ReLU 
+normalization = ReZero
+input embedding layer
+Linear
+softmax
+dest. 
+emb.
+origin 
+emb.
++
++
+...
+...
+Figure 4: Computation flow at the t-th time step, when a partial solution of length t − 1 already
+exists. The input state consist of the destination node (i.e. the first and last node in the TSP tour),
+the origin node (i.e., the most recent node in the tour), and the set of remaining nodes. After passing
+all input nodes through an embedding layer, we add special, learnable vector embeddings to the
+origin and current node to signal their special meaning. Finally, a Transformer encoder followed by
+a linear classifier head selects the next node at step t.
+• For any M-trajectory τ=s0t1r1s1 · · · tnrnsn in Yf, define ψ(τ) =def SOL(sn). Since
+τ∈Yf, we have s0=(f, ϵ) and sn is a stop state, i.e. SOL(sn)=ψ(τ)∈X, and by Equa-
+tion 2a, f(ψ(τ))<∞. Hence ψ:Yf �→ Xf.
+• By construction, sm=(f, t1:m) for all m∈1:n and each transition in τ has a finite reward
+VAL(sm−1)−VAL(sm) (condition for it to be valid). Hence the cumulated reward is given
+by R(τ)=VAL(s0)−VAL(sn). Now, VAL(s0)=vf which is independent of τ and by Equa-
+tion 2c, VAL(sn)=f(ψ(τ)). Hence f(ψ(τ))=vf−R(τ).
+• Let’s show that ψ is surjective. Let x∈Xf. Equation 2a ensures that x=SOL(f, t1:n)
+for some t1:n∈T ∗.
+For each m∈{0:n}, let sm=(f, t1:m) and consider the sequence
+τ=s0t1r1s1 · · · tnrnsn. Now, SOL(sn)=x̸=⊥ hence τ ends in a stop state and starts at
+(f, ϵ). By Equation 2c we have VAL(sn)=f(x), hence VAL(sn)<∞, and VAL(sm)<∞ for
+all m∈{0:n−1}. And by Equation 2b SOL(sm)=⊥, hence all the transitions in τ are valid
+in M. Hence τ∈Yf and by definition, ψ(τ)=x.
+Therefore we can apply the lemma proved above:
+arg min
+x∈Xf f(x) = ψ(arg min
+τ∈Yf f(ψ(τ))) = ψ(arg min
+τ∈Yf vf−R(τ))
+= ψ(arg max
+τ∈Yf R(τ)) = ψ(SOLVEMDP
+M (f, ϵ)) = SOLVE(f)
+Now, obviously, arg minx∈X f(x) = arg minx∈Xf f(x), since by definition f is infinite on X\Xf.
+18
+
+Published as a conference paper at ICLR 2023
+G
+PLOTS OF SOME TSPLIB AND CVRPLIB SOLUTIONS
+(a) Optimal solution
+(b) Our model (BS16), opt_gap 0.549%
+(c) MDAM (BS50), opt_gap 11.501%
+(d) POMO (x8), opt_gap 18.614%
+Instance pcb442
+(a) Optimal solution
+(b) Our model (BS16), opt_gap 4.253%
+(c) MDAM (BS50), opt_gap 20.916%
+(d) POMO (x8), opt_gap 44.664%
+Instance pr1002
+19
+
+Published as a conference paper at ICLR 2023
+(a) Optimal solution
+(b) Our model (BS16), opt_gap 3.464%
+(c) MDAM (BS50), opt_gap 45.669%
+(d) POMO (x8), opt_gap 11.416%
+Instance X-n284-k15
+(a) Best known solution
+(b) Our model (BS16), opt_gap 2.667%
+(c) MDAM (BS50), opt_gap 19.739%
+(d) POMO (x8), opt_gap 46.603%
+Instance X-n513-k21
+20
+
+Published as a conference paper at ICLR 2023
+H
+BACKGROUND ON BISIMULATION-BISIMILARITY
+H.1
+BISIMULATION IN LABELLED TRANSITION SYSTEMS
+Bisimulation is a very broad concept which applies to arbitrary Labelled Transition Systems (LTS). It
+has been declined in various flavours of LTS, such as Process Calculi, Finite State Automata, Game
+theory, and of course MDP (initially deterministic MDP such as those used here, later extended
+to stochastic MDP which we are not concerned with here). A bisimulation is a binary relation R
+among states which “commutes” with the transitions of the LTS in the following diagram, which
+should informally be read as follows: if the pair of arrows connected to p (resp. q) exists then so
+does the “opposite” pair (w.r.t. the centre of the diagram).
+p
+q
+p′
+q′
+ℓ
+ℓ
+R
+R
+The notation p
+ℓ
+−−→ p′ means the transition from p to p′ with label ℓ is valid. Thus, formally,
+Definition 1. A binary relation R on states is a bisimulation if for all label ℓ and states p, q such
+that pRq
+∀p′ s.t. p
+ℓ
+−−→ p′ ∃q′ s.t. q
+ℓ
+−−→ q′ , p′Rq′
+∀q′ s.t. q
+ℓ
+−−→ q′ ∃p′ s.t. p
+ℓ
+−−→ p′ , p′Rq′
+Note that this definition is extended to the “heterogeneous” case where R is bi-partite, relating
+the state spaces of two LTS L1, L2 sharing the same label space. One just forms a new LTS L
+whose state space is the disjoint union of the state spaces of L1, L2 and the transitions are those of
+L1, L2 in their respective (disjoint) component. An heterogeneous bisimulation on L1, L2 is then a
+(homogeneous) bisimulation on L. Most results below also have a heterogeneous version.
+Proposition 2. The set of bisimulations (subset of the set of binary relations on states) is stable by
+union, composition, and inversion, hence also by reflexive-symmetric-transitive closure.
+In particular, the union of all bisimulations, called the bisimilarity of the LTS, is itself a bisimulation,
+and it is also an equivalence relation.
+Proof. (outline) Let’s detail stability by composition, the other cases are similarly obvious.
+If
+R1, R2 are the two bisimulations being composed, apply the commutation property to each cell
+of the following diagram (from top to bottom).
+p
+r
+q
+p′
+r′
+q′
+ℓ
+ℓ
+ℓ
+R1
+R2
+R1
+R2
+Definition 2. Given an LTS L, its transitive closure is another LTS denoted L∗ on the same state
+space, where the labels are the sequences of labels of L and the transitions are defined by
+p
+ℓ1:n
+−−−−→
+(L∗)
+p′
+if
+∃p0:n such that p = p0
+ℓ1
+−−−→
+(L)
+p1 · · ·
+ℓn−1
+−−−−→
+(L)
+pn−1
+ℓn
+−−−→
+(L)
+pn = p′
+Proposition 3. If R is a bisimulation on L, then it is also a bisimulation on L∗.
+Proof. (outline) This is essentially shown by successively applying the commutation property to
+each cell of the following diagram (from left to right):
+21
+
+Published as a conference paper at ICLR 2023
+p0
+q0
+p1
+q1
+pn−1
+qn−1
+pn
+qn
+ℓ1
+ℓn
+ℓ1
+ℓn
+R
+R
+R
+R
+Definition 3. Given an LTS L and an equivalence relation R on its state space, we can define the
+quotient LTS L/R with the same label space, where the states are the R-equivalence classes and
+the transitions are defined, for any classes ˙p, ˙p′, by
+˙p
+ℓ
+−−−−→
+L/R
+˙p′
+if
+∀p ∈ ˙p ∃p′ ∈ ˙p′
+p
+ℓ
+−−→
+L
+p′
+Proposition 4. Let R be an equivalence on the state space of L. R is a bisimulation on L if and
+only if ∈ is a (heterogeneous) bisimulation on L, L/R.
+Proof. We show both implications:
+• Assume R is a bisimulation on L.
+– Let p ∈ ˙q and p
+ℓ−→ p′. Let q ∈ ˙q. Hence pRq and p
+ℓ−→ p′. Since R is a bisimulation,
+there exists q′ such that q
+ℓ−→ q′ and p′Rq′. Hence for all q ∈ ˙q, there exists q′ ∈ ¯p′
+such that q
+ℓ−→ q′. Hence by definition ˙q
+ℓ−→ ¯p′ while p′ ∈ ¯p′.
+– Let p ∈ ˙q and ˙q
+ℓ−→ ˙q′. Hence by definition, there exists p′ ∈ ˙q′ such that p
+ℓ−→ p′.
+• Assume ∈ is a (heterogeneous) bisimulation on L, L/R.
+– Let pRq and p
+ℓ−→ p′. Hence p ∈ ¯q and p
+ℓ−→ p′. Since ∈ is a bisimulation, there exists
+˙q′ such that p′ ∈ ˙q′ and ¯q
+ℓ−→ ˙q′. Now q ∈ ¯q, hence, by definition, there exists q′ ∈ ˙q′
+such that q
+ℓ−→ q′. And p′Rq′ since p′, q′ ∈ ˙q′.
+– Let pRq and q
+ℓ−→ q′. Hence qRp and q
+ℓ−→ q′, and we are in the previous case up to a
+permutation of variables.
+Proposition 5. Let R be an equivalence relation on the state space of L. If R is a bisimulation on
+L, then for any L-state p, L/R-state ˙p and L∗-label ℓ
+¯p
+ℓ
+−−−−−−→
+(L/R)∗
+˙p′
+if and only if
+∃p′ ∈ ˙p′
+p
+ℓ
+−−→
+L∗
+p′
+Proof. Simple combination of Propositions 4 and 3. R is a bisimulation on L, hence ∈ is a het-
+erogeneous bisimulation on L, L/R (Proposition 4), hence also a heterogeneous bisimulation on
+L∗, (L/R)∗ (Proposition 3, heterogeneous version).
+• If ¯p
+ℓ
+−−−−−−→
+(L/R)∗
+˙p′, since p∈¯p and ∈ is a bisimulation, we have p
+ℓ
+−−→
+L∗
+p′ for some p′∈ ˙p′.
+• Conversely, if p
+ℓ
+−−→
+L∗
+p′ for some p′∈ ˙p′, since p∈¯p and ∈ is a bisimulation, we have
+¯p
+ℓ
+−−−−−−→
+(L/R)∗
+˙q′ and p′∈ ˙q′ for some ˙q′. Now p′∈ ˙p′∩ ˙q′ hence ˙p′= ˙q′ and ¯p
+ℓ
+−−−−−−→
+(L/R)∗
+˙p′.
+H.2
+BISIMULATION IN DETERMINISTIC MDP
+Definition 4. An MDP is a pair (L, ⊤) where L is a LTS with label space A × R for some action
+space A (action-reward pairs denoted a|r) and ⊤ is a subset of states (the stop states). It is said to
+be deterministic if
+if s
+a|r1
+−−→ s′
+1 and s
+a|r2
+−−→ s′
+2 then r1 = r2 and s′
+1 = s′
+2
+22
+
+Published as a conference paper at ICLR 2023
+Given an L-trajectory τ, i.e. a sequence s0a1r1s1 · · · anrnsn where si−1
+ai|ri
+−−−→ si for all i∈{1:n},
+its cumulated reward is defined by R(τ)= �n
+i=1 ri. The generic problem statement of the MDP
+solution framework is, given an MDP (L, ⊤) and one of its states so, to solve the following optimi-
+sation:
+SOLVEMDP((L, ⊤), so) = arg max
+τ
+R(τ) | τ is a L-trajectory starting at so and ending in ⊤
+This definition of MDP and the standard textbook one coincide only in the deterministic case (in
+the standard definition, an MDP is deterministic if the distribution of output state-reward pairs for a
+given input state and allowed action is “one-hot”). The non deterministic case in the definition above
+does not match the standard definition: it would be wrong to interpret two distinct transitions for the
+same input state s and action a as meaning that the outcome of applying a to state s is distributed
+between the two output reward-state pairs according to a specific distribution (e.g. uniform). Also,
+in the problem statement, the objective R(τ) has no expectation, which, with the standard definition,
+only makes sense in the case of a deterministic MDP. Similarly, the standard problem statement is
+expressed in terms of policies rather than trajectories directly, but in the deterministic case, the two
+are equivalent. Observe that there is a one-to-one correspondence between trajectories in L and
+transitions in the LTS L∗, so the problem statement can be formulated equivalently as
+SOLVEMDP((L, ⊤), so) = arg max
+ℓ
+R(ℓ) | ∃s ∈ ⊤, so
+ℓ
+−−→
+L∗
+s
+(4)
+Proposition 6. Let (L, ⊤) be an MDP and R an equivalence relation on its state space.
+1. (L/R, ¯⊤) is also an MDP, where ¯⊤={¯s|s∈⊤}, and if L is deterministic, so is L/R.
+2. If R is a bisimulation on L preserving ⊤ (i.e. �
+s∈⊤ ¯s = ⊤), then for any state so and label
+ℓ in L∗ we have
+∃s ∈ ⊤, so
+ℓ
+−−→
+L∗
+s
+if and only if
+∃ ˙s ∈ ¯⊤, ��so
+ℓ
+−−−−−−→
+(L/R)∗
+˙s
+Proof. The second property is a direct consequence of Proposition 5 and the assumption that ⊤ is
+preserved by R. For the first, assume that L is deterministic. Let ˙s, ˙s1, ˙s2 be L/R states, such that
+˙s
+a|r1
+−−→ ˙s1 and ˙s
+a|r2
+−−→ ˙s2. Choose s ∈ ˙s. Hence, by definition, there exist s1∈ ˙s1 and s2∈ ˙s2 such
+that s
+a|r1
+−−→ s1 and s
+a|r2
+−−→ s2. Since L is deterministic, we have r1=r2 and s1=s2∈ ˙s1∩ ˙s2, hence
+˙s1 = ˙s2. Hence L/R is also deterministic.
+Therefore, when R is a bisimulation equivalence on L preserving ⊤, the generic MDP problem
+statement of Eq. equation 4 can be reformulated as
+SOLVEMDP((L, ⊤), so) = SOLVEMDP((L/R, ¯⊤), ¯so) = arg max
+ℓ
+R(ℓ) | ∃ ˙s ∈ ¯⊤, ¯so
+ℓ
+−−−−−−→
+(L/R)∗
+˙s
+(5)
+Note that a bisimulation on L preserving ⊤ is simply a bisimulation on the LTS ˙L defined as follows:
+˙L has the same state space as L and an additional transition s
+·−→ s for each s∈⊤, where “·” is a
+distinguished label not present in L.
+A bisimulation R on ˙L captures some symmetries of the state space of ˙L. If R is taken to be the
+bisimilarity of ˙L, i.e. the union of all the bisimulations on ˙L, i.e. the union of all the bisimulations
+on L preserving ⊤, then it captures all the possible symmetries of the state space. This should be
+seen as an asymptotic result, since constructing and working with the full bisimilarity of ˙L is not
+feasible. But Proposition 6 remains valuable as it applies to all bisimulation, not just the maximal
+bisimulation of ˙L (its bisimilarity).
+23
+

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@@ -0,0 +1,807 @@
+REDUCED CLIQUE GRAPHS: A CORRECTION TO
+“CHORDAL GRAPHS AND THEIR CLIQUE GRAPHS”
+DILLON MAYHEW AND ANDREW PROBERT
+Abstract. Galinier, Habib, and Paul introduced the reduced clique
+graph of a chordal graph G.
+The nodes of the reduced clique graph
+are the maximal cliques of G, and two nodes are joined by an edge if
+and only if they form a non-disjoint separating pair of cliques in G. In
+this case the weight of the edge is the size of the intersection of the two
+cliques. A clique tree of G is a tree with the maximal cliques of G as
+its nodes, where for any v ∈ V (G), the subgraph induced by the nodes
+containing v is connected. Galinier et al. prove that a spanning tree of
+the reduced clique graph is a clique tree if and only if it has maximum
+weight, but their proof contains an error. We explain and correct this
+error.
+In addition, we initiate a study of the structure of reduced clique
+graphs by proving that they cannot contain any induced cycle of length
+five (although they may contain induced cycles of length three, four, or
+six). We show that no cycle of length four or more is isomorphic to a
+reduced clique graph. We prove that the class of clique graphs of chordal
+graphs is not comparable to the class of reduced clique graphs of chordal
+graphs by providing examples that are in each of these classes without
+being in the other.
+1. Introduction
+We consider only simple graphs. A chord of a cycle is an edge that joins
+two vertices of the cycle without being in the cycle itself. A graph is chordal
+if any cycle with at least four vertices has a chord. A clique is a set of
+pairwise adjacent vertices. If S is a set of vertices and P is a path, then P
+is S-avoiding if no internal vertex of P is in S. Assuming that a and b are
+distinct vertices, an ab-separator is a set S of vertices not containing either
+a or b such that there is no S-avoiding path from a to b. If, in addition, S
+does not properly contain an ab-separator then it is a minimal ab-separator.
+If G is a chordal graph, then C(G) is the corresponding clique graph
+(also known as the clique intersection graph).
+The vertices of C(G) are
+the maximal cliques of G, and two maximal cliques are adjacent in C(G) if
+and only if they have a non-empty intersection. The vertices of the reduced
+clique graph, CR(G), are again the maximal cliques of G, but C and C′ are
+adjacent in CR(G) if and only if C ∩ C′ ̸= ∅ and C and C′ form a separating
+pair: that is, there is no (C ∩ C′)-avoiding path from a vertex in C − C′
+1
+arXiv:2301.03781v1  [math.CO]  10 Jan 2023
+
+2
+MAYHEW AND PROBERT
+to a vertex in C′ − C. Note that the vertices of CR(G) are identical to the
+vertices of C(G), and every edge of CR(G) is an edge of C(G).
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+234589
+234689
+235789
+123
+8910
+G
+C(G)
+CR(G)
+234589
+234689
+235789
+123
+8910
+Figure 1. A chordal graph, its clique graph, and its reduced
+clique graph.
+The reduced clique graph was introduced in [3] (where it is called a clique
+graph) and studied further in [5–8].
+Let G be a graph, and let T be a tree whose vertices are the maximal
+cliques of G. If, for every v ∈ V (G), the maximal cliques of G that contain
+v induce a subtree of T, then T is a clique tree. Clique trees were introduced
+by Gavril [4], who proved that a graph has a clique tree exactly when it is
+chordal.
+We weight each edge of CR(G) as follows: the edge joining cliques C and
+C′ is weighted with |C ∩ C′|. The following result is [3, Theorem 6].
+Theorem 1.1. Let G be a connected chordal graph. Let T be a spanning
+tree of CR(G). Then T is a clique tree if and only if it is a maximum-weight
+spanning tree.
+Although the statement of Theorem 1.1 is correct, it is not proved in
+[3, Theorem 6] because of a flaw in the argument. The issue arises in the
+proof that a maximum-weight spanning tree must be a clique tree.
+We
+illustrate the error by using the same argument to prove a false statement.
+Non-theorem 1.2. Let G be a chordal graph. Let C0, C1, . . . , Cn be the
+sequence of maximal cliques in a path of CR(G) where n > 1. Assume that
+there is a vertex v of G such that v is in C0 ∩ Cn, but in none of the cliques
+C1, . . . , Cn−1. Then C0 and Cn are adjacent in CR(G).
+Non-proof. Consider the subgraph G′ of G induced by C0 ∪ C1 ∪ · · · ∪ Cn.
+Thus G′ is chordal. From [10, Corollary 2] we see that either v is a simplicial
+vertex (meaning that the neighbours of v in G′ form a clique), or there is a
+pair, a, b, of vertices such that v belongs to a minimal ab-separator of G′.
+In the former case v is in a unique maximal clique of G′ ([1, Theorem 3.1]).
+But C0 and Cn are distinct maximal cliques of G′ that contain v. Therefore
+
+REDUCED CLIQUE GRAPHS
+3
+we can let S be a minimal ab-separator of G′, where v is in S. The proof
+of [2, Lemma 2.3] shows that there are two distinct maximal cliques, Da
+and Db, of G′ such that Da and Db properly contain S, and Da − S is in
+the same connected component of G′ − S as a, while Db �� S is in the same
+component as b. Thus Da and Db are maximal cliques of G′ that contain v.
+But the only maximal cliques of G′ that contain v are C0 and Cn. Therefore
+we can assume without loss of generality that Da = C0 and Db = Cn. Any
+path from a vertex of C0 − Cn to a vertex of Cn − C0 must contain a vertex
+in S = C0 ∩ Cn. Therefore C0 and Cn form a non-disjoint separating pair,
+so C0 and Cn are adjacent in CR(G), as claimed.
+□
+We can see that this non-theorem is, indeed, not a theorem by examining
+Figure 1. Set C0, C1, and C2 to be the maximal cliques {2, 3, 4, 6, 8, 9},
+{1, 2, 3}, and {2, 3, 5, 7, 8, 9}, respectively.
+Thus C0, C1, C2 is the vertex
+sequence of a path in CR(G). The vertex 8 is in C0 ∩ C2, but not in C1.
+However C0 and C2 are not adjacent in CR(G). The error in the “proof”
+lies in the claim that “the only maximal cliques of G′ that contain v are C0
+and Cn”. This need not be true. Indeed, {2, 3, 4, 5, 8, 9} is a maximal clique
+in the subgraph induced by C0 ∪ C1 ∪ C2, and it contains 8, but it is not
+equal to either C0 or C2. Exactly the same error appears in the proof of
+[3, Theorem 6]. Nonetheless, Theorem 1.1 is true, and we prove it in the
+next section.
+2. Reduced clique graphs and clique trees
+In [9] we will apply our main theorem to some matroid problems. For
+these purposes we would like to extend its scope somewhat.
+Instead of
+weighting the edges of CR(G) with sizes of intersections, we consider more
+general weightings.
+Definition 2.1. Let G be a chordal graph. We consider a function σ which
+takes
+{∅} ∪ {C ∩ C′ : C, C are distinct maximal cliques of G}
+to non-negative integers. We insist that σ(∅) = 0 and if X and X′ are in the
+domain of σ and X ⊂ X′, then σ(X) < σ(X′). In such a case the function
+σ is a legitimate weighting of G.
+Theorem 2.2. Let G be a connected chordal graph and let σ be a legitimate
+weighting of G. Every clique tree is a spanning tree of CR(G) and every edge
+of CR(G) is contained in a clique tree. Moreover, a spanning tree of CR(G)
+is a clique tree if and only if it has maximum weight amongst all spanning
+trees.
+Note that the function that takes each intersection C ∩ C′ to |C ∩ C′| is
+a legitimate weighting, so Theorem 2.2 does indeed imply Theorem 1.1. We
+now start proving the intermediate results required for the proof of Theorem
+2.2.
+
+4
+MAYHEW AND PROBERT
+Proposition 2.3. Let G be a chordal graph, and let C and C′ be maximal
+cliques of G. Let S be a set of vertices that contains C∩C′. Let v0, v1, . . . , vk
+be the vertex sequence of P, a shortest-possible S-avoiding path from a vertex
+in C − C′ to a vertex in C′ − C. Then (C ∩ C′) ∪ {vi, vi+1} is a clique for
+each i = 0, 1, . . . , k − 1.
+Proof. If C ∩ C′ = ∅ then the result holds trivially, so we assume C ∩ C′
+is non-empty. Note that every vertex in C ∩ C′ is adjacent to v0, and also
+to vk, since these vertices are in C − C′ and C′ − C. Now the result can
+only fail if there is a vertex x ∈ C ∩ C′ that is not adjacent to vi for some
+i ∈ {1, . . . , k − 1}. Let p be the largest integer such that p < i and x is
+adjacent to vp. Similarly, let q be the smallest integer such that q > i and
+x is adjacent to vq. Consider the cycle obtained by adding the edges vpx
+and vqx to vp, vp+1, . . . , vq. This cycle contains the distinct vertices vp, vi,
+vq, and x, so it must contain a chord. No chord can join two vertices in the
+path P, since P is as short as possible. Thus any chord is incident with x.
+But x is not adjacent to any of the vertices in vp+1, . . . , vq−1 by the choice
+of p and q, so we have a contradiction.
+□
+Proposition 2.4. Let G be a chordal graph, and let C and C′ be maximal
+cliques of G where C ∩ C′ ̸= ∅. If C and C′ are not adjacent in CR(G),
+then they are joined by a path of CR(G) with vertex sequence C0, C1, . . . , Cs,
+where each Ci ∩ Ci+1 properly contains C ∩ C′.
+Proof. Assume this fails for C and C′, and they have been chosen so that
+C ∩ C′ is as large as possible. Let S be C ∩ C′. Because C and C′ are not
+adjacent in CR(G), but S ̸= ∅, it follows that there is an S-avoiding path
+from a vertex in C − C′ to a vertex in C′ − C. Let v0, v1, . . . , vk be the
+vertex sequence of such a path, where k is as small as possible. We assume
+v0 is in C − C′ while vk is in C′ − C. We apply Proposition 2.3 and for each
+i = 1, . . . , k, we let Di be a maximal clique of G that contains S ∪{vi−1, vi}.
+Set D0 to be C and set Dk+1 to be C′. Note that Di ̸= Dj when i < j,
+because vi−1 is not adjacent to vj. For each i = 0, 1, . . . , k, the intersection
+of Di and Di+1 contains S as well as vi. If Di and Di+1 are adjacent in
+CR(G) then we let Pi be the path of CR(G) consisting of Di, Di+1, and
+the edge between them. Otherwise Di and Di+1 are not adjacent in CR(G)
+and the assumption on the cardinality of S means that there is a path Pi of
+CR(G) from Di to Di+1 such that every intersection of consecutive cliques
+in Pi properly contains S ∪ vi.
+We concatenate the paths P0, P1, . . . , Pk
+and obtain a walk of CR(G) from C to C′. The intersection of any two
+consecutive cliques in this walk properly contains S. It follows that there is
+a path of CR(G) from C to C′ with exactly the same property, and now C
+and C′ fail to provide a counterexample after all.
+□
+Figure 2 illustrates Proposition 2.4.
+The intersection of cliques C =
+{1, 2, 3} and C′ = {3, 5, 7, 8} is {3} ̸= ∅, but C and C′ are not adjacent
+
+REDUCED CLIQUE GRAPHS
+5
+in CR(G). However, there is a path between C and C′ in CR(G), and the
+intersection of any consecutive two cliques in the path properly contains {3}.
+1
+6
+4
+7
+5
+2
+3
+8
+123
+2345
+3567
+3456
+3578
+G
+CR(G)
+Figure 2.
+Proposition 2.5. Let G be a connected chordal graph. Let T be a clique
+tree of G. Assume that C and C′ are maximal cliques of G that are adjacent
+in T. Then C and C′ are adjacent in CR(G).
+Proof. Assume C and C′ are adjacent in T, but not in CR(G). We partition
+the maximal cliques of G as follows. Let U be the set of maximal cliques of
+G such that D is in U if and only if the path of T from D to C does not
+contain C′. Similarly, define U′ so that D′ is in U′ if and only if the path
+of T from D′ to C′ does not contain C. Note that every maximal clique
+of G is in exactly one of U or U′, since T is a tree. Furthermore C is in U
+and C′ is in U′. Let U be the union of the cliques in U, and let U ′ be the
+union of the cliques in U′. Every vertex is in at least one maximal clique
+so U ∪ U ′ = V (G). Note that C ⊆ U and C′ ⊆ U ′, so neither U nor U ′ is
+empty.
+If U ∩ U ′ = ∅, then we choose u ∈ U and u′ ∈ U ′ so that u and u′ are
+adjacent in G. (We are able to do so because G is connected.) The edge
+between u and u′ is contained in a maximal clique. If this maximal clique
+is in U then u′ is in U ∩ U ′, and if it is in U′ then u is in U ∩ U ′. In either
+case we have a contradiction, so U ∩ U ′ ̸= ∅.
+Choose an arbitrary vertex v in U ∩ U ′. Choose D ∈ U and D′ ∈ U′ such
+that v is in D ∩ D′. Because T is a clique tree, it follows that v is contained
+in all the cliques belonging to the path of T from D to D′. In particular, v
+is contained in C and C′. Thus U ∩ U ′ ⊆ C ∩ C′ and C ∩ C′ is non-empty.
+Let S be C ∩ C′. Since C and C′ are not adjacent in CR(G), we can
+apply Proposition 2.4 and find a path P of CR(G) from C to C′, where the
+intersection of each pair of consecutive cliques in this path properly contains
+S. Since C is in U and C′ is in U′, there is an edge of P that joins a clique
+D ∈ U to a clique D′ ∈ U′. Then D∩D′ properly contains S, so we choose v
+in (D ∩D′)−S. Again using the fact that T is a clique tree, we see that the
+path of T from D to D′ consists of cliques that contain v. In particular, v
+is in C ∩ C′ = S, and we have a contradiction that completes the proof.
+□
+
+6
+MAYHEW AND PROBERT
+It follows from Proposition 2.5 that every clique tree of G is a spanning
+tree of CR(G).
+Proposition 2.6. Let G be a connected chordal graph and let σ be a legiti-
+mate weighting of G. Let T be a clique tree of G. Let C and C′ be maximal
+cliques of G that are adjacent in C(G) and let P be the path of T between C
+and C′. The weight of any edge in P is at least σ(C ∩ C′). Moreover, if C
+and C′ are adjacent in CR(G), then at least one edge in P has weight equal
+to σ(C ∩ C′).
+Proof. Let S be C ∩ C′. Let P be the path of T from C to C′, and let the
+cliques in this path be C0, C1, . . . , Cn, where C0 = C and Cn = C′. Note that
+P is a path of CR(G) by Proposition 2.5. Thus any two consecutive cliques
+in the path have a non-empty intersection. Assume σ(Ci ∩ Ci+1) < σ(S)
+for some i. If S were a subset of Ci ∩ Ci+1, then we would have σ(S) ≤
+σ(Ci ∩Ci+1) by the definition of a legitimate weighting, but this is not true.
+Therefore we can choose v to be a vertex in S − (Ci ∩ Ci+1). Now v is a
+vertex of both C and C′, but the path of T between C and C′ contains at
+least one maximal clique (either Ci or Ci+1) that does not contain v. This
+contradicts the fact that T is a clique tree. Therefore the weight of any edge
+in P is at least equal to σ(S).
+Now assume that C and C′ are adjacent in CR(G), so that they form a
+separating pair. That is, there are distinct connected components of G − S
+that contain, respectively, C −S and C′−S. There must be maximal cliques
+D and D′ that are adjacent in P, where D − S is in the same connected
+component of G−S as C−S, and D′−S is not in this connected component.
+This means that D ∩ D′ is contained in S. Hence σ(D ∩ D′) ≤ σ(S). The
+previous paragraph shows that σ(D ∩ D′) ≥ σ(S), so the result follows.
+□
+The proof of the next result is a straightforward adaptation of a proof
+given by Blair and Peyton [1, Theorem 3.6].
+Lemma 2.7. Let G be a connected chordal graph. Let σ be a legitimate
+weighting of G and let T be a spanning tree of C(G). Then T is a clique
+tree of G if and only if it is a maximum-weight spanning tree of C(G).
+Proof. If T is a clique tree, then for any pair of maximal cliques, C and C′,
+such that C and C′ are adjacent in C(G), the weight of the edge between C
+and C′ is no greater than the weight of any edge in the path of T between
+C and C′ (Proposition 2.6). It immediately follows that T has maximum
+weight.
+For the other direction, we assume that T is a maximum-weight spanning
+tree. Because every chordal graph has a clique tree, and any clique tree is a
+spanning tree of CR(G) (and hence of C(G)), we can choose a clique tree T ′
+so that T and T ′ have as many edges in common as possible. We can choose
+an edge in T that is not in T ′, because otherwise there is nothing left for us
+to prove. So let e be such an edge, and assume that e joins maximal cliques
+C and C′. There are two connected components of T\e, one containing C
+
+REDUCED CLIQUE GRAPHS
+7
+and the other containing C′. Let P be the path of T ′ from C to C′. We let f
+be an edge of P which joins two cliques that are not in the same component
+of T\e. Note that f is an edge of T ′, and hence an edge of C(G).
+If (T − e) ∪ f is not a spanning tree of C(G), then there is a path of T
+between the end-vertices of f that does not use e. But the end-vertices of
+f are in different connected components of T\e, so (T − e) ∪ f is indeed a
+spanning tree. Similarly, if (T ′ − f) ∪ e is not a spanning tree, then there
+is a path of T ′ between C and C′ that does not contain f. But P is the
+unique path of T ′ between C and C′, and f is an edge of P. So (T − e) ∪ f
+and (T ′ − f) ∪ e are both spanning trees of C(G).
+Applying Proposition 2.6 to the clique tree T ′ shows that the weight of f
+is at least the weight of e. Since T is a maximum-weight spanning tree, and
+(T − e) ∪ f is a spanning tree it follows that the weights on e and f must
+be equal. Let D and D′ be the maximal cliques joined by f. Any element
+that is in both C and C′ must be in all the cliques in P, since T ′ is a clique
+tree. This shows that C ∩ C′ ⊆ D ∩ D′. If C ∩ C′ were a proper subset of
+D ∩ D′, then the definition of a legitimate weighting would mean that the
+weight of e is strictly less than the weight of f, which is not true. Therefore
+C ∩ C′ = D ∩ D′.
+We note that (T ′−f)∪e cannot be a clique tree, since it has one more edge
+in common with T than T ′ does. Therefore we choose a vertex v ∈ V (G) so
+that the maximal cliques containing v do not induce a subtree of (T ′−f)∪e.
+Let T ′′ be the subtree of T ′ induced by the maximal cliques containing v.
+Then f is in T ′′, or else T ′′ would be a subtree of (T ′ − f) ∪ e. This means
+that v is in D ∩ D′ = C ∩ C′.
+So both C and C′ are in T ′′, but they
+are not in the same component of T ′′\f, because in that case (T ′ − f) ∪ e
+would contain a cycle. So e joins two vertices of T ′′ that are in different
+components of T ′′\f. Thus (T ′′ − f) ∪ e is a subtree of (T ′ − f) ∪ e, and we
+have a contradiction that completes the proof.
+□
+Proof of Theorem 2.2. We have already noted that every clique tree is a
+spanning tree of CR(G). Let T be a clique tree of G. Then T is a maximum-
+weight spanning tree of C(G) by Lemma 2.7. But every edge of T is an edge
+of CR(G), by Proposition 2.5. Since CR(G) is a subgraph of C(G) it follows
+that T is a maximum-weight spanning tree of CR(G).
+For the other direction, we let T be a maximum-weight spanning tree
+of CR(G).
+We claim that T is also a maximum-weight spanning tree of
+C(G). To prove this claim, let e be an arbitrary edge of C(G) that is not
+in T, let C and C′ be the maximal cliques of G joined by e, and let P
+be the path of T that joins C and C′. If e is an edge of CR(G), then the
+weight of e is no greater than the weight of any edge in P, since T is a
+maximum-weight spanning tree of CR(G). Therefore we assume that e is
+not an edge of CR(G). Now it follows from Proposition 2.4 and the definition
+of a legitimate weighting that the edges in P all have weight strictly greater
+than the weight of e. In either case, the weight of e does not exceed the
+
+8
+MAYHEW AND PROBERT
+weight of any edge in P. This implies that T is indeed a maximum-weight
+spanning tree of C(G), and thus T is a clique tree of G by Lemma 2.7.
+To complete the proof, we let e be an arbitrary edge of CR(G). We will
+prove that e is in a maximum-weight spanning tree of CR(G). We let C and
+C′ be the maximal cliques joined by e. Let T be an arbitrary maximum-
+weight spanning tree of CR(G), so that T is a clique tree by the previous
+paragraph. If e is in T then we have nothing left to prove, so assume that
+P is the path of T joining C to C′, where P contains more than one edge.
+Proposition 2.6 shows that P contains an edge, f, with weight equal to the
+weight of e. Now (T − f) ∪ e is a maximum-weight spanning tree of CR(G)
+that contains e, and we are done.
+□
+From the previous arguments we can deduce further additional facts, both
+noted in [3]: any edge that is in C(G) but not CR(G) cannot be in any
+maximum-weight spanning tree of C(G). Secondly, CR(G) is in fact the
+union of all clique trees of G.
+Although the next fact is incidental to our main results here, we note it
+for a future application in [9].
+Proposition 2.8. Let G be a connected chordal graph, and let T be a clique
+tree of G. Let C and C′ be adjacent in T and let S be C ∩ C′. Assume
+that D and D′ are maximal cliques of G and the path of T from D to D′
+contains both C and C′. Then D − S and D′ − S are in different connected
+components of G − S.
+Proof. Let U be the family of maximal cliques of G such that D is in U if
+and only if the path of T from D to C does not contain C′. Similarly, we let
+U′ be the family of maximal cliques where D′ is in U′ if and only if the path
+of T from D′ to C′ does not contain C. Note that every maximal clique of
+G belongs to exactly one of U and U′. We are asserting that if D ∈ U and
+D′ ∈ U′, then D − S and D′ − S are in different connected components of
+G − S. Assume that this fails for D and D′, where D ∩ D′ is as large as
+possible. Let H be the connected component of G − S that contains both
+D − S and D′ − S.
+Let P be the path of T from D to D′. Therefore P contains both C and
+C′. Let v be an arbitrary vertex of D ∩ D′. Then v is in every maximal
+clique that appears in P, since T is a clique tree. In particular, v is in C
+and C′. Thus v is in S, and this shows that D ∩ D′ is contained in S.
+Let v0, v1, . . . , vk be the vertex sequence of a shortest-possible path of H
+from a vertex v0 ∈ D − S to a vertex vk ∈ D′ − S. This is an S-avoiding
+path, where S contains D ∩ D′. Thus we can apply Proposition 2.3. For
+i = 1, 2, . . . , k we let Di be a maximal clique of G that contains (D ∩ D′) ∪
+{vi−1, vi}. Let D0 be D and let Dk+1 be D′. Note that each Di − S is
+contained in H. This is true for D0 and Dk+1 by definition, and every other
+Di contains the edge vi−1vi, which is in the path of H from v0 to vk. Since
+D0 is in U and Dk+1 is in U′, we can choose i so that Di is in U and Di+1
+is in U′. The intersection of Di and Di+1 is larger than D ∩ D′, since it
+
+REDUCED CLIQUE GRAPHS
+9
+contains (D ∩ D′) ∪ vi. As Di − S and Di+1 − S are both contained in H
+we have a contradiction to the choice of D and D′.
+□
+3. The structure of reduced clique graphs
+Habib and Stacho comment on the possibility of investigating the struc-
+ture of graphs that are isomorphic to reduced clique graphs [6, p. 714].
+In this section we make a contribution to this investigation. We start by
+answering an obvious question that requires a non-trivial proof.
+Corollary 3.1. Let G be a chordal graph. Then CR(G) is connected if and
+only if G is connected.
+Proof. Assume that H and H′ are distinct connected components of G. No
+maximal clique of H can share a vertex with a maximal clique of H′. It
+follows that there be no path of CR(G) that joins two such cliques. Thus
+CR(G) is not connected.
+The other direction is stated without proof in [6, p. 716]. Assume that
+G is connected. Since G is chordal it has a clique tree [4, Theorem 2], and
+Proposition 2.5 shows that every edge of the clique tree is an edge of CR(G).
+Thus CR(G) has a spanning tree, so it is connected.
+□
+Next we note a characterisation of clique graphs due to Szwarcfiter and
+Bornstein.
+Theorem 3.2 ([11, Theorem 2.1]). The graph H is isomorphic to C(G) for
+some connected chordal graph G if and only if H has a spanning tree T such
+that whenever u and v are adjacent in H, the path of T from u to v induces
+a clique of H.
+3.1. Induced cycles. Next we observe that clique graphs can have induced
+cycles of any length. We will later show that this is not true for reduced
+clique graphs. For an integer n ≥ 3 the wheel graph with n spokes is obtained
+from a cycle of n vertices by adding a new vertex and making it adjacent to
+all vertices of the cycle. Thus the wheel graph with n spokes has an induced
+cycle of n vertices.
+Proposition 3.3. For each integer n ≥ 3 the wheel graph with n spokes is
+isomorphic to the clique graph of a chordal graph.
+Proof. This is easy to prove using Theorem 3.2, but we will give a direct
+construction. Start with a clique on the n + 1 vertices u0, u1, . . . , un−1, x.
+For each i ∈ Z/nZ, add a new vertex vi and make it adjacent to ui and ui+1.
+Call the resulting graph G. It is easy to verify that G is chordal, and its
+maximal cliques are {u0, u1, . . . , un−1, x} along with {vi, ui, ui+1} for each
+i ∈ Z/nZ. The result follows.
+□
+Definition 3.4. Let G be a chordal graph. Let C0, C1, . . . , Cn−1 be a cyclic
+ordering of the maximal cliques in an induced cycle of CR(G). We take the
+indices to be from Z/nZ, so Ci and Cj are adjacent in CR(G) if and only if
+
+10
+MAYHEW AND PROBERT
+j ∈ {i − 1, i + 1}. If |Ci ∩ Ci+1| ≤ |Cj ∩ Cj+1| for every j ∈ Z/nZ, then we
+say that the edge between Ci and Ci+1 is a minimal edge of the cycle.
+Lemma 3.5. Let G be a chordal graph. Let C0, C1, . . . , Cn−1 be a cyclic
+ordering of the maximal cliques in an induced cycle of CR(G), where n ≥ 4
+and the indices are from Z/nZ. Assume that the edge between C0 and C1 is
+a minimal edge of the induced cycle. Let S be C0 ∩ C1 and for i = 0, 1 let
+Hi be the connected component of G−S that contains Ci −S. Then H0 and
+H1 are distinct connected components and Ci − S is contained in H0 or H1
+for every i ∈ Z/nZ. Furthermore, either:
+(i) H0 contains all of C0 − S, C2 − S, . . . , Cn−1 − S,
+(ii) H1 contains all of C1 − S, C2 − S, . . . , Cn−1 − S, or
+(iii) n = 4, and H0 contains C0 −S and C2 −S while H1 contains C1 −S
+and C3 − S.
+Proof. Note that because C0, C1, . . . , Cn−1 are distinct maximal cliques of
+G, none of them is contained in S. Thus Ci − S is non-empty for all i. We
+consider the connected components of G − S. Any set Ci − S is contained
+in such a component. Because C0 and C1 form a separating pair, C0 − S
+and C1 − S are contained in different connected components of G − S, so
+H0 and H1 are distinct components.
+Claim 3.5.1. Assume that i and j are distinct indices in Z/nZ such that
+there are distinct connected components of G−S, call them Hi and Hj, that
+contain Ci − S and Cj − S respectively. Assume also that Ci is adjacent in
+CR(G) to Cp, where Cp − S is not contained in Hi and that Cj is adjacent
+to Cq, where Cq − S is not contained in Hj. Then Ci and Cj are adjacent
+in CR(G).
+Proof. Note that because the cycle of CR(G) is induced, p is in {i − 1, i + 1}
+and q is in {j − 1, j + 1}. Note also that Ci ∩ Cp is contained in S. If this
+containment is proper then |Ci ∩ Cp| < |S| = |C0 ∩ C1| and we have violated
+our assumption that the edge between C0 and C1 is minimal. Therefore Ci
+and Cp both contain S. The same argument shows S ⊆ Cj ∩Cq. Now Ci∩Cj
+is equal to S. Moreover Ci − S and Cj − S are in different components of
+G − S, so Ci and Cj form a separating pair of maximal cliques. Hence they
+are adjacent in CR(G).
+□
+We colour the cliques of C0, C1, . . . , Cn−1 in the following way. For each
+i ∈ Z/nZ, if Ci − S is contained in H0 we colour Ci red, and if Cj − S is in
+H1 we colour Ci blue. Thus C0 is red and C1 is blue.
+Claim 3.5.2. Any maximal clique Ci is either red or blue.
+Proof. If the claim fails then there is some i ∈ Z/nZ−{0, 1} such that Ci−S
+is contained in neither H0 nor H1. Let Hi be the connected component of
+G − S that contains Ci − S. We colour any clique Cj in C0, C1, . . . , Cn−1
+green if Cj −S is contained in Hi. We know that the collections of red, blue,
+
+REDUCED CLIQUE GRAPHS
+11
+and green cliques are all non-empty. So therefore we can find a red clique,
+Cred, adjacent to a clique that is not red. We can similarly find Cblue, a blue
+clique that is adjacent to a non-blue clique, and Cgreen, a green clique that
+is adjacent to a clique that is not green. Now Claim 3.5.1 implies that Cred,
+Cblue, and Cgreen are adjacent to each other in CR(G). As they are three
+distinct vertices in an induced cycle of CR(G) with at least four vertices,
+this is an immediate contradiction.
+□
+If C1 is the only blue clique, then statement (i) holds and we have nothing
+left to prove. Similarly, if C0 is the only red clique, then (ii) holds and we
+are done. So we assume there are at least two red cliques and at least two
+blue cliques. We can choose Cred and C′
+red to be distinct red cliques that are
+adjacent to blue cliques, and we can choose Cblue and C′
+blue to be two distinct
+blue cliques that are adjacent to red cliques. Now Claim 3.5.1 implies that
+Cred and C′
+red are adjacent to both Cblue and C′
+blue. Thus the four cliques
+induce a cycle in CR(G). This is impossible if n ≥ 5, so we conclude that
+n = 4. Now C0 is a red clique and it is adjacent to two blue cliques. Thus
+C1 and C3 are blue, C2 is red, and we are finished.
+□
+The example in Figure 1 shows that a reduced clique graph may contain
+an induced cycle with four vertices.
+We will next show that there is no
+example with an induced cycle of five vertices.
+Lemma 3.6. There is no chordal graph G such that CR(G) has an induced
+cycle with exactly five vertices.
+Proof. Assume otherwise and let G be a chordal graph such that CR(G)
+contains an induced cycle with five vertices. Let C0, C1, C2, C3, C4 be the
+maximal cliques in this cycle, where the indices are from Z/5Z and Ci is
+adjacent to Cj if and only if j ∈ {i−1, i+1}. By adding a constant to these
+indices as necessary, we may assume that
+|C0 ∩ C1| ≤ |Ci ∩ Ci+1|
+for all i ∈ Z/5Z, so that the edge between C0 and C1 is a minimal edge of
+the cycle. Let S be C0 ∩ C1. Note that S is non-empty.
+Now we apply Lemma 3.5. By applying the permutation ρ: i �→ 1 − i
+as necessary, we may assume that statement (ii) in Lemma 3.5 applies.
+Therefore we let H0 and H1 be connected components of G − S such that
+H0 contains C0 − S and H1 contains C1 − S, C2 − S, C3 − S, and C4 − S.
+Claim 3.6.1. C0 ∩ C4 = S = C0 ∩ C1.
+Proof. Because C0 − S and C4 − S are contained in different components of
+G − S, it follows that C0 ∩ C4 ⊆ S. All we have left to prove is that this
+containment is not proper. If it were proper, then we would contradict the
+assumption that the edge between C0 and C1 is minimal.
+□
+Claim 3.6.2. Neither C2 nor C3 contains S.
+
+12
+MAYHEW AND PROBERT
+Proof. Note that C0 ∩ C2 ⊆ S because C0 − S and C2 − S are contained
+in different components of G − S.
+Certainly any path from a vertex of
+C0 − C2 to a vertex of C2 − C0 must use a vertex of S. If C0 ∩ C2 = S,
+then C0 and C2 form a separating pair, so C0 and C2 are adjacent in CR(G).
+This contradicts the fact that C0 and C2 are non-consecutive vertices in an
+induced cycle. The same argument shows that C3 does not contain S.
+□
+Claim 3.6.3. C2 ∩ C4 ⊆ C1 and C3 ∩ C1 ⊆ C4.
+Proof. Assume that x is a vertex of C2 ∩ C4 that is not in C1. By Claim
+3.6.2 we can let y be a vertex in S − C2. Thus y is in C1 − C2. So x is in
+C2 −C1 and y is in C1 −C2. Claim 3.6.1 implies that y is in C4. As x is also
+in C4 we see that x and y are adjacent. Because C1 and C2 are adjacent in
+CR(G) they have a non-empty intersection, but now the edge xy shows that
+C1 and C2 do not form a separating pair and we have a contradiction. A
+symmetric argument shows C3 ∩ C1 ⊆ C4.
+□
+Claim 3.6.4. C2 contains a vertex of C1 − C4 and C3 contains a vertex of
+C4 − C1.
+Proof. By symmetry it suffices to prove the first statement. Assume that C2
+contains no vertex of C1 − C4. Because C1 and C2 are adjacent in CR(G),
+they have at least one vertex in common. By our assumption, no vertex of
+C1 ∩ C2 is in C1 − C4, so any such vertex must be in C1 ∩ C4. Therefore C2
+and C4 are not disjoint. Since C2 and C4 are not adjacent in CR(G), we can
+let P be a (C2 ∩C4)-avoiding path from a vertex x ∈ C2 −C4 to y ∈ C4 −C2.
+Our assumption means that x is not in C1 − C4, so it is in C2 − C1. Our
+assumption and Claim 3.6.3 imply that C2 ∩ C4 = C2 ∩ C1. Therefore P
+is a (C2 ∩ C1)-avoiding path. But Claim 3.6.2 shows that we can choose a
+vertex z in S − C2. Thus z is in C1 − C2 and Claim 3.6.1 shows that z is
+in C4. Assuming that z and y are not equal, they are adjacent, as both are
+in C4. By appending (if necessary) the edge yz to the end of P we obtain
+a (C1 ∩ C2)-avoiding path from a vertex in C2 − C1 to a vertex in C1 − C2.
+Hence C1 and C2 do not form a separating pair and this contradicts the fact
+that they are adjacent in CR(G).
+□
+Claim 3.6.5. Either C2 ∩ (C1 ∩ C4) ⊆ C3 or C3 ∩ (C1 ∩ C4) ⊆ C2.
+Proof. Note that C2 ∩ C3 is non-empty, since C2 and C3 are adjacent in
+CR(G). If the claim fails, then we choose x ∈ (C2 ∩ C1 ∩ C4) − C3 and
+y ∈ (C3 ∩ C1 ∩ C4) − C2. Now x and y are both in C1 ∩ C4, so they are
+adjacent. Moreover x is in C2 − C3 and y is in C3 − C2. Thus C2 and C3 do
+not form a separating pair and we have a contradiction.
+□
+By using Claim 3.6.5, we will assume that C2 ∩ (C1 ∩ C4) is a subset of
+C3. The other outcome from Claim 3.6.5 yields to a symmetric argument.
+Using Claim 3.6.2 we choose a vertex x ∈ S that is not in C3. Note that
+Claim 3.6.1 implies that S is contained in C1 ∩C4. So x is in (C1 ∩C4)−C3.
+Our assumption therefore implies that x is not in C2.
+
+REDUCED CLIQUE GRAPHS
+13
+By Claim 3.6.4 we can also choose y in C2 ∩(C1 −C4) and z in C3 ∩(C4 −
+C1). Claim 3.6.3 implies that y is in C2 −C3 and z is in C3 −C2. Now x and
+y are adjacent as they are both in C1, and x and z are adjacent as they are
+both in C4. Note that C2 ∩ C3 is non-empty as C2 and C3 are adjacent in
+CR(G). But the path with vertex sequence y, x, z is (C2 ∩ C3)-avoiding, so
+C2 and C3 do not form a separating pair. This final contradiction completes
+the proof.
+□
+Lemma 3.6 shows that the class of reduced clique graphs is contained in
+the class of graphs with no length-five induced cycle. We next show that
+this containment is proper.
+Proposition 3.7. Let n ≥ 4 be an integer. There is no chordal graph G
+such that either C(G) or CR(G) is a cycle with n vertices.
+Proof. Szwarcfiter and Bornstein characterise the clique graphs of chordal
+graphs [11]. In particular H is isomorphic to C(G) for some chordal graph
+G if and only if H has a spanning tree T such that whenever u and v are
+adjacent in H, the path of T from u to v induces a clique of H.
+Now
+assume H is a cycle with at least four vertices. Any spanning tree of H is a
+Hamiltonian path. The end vertices of this path are adjacent in H, but the
+path of the spanning tree between these vertices does not induce a clique.
+Therefore H is not isomorphic to C(G) for any chordal graph G.
+We turn to reduced chordal graphs. Assume for a contradiction that G is
+a chordal graph with C0, C1, . . . , Cn−1 as its list of maximal cliques, where
+the indices are from Z/nZ, and Ci is adjacent to Cj in CR(G) if and only if
+j ∈ {i − 1, i + 1}. We can assume without loss of generality that the edge
+between C0 and C1 is a minimal edge of CR(G). Let S be C0 ∩ C1. Assume
+that statement (iii) in Lemma 3.5 holds. Thus n = 4 and there are distinct
+connected components, H0 and H1, of G − S such that H0 contains C0 − S
+and C2 − S while H1 contains C1 − S and C3 − S. Note that C0 ∩ C3 ⊆ S,
+and in fact C0 ∩ C3 is equal to S, or else the minimality of the C0-C1 edge
+is contradicted.
+Either C0 ∩ C2 is empty, or it is not. In the latter case, we can apply
+Proposition 2.4 to C0 and C2.
+We see that either C0 ∩ C1 or C0 ∩ C3
+properly contains C0 ∩C2. By symmetry, we can assume C0 ∩C2 is a proper
+subset of C0 ∩ C1 = S. Thus C0 − S and C2 − S are disjoint sets. They are
+contained in the same connected component of G − S, so we can let P be a
+shortest-possible path of H0 from a vertex of C0 − S to a vertex of C2 − S.
+On the other hand, if C0 ∩ C2 is empty, then C0 − S and C2 − S are again
+disjoint subsets in H0, so we again let P be a shortest-possible path of H0
+from C0 − S to C2 − S. In either case, P contains exactly one vertex of C0
+and exactly one vertex of C2. Then P must contain at least one edge, and
+this edge is in a maximal clique that is equal to neither C0 nor C2. Nor can
+this maximal clique be C1 or C3, because any edge of P is contained in H0.
+So we have a contradiction in the case that (iii) in Lemma 3.5 holds.
+
+14
+MAYHEW AND PROBERT
+Now we assume that either (i) or (ii) holds. By applying the permutation
+ρ: i �→ 1 − i as necessary, we will assume that H0 and H1 are distinct
+connected components of G − S, and that H0 contains C0 − S while H1
+contains Ci − S for i = {1, 2, . . . , n − 1}. By the same argument as earlier,
+we can see that Cn−1 contains S, or else the choice of the C0 − C1 edge is
+contradicted.
+Now C1 ∩ Cn−1 contains S, and C1 and Cn−1 are non-adjacent in CR(G).
+We apply Proposition 2.4 and see that there is a path of CR(G) from C1 to
+Cn−1 such that every intersection of consecutive cliques in the path properly
+contains C1∩Cn−1. This path is either C1, C0, Cn−1, or it is C1, C2, . . . , Cn−1.
+Assume the former. Then C1 ∩ C0 = S properly contains C1 ∩ Cn−1 ⊇ S
+and we have a contradiction. Hence any intersection of consecutive cliques
+in C1, C2, . . . , Cn−1 properly contains C1 ∩ Cn−1, and hence contains S. It
+follows that C2 contains S and thus C0 ∩ C2 is non-empty.
+Since C0 − S and C2 − S are contained in different components of G − S,
+any path of a vertex from C0 − C2 to a vertex of C2 − C0 must contain a
+vertex of S = C0 ∩ C2. Thus C0 and C2 form a separating pair in G, and
+hence they are adjacent in CR(G), which is a contradiction.
+□
+3.2. Clique graphs vs. reduced clique graphs. Consider the classes
+{C(G)} and {CR(G)}, where G ranges over all chordal graphs. Proposition
+3.3 and Lemma 3.6 show that the wheel with five spokes is isomorphic to
+a graph in the former class but not the latter.
+Is there a graph that is
+isomorphic to a graph in the latter class but not the former? We will show
+that the answer is, once again, yes. Recall that if G and G′ are disjoint
+graphs, then G ⊠ G′ is obtained from the union of G and G′ by making
+every vertex of G adjacent to every vertex of G′. We use Pn to denote the
+path of length n.
+Lemma 3.8. Let m, n ≥ 1 be integers. Then Pm ⊠ Pn is isomorphic to
+the reduced clique graph of a chordal graph. If n ≥ 22, then Pn ⊠ Pn is not
+isomorphic to the clique graph of a chordal graph.
+Proof. Let G be the graph obtained from the disjoint union of Pm and Pn
+and adding a new vertex that is adjacent to every vertex of the disjoint
+union. It is easy to confirm that G is chordal, and that CR(G) is isomorphic
+to Pm ⊠ Pn.
+For the second statement, we let H be a graph with disjoint induced paths
+Pu = u0, u1, . . . , un−1 and Pv = v0, v1, . . . , vn−1, where n ≥ 22 and every ui
+is adjacent to every vj. Thus H is isomorphic to Pn ⊠ Pn. We will assume
+for a contradiction that H is isomorphic to C(G) for some chordal graph G.
+Because C(G) is connected it follows easily that G is connected, so we can
+apply Theorem 3.2 and deduce that H has a spanning tree T, where the
+path of T from u to v induces a clique of H whenever u and v are adjacent
+in H.
+
+REDUCED CLIQUE GRAPHS
+15
+Claim 3.8.1. Let i and j be integers satisfying 0 < i, j < n−1. The path of
+T from ui to vj is contained in one of: {ui, ui+1, vj, vj+1}, {ui, ui+1, vj−1, vj},
+{ui−1, ui, vj, vj+1}, {ui−1, ui, vj−1, vj}.
+Proof. Let P be the path of T from ui to vj. Since ui is adjacent to vj it
+follows that P induces a clique of H. As ui is not adjacent to any of the
+vertices in u0, . . . , ui−2, ui+2, . . . , un−1, it follows that the vertices of P that
+are in Pu belong to {ui−1, ui, ui+1}. Similarly, the vertices of P that are in
+Pv belong to {vj−1, vj, vj+1}. But ui−1 is not adjacent to ui+1, so P does
+not contain both. The claim follows by symmetry.
+□
+Claim 3.8.1 implies that the path of T between ui and vj has at most
+three edges.
+Let P be a longest-possible path of T and let p0, p1, . . . , pk−1 be the ver-
+tices of P. For i = 0, 1, . . . , k − 1, let Ui be the set of vertices in Pu such
+that u is in Ui if and only if the shortest path of T from u to a vertex in P
+contains pi. We define Vi to be the analogous set of vertices in Pv. Note that
+(U0, U1, . . . , Uk−1) is a partition of the vertices of Pu, and (V0, V1, . . . , Vk−1)
+is a partition of the vertices of Pv.
+Claim 3.8.2. Either
+max{|Ui|: 0 ≤ i ≤ k − 1} ≤ 3
+or
+max{|Vi|: 0 ≤ i ≤ k − 1} ≤ 3.
+Proof. Assume for a contradiction that |Ui| ≥ 4 and |Vj| ≥ 4. Let p and q,
+respectively, be the smallest (largest) integers such that up, uq ∈ Ui. Then
+q − 1 > p + 1 because |Ui| ≥ 4. In the same way, let s and t be the smallest
+(largest) integers such that vs, vt ∈ Vj. Then t−1 > s+1. It is simple to see
+from Claim 3.8.1 that the path of T from up to vs has no vertex in common
+with the path of T from uq to vt. But this contradicts the fact that both
+paths contain pi and pj.
+□
+By using Claim 3.8.2, we will assume without loss of generality that |Vi| ≤
+3 for each i = 0, 1, . . . , k − 1. Since (U0, U1, . . . , Uk−1) is a partition of the
+vertices in Pu we can choose i so that Ui contains a vertex x. We claim that
+if j ≤ i − 4 or j ≥ i + 4, then Vj = ∅. If this fails, then the path of T from a
+vertex in Vj to x contains at least four edges of P. But this contradicts our
+earlier conclusion that any path of T from a vertex of Pu to a vertex of Pv
+contains at most three edges. So now the vertices of Pv belong to
+Vi−3 ∪ Vi−2 ∪ · · · ∪ Vi+2 ∪ Vi+3
+and this union has cardinality at most 7 × 3. Thus Pv contains at most 21
+vertices and this contradicts n ≥ 22.
+□
+4. Conclusions and open problems
+Given Lemma 3.6 it might be natural to believe that reduced clique graphs
+cannot have any induced cycles with five or more vertices. But Figure 3
+shows a chordal graph G where CR(G) has an induced cycle with six vertices.
+
+16
+MAYHEW AND PROBERT
+5
+10
+235
+346
+1234
+G
+C(G)
+CR(G)
+2
+4
+3
+6
+1
+8
+9
+3479
+2378
+710
+2347
+2347
+235
+346
+1234
+3479
+2378
+710
+2347
+2347
+7
+Figure 3.
+Nonetheless we believe the following to be true.
+Conjecture 4.1. There is no chordal graph G such that CR(G) contains
+an induced cycle with seven or more vertices.
+So far as we have been able to tell, every chordal graph is isomorphic
+to both a clique graph, and to a reduced clique graph. We conjecture this
+holds generally.
+Conjecture 4.2. Let H be a chordal graph. There are chordal graphs G
+and G′ such that H is isomorphic to both C(G) and CR(G′).
+Szwarcfiter and Bornstein present a polynomial-time algorithm for decid-
+ing whether a given graph is isomorphic to C(G) for some chordal graph G
+[11]. Their techniques do not obviously extend to recognising reduced clique
+graphs. Nonetheless, we will make the following conjecture.
+Conjecture 4.3. There is a polynomial-time algorithm for deciding whether
+a given graph is isomorphic to CR(G) for some chordal graph G.
+More informally, we ask if there is a structural description for reduced
+clique graphs that is analogous to Theorem 3.2.
+References
+[1] Jean R. S. Blair and Barry Peyton, An introduction to chordal graphs and clique
+trees, Graph theory and sparse matrix computation, IMA Vol. Math. Appl., vol. 56,
+Springer, New York, 1993, pp. 1–29.
+[2] Peter Buneman, A characterisation of rigid circuit graphs, Discrete Math. 9 (1974),
+205–212.
+[3] Philippe Galinier, Michel Habib, and Christophe Paul, Chordal graphs and their clique
+graphs, Graph-theoretic concepts in computer science (Aachen, 1995), Lecture Notes
+in Comput. Sci., vol. 1017, Springer, Berlin, 1995, pp. 358–371.
+[4] F˘anic˘a Gavril, The intersection graphs of subtrees in trees are exactly the chordal
+graphs, J. Combinatorial Theory Ser. B 16 (1974), 47–56.
+
+REDUCED CLIQUE GRAPHS
+17
+[5] Michel Habib and Vincent Limouzy, On some simplicial elimination schemes for
+chordal graphs, DIMAP Workshop on Algorithmic Graph Theory, Electron. Notes
+Discrete Math., vol. 32, Elsevier Sci. B. V., Amsterdam, 2009, pp. 125–132.
+[6] Michel Habib and Juraj Stacho, Reduced clique graphs of chordal graphs, European
+J. Combin. 33 (2012), no. 5, 712–735.
+[7] Terry A. McKee, Minimal weak separators of chordal graphs, Ars Combin. 101 (2011),
+321–331.
+[8] Yasuko Matsui, Ryuhei Uehara, and Takeaki Uno, Enumeration of the perfect se-
+quences of a chordal graph, Theoret. Comput. Sci. 411 (2010), no. 40-42, 3635–3641.
+[9] Dillon Mayhew and Andrew Probert, Supersolvable saturated matroids and chordal
+graphs. In preparation.
+[10] Donald J. Rose, Triangulated graphs and the elimination process, J. Math. Anal. Appl.
+32 (1970), 597–609.
+[11] Jayme L. Szwarcfiter and Claudson F. Bornstein, Clique graphs of chordal and path
+graphs, SIAM J. Discrete Math. 7 (1994), no. 2, 331–336.
+

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@@ -0,0 +1,1715 @@
+arXiv:2301.05540v1  [math.NA]  13 Jan 2023
+SOLVING PDES WITH INCOMPLETE INFORMATION
+PETER BINEV, ANDREA BONITO, ALBERT COHEN, WOLFGANG DAHMEN
+RONALD DEVORE, AND GUERGANA PETROVA
+Abstract. We consider the problem of numerically approximating the solutions to a partial differential
+equation (PDE) when there is insufficient information to determine a unique solution. Our main example
+is the Poisson boundary value problem, when the boundary data is unknown and instead one observes
+finitely many linear measurements of the solution. We view this setting as an optimal recovery problem
+and develop theory and numerical algorithms for its solution. The main vehicle employed is the derivation
+and approximation of the Riesz representers of these functionals with respect to relevant Hilbert spaces of
+harmonic functions.
+1. Introduction
+The questions we investigate sit in the broad research area of using measurements to enhance the numer-
+ical recovery of the solution u to a PDE. The particular setting addressed in this paper is to numerically
+approximate the solution to an elliptic boundary value problem when there is insufficient information on the
+boundary value to determine a unique solution to the PDE. In place of complete boundary information, we
+have a finite number of data observations of the solution u. This data serves to narrow the set of possible
+solutions. We ask what is the optimal accuracy to which we can recover u and what is a near optimal
+numerical algorithm to approximate u. Problems of this particular type arise in several fields of science and
+engineering (see e.g. [28, 3, 7] for examples in fluid dynamics), where a lack of full information on boundary
+conditions arises for various reasons. For example, the correct physics might not be fully understood [22, 24],
+or the boundary values are not accessible [11], or they must be appropriately modified in numerical schemes
+[8, 23]. Other examples of application domains for the results of the present paper can be found in the
+introduction of [9].
+1.1. A model for PDEs with incomplete data. In this paper, we consider the model elliptic problem
+(1.1)
+− ∆u = f
+in Ω,
+u = g
+on Γ := ∂Ω,
+where Ω ⊂ Rd is a bounded Lipschitz domain with d = 2 or 3. The Lax-Milgram theorem [29] implies the
+existence and uniqueness of a solution u from the Sobolev space H1(Ω) to (1.1), once f and g are prescribed
+in H−1(Ω) (the dual of H1
+0(Ω)) and in H1/2(Γ) (the image of H1(Ω) by the trace operator), respectively.
+Recall that the trace operator T is defined on a function w ∈ C(¯Ω) as the restriction of w to Γ and this
+definition is then generalized to functions in Sobolev spaces by a denseness argument. In particular, the
+trace operator is well defined on H1(Ω). For any function v in H1(Ω) we denote by vΓ its trace,
+(1.2)
+vΓ := T (v) = v|Γ,
+v ∈ H1(Ω).
+The Lax-Milgram analysis also yields the inequalities
+(1.3)
+c0∥v∥H1(Ω) ≤ ∥∆v∥H−1(Ω) + ∥vΓ∥H1/2(Γ) ≤ c1∥v∥H1(Ω),
+v ∈ H1(Ω).
+Here the constants c0, c1 depend on Ω and on the particular choice of norms employed on H1(Ω) and H1/2(Γ).
+Our interest centers on the question of how well we can numerically recover u in the H1 norm when we
+do not have sufficient knowledge to guarantee a unique solution to (1.1). There are many possible settings
+to which our techniques apply, but we shall focus on the following scenario:
+Date: January 16, 2023.
+This research was supported by the NSF Grants DMS 2110811 (AB), DMS 2038080 (PB and WD), DMS-2012469 (WD),
+DMS 21340077 (RD and GP), the MURI ONR Grant N00014-20-1-278 (RD and GP), the ARO Grant W911NF2010318 (PB),
+and the SFB 1481, funded by the German Research Foundation (WD).
+1
+
+(i) We have a complete knowledge of f but we do not know g.
+(ii) The function g belongs to a known compact subset KB of H
+1
+2 (Γ).
+Thus, membership in KB
+describes our knowledge of the boundary data. The function u we wish to recover comes from the
+set
+(1.4)
+K := {u : u solves (1.1) for some g ∈ KB},
+which is easily seen from (1.3) to be a compact subset of H1(Ω).
+(iii) We have access to finitely many data observations of the unknown solution u, in terms of a vector
+(1.5)
+λ(u) := (λ1(u), . . . , λm(u)) ∈ Rm,
+where the λj are fixed and known linear functionals defined on the functions from K.
+Natural candidates for the compact set KB are balls of Sobolev spaces that are compactly embedded in
+H
+1
+2 (Γ). We thus restrict our attention for the remainder of this paper to the case
+(1.6)
+KB := U(Hs(Γ)),
+for some s > 1
+2,where the precise definition of Hs(Γ) and its norm ∥ · ∥Hs(Γ) is described later. Note that
+U(Y ) denotes the unit ball of a Banach space Y with respect to the norm ∥ · ∥Y .
+1.2. The optimal recovery benchmark. Let wj := λj(u), j = 1, . . . , m, and
+(1.7)
+w := (w1, . . . , wm) = λ(u) ∈ Rm,
+be the vector of data observations. Therefore, the totality of information we have about u is that it lies in
+the compact set
+(1.8)
+Kw := {u ∈ K : λ(u) = w}.
+Our problem is to numerically find a function ˆu ∈ H1(Ω) which approximates simultaneously all the
+u ∈ Kw. This is a special case of the problem of optimal recovery from data (see [15, 27, 19]). The optimal
+recovery, i.e. the best choice for ˆu, has the following well known theoretical description. Let B(Kw) be a
+smallest ball in H1(Ω) which contains Kw and let R(Kw) := R(Kw)H1(Ω) be its radius. Then, R(Kw) is the
+optimal recovery error, that is, the smallest error we can have for recovering u in the norm of H1(Ω), and
+the center of B(Kw) is an optimal recovery of u.
+We are interested in understanding how small R(Kw) is and what are the numerical algorithms which are
+near optimal in recovering u from the given data w. We say that an algorithm w �→ ˆu = ˆu(w) delivers near
+optimal recovery with constant C if
+(1.9)
+∥u − ˆu(w)∥H1(Ω) ≤ CR(Kw),
+w ∈ Rm.
+Of course, we want C to be a reasonable constant independent of m. Our results actually deliver a recovery
+estimate of the form
+(1.10)
+∥u − ˆu(w)∥H1(Ω) ≤ R(Kw) + ε,
+w ∈ Rm,
+where ε > 0 can made arbitrarily small at the price of higher computational cost. In this sense, the recovery
+is near optimal with constant C > 1 in (1.9) that can be made arbitrarily close to 1.
+1.3. A connection with the recovery of harmonic functions. There is a natural restatement of our
+recovery problem in terms of harmonic functions. Let f be the right side of (1.1), where f is a known fixed
+element of H−1(Ω). Let u0 be the function in H1(Ω) which is the solution to (1.1) with g = 0. Then, we
+can write any function u ∈ K as
+(1.11)
+u = u0 + uH,
+where uH is a harmonic function in H1(Ω) which has boundary value g = T (uH) with g ∈ KB. Recall our
+assumption that KB is the unit ball of Hs(Γ) with s > 1
+2.
+2
+
+Let Hs(Ω) denote the set of harmonic functions v defined on Ω for which vΓ ∈ Hs(Γ). We refer the reader
+to [2], where a detailed study of spaces like Hs(Ω) is presented. We define the norm on Hs(Ω) to be the one
+induced by the norm on Hs(Γ), namely,
+(1.12)
+∥v∥Hs(Ω) := ∥vΓ∥Hs(Γ),
+v ∈ Hs(Ω).
+There exist several equivalent definitions of norms on Hs(Γ), as discussed later. For the moment, observe
+that from (1.3) it follows the existence of a constant Cs such that
+(1.13)
+∥v∥H1(Ω) ≤ Cs∥v∥Hs(Ω),
+v ∈ Hs(Ω).
+Indeed, the space Hs(Ω) is a Hilbert space that is compactly embedded in H1(Ω), as a consequence of the
+compact embedding of Hs(Γ) in H1/2(Γ). We denote by KH the unit ball of Hs(Ω),
+(1.14)
+KH := U(Hs(Ω)).
+Since the function u0 in (1.11) is fixed, it follows from (1.6) that
+(1.15)
+R(Kw) = R(KH
+w′)H1(Ω),
+w′ := λ(uH) = w − λ(u0).
+There are two conclusions that can be garnered from this reformulation. The first is that the optimal
+error in recovering u ∈ Kw is the same as that in recovering the harmonic function uH ∈ KH
+w′ in the H1(Ω)
+norm. The harmonic recovery problem does not involve f except in determining w′. The second point is
+that one possible numerical algorithm for our original problem is to first construct a sufficiently accurate
+approximation ˆu0 to u0 and then to numerically implement an optimal recovery of a harmonic function in
+KH from data observations. This numerical approach requires the computation of w′. In theory, u0 is known
+to us since we have a complete knowledge of f. However, u0 must be computed and any approximation ˆu0
+will induce an error. Although this error can be made arbitrarily small, it means that we only know w′ up to
+a certain numerical accuracy. One can thus view the harmonic reformulation as an optimal recovery problem
+with perturbed observations of w′. The numerical algorithm presented here follows this approach. Its central
+constituent, namely the recovery of harmonic functions from a finite number of noisy observations, can be
+readily employed as well in a number of different application scenarios described e.g. in [9].
+1.4. Objectives and outline. Our main goal is to create numerical algorithms which are guaranteed to
+produce a function ˆu which is near optimal and to discuss their practical implementation. We begin in §2
+with some remarks on the definition of the space Hs(Γ) and its norm, which are of importance both in the
+accuracy analysis and the practical implementation of recovery algorithms.
+The general approach for optimal recovery that was introduced in [15, 14] is recalled in §3. We describe
+a solution algorithm which takes into consideration the effect of numerical perturbations. We first consider
+the case when the linear functionals λj are defined on all of H1(Ω) and then adapt this algorithm to the
+case when the linear functionals are point evaluations
+(1.16)
+λj(u) := u(xj),
+xj ∈ Ω,
+j = 1, . . . , m.
+Point evaluations are not defined on all of H1(Ω) when d > 1, however, they are defined on K when the
+smoothness order s is large enough.
+The critical ingredient in our proposed algorithm is the numerical computation of the Riesz representers
+φj of the restrictions of λj to the Hilbert space Hs(Ω). Each of these Riesz representers is characterized
+as a solution to an elliptic problem and can be computed offline since it does not involve the measurement
+vector w. Our suggested numerical method for approximating φj is based on finite element discretizations
+and is discussed in §4. We establish quantitative error bounds for the numerical approximation in terms of
+the mesh size. Numerical illustrations of the optimal recovery algorithm are given in §5.
+Note that the optimal recovery error over the class K strongly depends on the choice of the linear
+functionals λj. For example, in the case of point evaluation, this error can be very large if the data sites
+{xj}m
+j=1 are poorly positioned, or small if they are optimally positioned. This points to the importance of
+the Gelfand widths and sampling numbers. They describe the optimal recovery error over K with optimal
+choice of functionals in the general case and the point evaluation case, respectively. The numerical behaviour
+of these quantities in our specific setting is discussed in §6.
+3
+
+2. The spaces Hs(Γ) and Hs(Ω)
+In this section, we discuss the definition and basic properties of the spaces Hs(Γ) and Hs(Ω). We refer
+to [1] for a general treatment of Sobolev spaces on domains D ⊂ Rd. Recall that for fractional orders r > 0,
+the norm of Hr(D) is defined as
+∥v∥2
+Hr(D) := ∥v∥2
+Hk(D) +
+�
+|α|=k
+�
+D×D
+|∂αv(x) − ∂αv(y)|2
+|x − y|d+2(r−k)
+dxdy,
+where k is the integer such that k < r < k+1, and ∥v∥2
+Hk(D) := �
+|α|≤k ∥∂αv∥2
+L2(D) is the standard Hk-norm.
+2.1. Equivalent definitions of Hs(Γ). Let Ω be any bounded Lipschitz domain in Rd. We recall the trace
+operator T introduced in §1.1. One first possible definition of the space Hs(Γ), for any s ≥ 1
+2, is as the
+restriction of Hs+ 1
+2 (Ω) to Γ, that is,
+Hs(Γ) = T (Hs+ 1
+2 (Ω)),
+with norm
+(2.1)
+∥g∥Hs(Γ) := min
+�
+∥v∥Hs+ 1
+2 (Ω) : vΓ = g
+�
+.
+The resulting norm is referred to as the trace norm definition for Hs(Γ).
+There is a second, more intrinsic way to define Hs(Γ), by properly adapting the notion of Sobolev
+smoothness to the boundary. This can be done by locally mapping the boundary onto domains of Rd−1
+and requiring that the pullback of g by such transformation have Hs smoothness on such domains. We refer
+the reader to [10] and [17] for the complete intrinsic definition, where it is proved to be equivalent to the
+trace definition for a range of s that depends on the smoothness of the boundary Γ.
+For small values of s, Sobolev norms for Hs(Γ) may also be equivalently defined without the help of local
+parameterizations, as contour integrals. For example, if 0 < s < 1 and Ω is a Lipschitz domain, we define
+∥g∥2
+Hs(Γ) := ∥g∥2
+L2(Γ) +
+�
+Γ×Γ
+|g(x) − g(y)|2
+|x − y|d−1+2s dxdy,
+and if s = 1 and Ω is a polygonal domain, we define
+(2.2)
+∥g∥2
+H1(Γ) := ∥g∥2
+L2(Γ) + ∥∇Γg∥2
+L2(Γ),
+where ∇Γ is the tangential gradient, and likewise
+∥g∥2
+Hs(Γ) := ∥g∥2
+H1(Γ) +
+�
+Γ×Γ
+|∇Γg(x) − ∇Γg(y)|2
+|x − y|d−1+2(s−1) dxdy,
+for 1 < s < 2. In the numerical illustration given in §5, we will specifically take the value s = 1 and a square
+domain, using the definition (2.2).
+When Ω has smooth boundary, it is known that the trace definition and intrinsic definition of the Hs(Γ)
+norms are equivalent for all s ≥ 1/2. On the other hand, when Ω does not have a smooth boundary, it is
+easily seen that the two definition are not equivalent unless restrictions are made on s. Consider for example
+the case of polygonal domains of R2: it is easily seen that the trace vΓ of a smooth function v ∈ C∞(Ω)
+has a tangential gradient ∇ΓvΓ that generally has jump discontinuities at the corner points and thus does
+not belong to H1/2(Γ). In turn, the equivalence between the trace and intrinsic norms only holds for s < 3
+2
+and in such case we limit the value of s to this range. The same restriction s < 3/2 applies to a polyhedral
+domain in the case d = 3.
+2.2. The regularity of functions in Hs(Ω). We next give some remarks on the Sobolev smoothness of
+functions from the space Hs(Ω) when s > 1/2. Clearly such harmonic functions are infinitely smooth inside
+Ω and also belong to H1(Ω), but one would like to know for which value of r they belong to Hr(Ω). To
+answer this question, we consider v ∈ Hs(Ω). By the definition of Hs(Ω), v is harmonic in Ω and vΓ ∈ Hs(Γ).
+4
+
+Having assumed that s in the admissible range where all above definitions of the Hs(Γ) norms are equivalent,
+and using the first one, we know that there exists a function ˜v ∈ Hs+ 1
+2 (Ω) such that ˜vΓ = vΓ
+∥˜v∥Hs+ 1
+2 (Ω) = ∥vΓ∥Hs(Γ) = ∥v∥Hs(Ω).
+We define v := v − ˜v so that v = ˜v +v. We are interested in the regularity of v since it will give the regularity
+of v. Notice that vΓ = 0 and
+−∆v = f := ∆˜v.
+The function f belongs to the Sobolev space Hs− 3
+2 (Ω) and we are left with the classical question of the
+regularizing effect in Sobolev scales when solving the Laplace equation with Dirichlet boundary conditions.
+Obviously, when Ω is smooth, we find that v ∈ Hs+ 1
+2 (Ω) and so we have obtained the continuous embedding
+Hs(Ω) ⊂ Hr(Ω),
+r = s + 1
+2.
+For less smooth domains, the smoothing effect is limited (in particular by the presence of singularities on
+the boundary of Ω), i.e., v is only guaranteed to be in Hr(Ω) where r may be less than s + 1/2, see [10].
+More precisely
+Hs(Ω) ⊂ Hr(Ω),
+where
+(2.3)
+r := min
+�
+s + 1
+2, r∗�
+,
+Here, r∗ = r∗(Ω) is the limiting bound for the smoothing effect:
+(i) For smooth domains r∗ = ∞.
+(ii) For convex domains r∗ = 2.
+(iii) For non-convex polygonal domains in R2, or a polyhedron in R3, one has 3/2 < r∗ < 2 where the
+value of r∗ depends on the reentrant angles.
+(iv) In particular for polygons, we can take r∗ = 1 + π
+ω − ε, for any ε > 0 where ω is the largest inner
+angle.
+Note that r∗ could be strictly smaller than s + 1
+2.
+In summary, for an admissible range of r > 1 that depends on s and Ω one has the continuous embedding
+Hs(Ω) ⊂ Hr(Ω), and so there exists a constant C1 that depends on (r, s) and Ω, such that
+(2.4)
+∥v∥Hr(Ω) ≤ C1∥v∥Hs(Ω) = C1∥vΓ∥Hs(Γ),
+v ∈ Hs(Ω).
+3. A near optimal recovery algorithm
+In this section, we present a numerical algorithm for solving (1.1) when the information about the bound-
+ary value g is incomplete. We first work under the assumption that the λj’s are continuous over H1(Ω), and
+assumed to be linearly independent (linear independence can be guaranteed by throwing away dependent
+functionals when necessary). We prove that the proposed numerical recovery algorithm is near optimal.
+We then adapt our approach to the case where the λj’s are point evaluations, see (1.16), and therefore not
+continuous over H1(Ω) when d ≥ 2.
+3.1. Minimum norm data fitting. As noted in §1.3, the problem of recovering u ∈ Kw is directly related
+to the problem of recovering the harmonic component uH ∈ KH from the given data observations w′. Note
+that KH is the unit ball of the Hilbert space Hs(Ω). There is a general approach for optimal recovery
+from data observations in this Hilbert space setting, as discussed e.g. in [15]. We first describe the general
+principles of this technique and then apply them to our specific setting.
+Let H be any Hilbert space and suppose that λ1, . . . , λm ∈ H∗ are linearly independent functionals from
+H∗. Let X be a Banach space such that H is continuously embedded in X. We are interested in optimal
+recovery of a function v in the norm ∥ · ∥X, knowing that v ∈ K := U(H), the unit ball of H. If w ∈ Rm is
+the vector of observations, we define the minimal norm interpolant as
+v∗(w) = argmin{∥v∥H : v ∈ H and λ(v) = w}.
+5
+
+It is easily checked that when Kw is non-empty, the function v∗(w) coincides with the Chebyshev center of
+Kw in X. To see this, first note that any v ∈ Kw may be written as v = v∗(w) + η where η belongs to the
+null space N of λ. Because v∗(w) has minimal norm, v − v∗(w) = η is orthogonal to v∗(w) and hence from
+the Pythagorean theorem
+∥v − v∗∥2
+H = ∥v∥2
+H − ∥v∗(w)∥2
+H ≤ 1 − ∥v∗(w)∥2
+H =: r2,
+because ∥v∥H ≤ 1. Notice that v∗(w) − η is also in Kw. It follows that Kw is precisely the ball in the affine
+space v∗(w) + N centered at v∗(w) and of radius r. In particular, Kw is centrally symmetric around v∗(w).
+Therefore, v∗(w) is the Chebyshev center for Kw for any norm, in particular for the ∥ · ∥X norm. Therefore,
+∥v − v∗(w)∥X ≤ R(Kw)X,
+v ∈ Kw,
+that is, the minimal norm interpolant gives optimal recovery with constant C = 1.
+Standard Hilbert space analysis shows that the mapping w �→ v∗(w) is a linear operator. More importantly,
+it has a natural expression that is useful for numerical computation. Namely, from the Riesz representation
+theorem each λj can be described as
+λj(v) = ⟨v, φj⟩H,
+v ∈ H,
+where φj ∈ H is called the Riesz representer of λj. The minimal norm interpolant has the representation
+(3.1)
+v∗ =
+m
+�
+j=1
+a∗
+jφj,
+where a∗ = (a∗
+1, . . . , a∗
+m) solves the system of equations
+Ga∗ = w,
+G := (⟨φi, φj⟩H)i,j=1,...,m,
+with G being the Gramian matrix associated to φ1, . . . , φm.
+Remark 3.1. In the case where H is a more general Banach space, we are still ensured that the minimal
+norm interpolation is a near-optimal recovery with constant C = 2. However, its dependence on the data w
+is no longer linear and the above observation regarding its computation does not apply.
+Let us now apply this general principle to our particular setting in which the Hilbert space H is Hs(Ω)
+and X = H1(Ω). Let φj ∈ Hs(Ω) be the Riesz representer of the functional λj when viewed as a functional
+on Hs(Ω). In other words
+λj(v) = ⟨v, φj⟩Hs(Ω),
+v ∈ Hs(Ω).
+We assume that the λj are linearly independent on Hs(Ω) and thus the Gramian matrix
+G =
+�
+gi,j
+�
+i,j=1,...,m,
+gi,j := ⟨φi, φj⟩Hs = λj(φi),
+is invertible.
+Now, let u = u0 + uH, with uH ∈ KH = U(Hs(Ω)) be the function in K that gave rise to our data
+observation w. So, we have
+w′ = w − λ(u0) = λ(uH).
+If a∗ is the vector in Rm which satisfies Ga∗ = w′, then u∗
+H := �m
+j=1 a∗
+jφj is the function of minimum Hs(Ω)
+norm which satisfies the data w′, i.e., λ(u∗
+H) = w′. We have seen that
+∥uH − u∗
+H∥H1(Ω) ≤ R(KH
+w′)H1(Ω),
+namely, u∗
+H is the optimal recovery of the functions in KH
+w′. Note that the recovery error is measured in H1
+not in Hs(Ω). In turn, see (1.15), the function u∗ := u∗
+H + u0 is the optimal recovery for functions in Kw:
+∥u − u∗∥H1(Ω) ≤ R(Kw)H1(Ω).
+The idea behind our proposed numerical method is to numerically construct a function ˆu ∈ H1 that
+approximates u∗ well. If, for example, we have for ε > 0 the bound
+∥u∗ − ˆu∥H1(Ω) ≤ ε,
+then for any u ∈ K, we have by the triangle inequality
+∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε.
+6
+
+Given any C > 1, by taking ε small enough, we have that ˆu is a near best recovery of the functions in Kw
+with constant C.
+3.2. The numerical recovery algorithm for H1-continuous functionals. Motivated by the above
+analysis, we propose the following numerical algorithm for solving our recovery problem. The algorithm
+involves approximations of the function u0 and the Riesz representers φj, typically computed by finite
+element discretizations, and the application of the linear functionals λj to these approximations. In order to
+avoid extra technicalities, we make here the assumption that the applications of the functionals to a known
+finite element function can be exactly computed.
+We first work under the additional assumption that the linear functionals λj are not only defined on K
+but that they are continuous over H1(Ω). We define Λ as the maximum of the norms of the λj on H1(Ω).
+In this case
+(3.2)
+|λj(v)| ≤ Λ∥v∥H1(Ω),
+v ∈ H1(Ω).
+In what follows, throughout this paper, we use the following weighted ℓ2 norm on Rm,
+∥z∥ :=
+
+ 1
+m
+m
+�
+j=1
+|zj|2
+
+
+1/2
+= m−1/2∥z∥ℓ2,
+z = (z1, . . . , zm) ∈ Rm.
+In particular, we have
+∥λ(v)∥ ≤ Λ∥v∥H1(Ω),
+v ∈ H1(Ω).
+Given a user prescribed accuracy ε > 0, our algorithm does the following four steps involving intermediate
+tolerances (ε1, ε2).
+Step 1: We numerically find an approximation ˆu0 to u0 which satisfies
+(3.3)
+∥u0 − ˆu0∥H1(Ω) ≤ ε1.
+To find such a ˆu0, we use standard or adaptive FEM methods. Given that ˆu0 has been constructed, we
+define ˆw := w − λ(ˆu0). Then, for w′ := w − λ(u0) we have, see (3.3),
+(3.4)
+∥w′ − ˆw∥ ≤ Λε1.
+On the other hand, since |λj(v)| ≤ Λ∥v∥H1(Ω) ≤ Λs∥v∥Hs(Ω) ≤ Λs, where
+Λs := CsΛ,
+see (3.2), (1.13) and (1.14), we derive that
+(3.5)
+∥w′∥ ≤ Λs.
+Thus by triangle inequality, we also find that
+(3.6)
+∥ ˆw∥ ≤ Λs + Λε1.
+Step 2: For each j = 1, . . . , m, we numerically compute an approximation ˆφj ∈ H1(Ω) to φj which satisfies
+(3.7)
+∥φj − ˆφj∥H1(Ω) ≤ ε2,
+j = 1, . . . , m.
+This numerical computation is crucial and is performed during the offline phase of the algorithm. We detail
+it in §4. Note that the ˆφj’s are not assumed to be in Hs(Ω), and in particular not assumed to be harmonic
+functions.
+Step 3: We define and compute the matrix
+ˆG = (ˆgi,j)i,j=1,...,m,
+ˆgi,j := λj(ˆφi),
+and thus |ˆgi,j − gi,j| ≤ Λε2 for all i, j.
+7
+
+It follows that for the matrix R := G − ˆG we have
+∥R∥1 ≤ mΛε2,
+where we use the shorthand notation ∥ · ∥1 := ∥ · ∥ℓ1→ℓ1 for matrices. Since G is invertible, we are ensured
+that ˆG is also invertible for ε2 small enough. We define
+M := ∥G−1∥1,
+ˆ
+M := ∥ ˆG−1∥1.
+While these two norms are finite, their size will depend on the nature and the positioning of the linear
+functionals λj, j = 1, . . . , m, as it will be seen in the section on numerical experiments. These two numbers
+are close to one another when ε2 is small since ˆ
+M converges towards the unknown quantity M as ε2 → 0.
+In particular, we have
+|M − ˆ
+M| = |∥G−1∥1 − ∥ ˆG−1∥1| ≤ ∥G−1 − ˆG−1∥1 = ∥ ˆG−1RG−1∥1 ≤ M ˆ
+MmΛε2,
+from which we obtain that
+(3.8)
+M ≤
+ˆ
+M
+1 − m ˆ
+MΛε2
+and
+ˆ
+M ≤
+M
+1 − mMΛε2
+,
+provided that mMΛε2 < 1 and m ˆ
+MΛε2 < 1. We also have the bound
+(3.9)
+∥ ˆG−1 − G−1∥1 ≤
+M 2
+1 − mMΛε2
+mΛε2 =: δ.
+It is important to observe that δ can be made arbitrarily small by diminishing ε2.
+Step 4: We numerically solve the m×m algebraic system ˆGˆa = ˆw, thereby finding a vector ˆa = (ˆa1, . . . , ˆam).
+We then define ˆuH := �m
+j=1 ˆaj ˆφj and our recovery of u is ˆu := ˆu0 + ˆuH.
+This step can be implemented by standard linear algebra solvers.
+One major advantage of the above algorithm is that Steps 1-2-3 can be performed offline since they do not
+involve the data w. That is, we can compute ˆu0, the approximate Riesz representers ˆφj and the approximate
+Gramian ˆG and its inverse without knowing w. In this way, the computation of ˆu from given data w can be
+done fast online by Step 4 which only involves solving an m × m linear system. This may be a significant
+advantage, for example, when having to process a large number of measurements for the same set of sensors.
+3.3. A near optimal recovery bound. The following theorem shows that a near optimal recovery of u
+can be reached provided that the tolerances in the above described algorithm are chosen small enough.
+Theorem 3.2. For any prescribed ε > 0, if the tolerances (ε1, ε2), are small enough such that mMΛε2 < 1
+and
+(3.10)
+ε1 + mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ) ≤ ε,
+where C0 := maxj=1,...,m ∥φj∥H1(Ω) and δ :=
+M2
+1−mMΛε2 mΛε2, then the function ˆu generated by the above
+algorithm satisfies
+∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε,
+for every
+u ∈ Kw.
+Thus, for any C > 1 it is a near optimal recovery of u with constant C provided ε is taken sufficiently small.
+Proof. Let u = u0 + v be our target function in Kw. We define w′ = w − λ(u0) and v∗ := v∗(w′) which is
+the Chebyshev center of KH
+w′. We recall the algebraic system Ga∗ = w′ associated to the characterization of
+v∗ (see (3.1)). We write
+(3.11) ∥u∗
+H− ˆuH∥H1(Ω) ≤
+���
+m
+�
+j=1
+a∗
+j(φj − ˆφj)
+���
+H1(Ω) +
+���
+m
+�
+j=1
+(a∗
+j −ˆaj)ˆφj
+���
+H1(Ω) ≤ ∥a∗∥ℓ1ε2+∥a∗−ˆa∥ℓ1(C0 +ε2),
+where we have used (3.7) and the fact that
+∥ˆφj∥H1(Ω) ≤ ∥φj∥H1(Ω) + ∥φj − ˆφj∥H1(Ω) ≤ C0 + ε2.
+8
+
+Note that
+(3.12)
+∥a∗∥ℓ1 = ∥G−1w′∥ℓ1 ≤ M∥w′∥ℓ1 ≤ Mm∥w′∥ ≤ mMΛs,
+where we have used that ∥w′∥ℓ1 ≤ m∥w′∥ and inequality (3.5). Therefore it follows from (3.11) and (3.12)
+that
+(3.13)
+∥u∗
+H − ˆuH∥H1 ≤ mMΛsε2 + ∥a∗ − ˆa∥ℓ1(C0 + ε2).
+For the estimation of ∥a∗ − ˆa∥ℓ1, we introduce the intermediate vector ˜a ∈ Rm, which is the solution to the
+system G˜a = ˆw. Clearly,
+∥˜a − a∗∥ℓ1 = ∥G−1( ˆw − w′)∥ℓ1 ≤ M∥ ˆw − w′∥ℓ1 ≤ Mm∥ ˆw − w′∥ ≤ mMΛε1,
+where we invoked (3.4). On the other hand, in view of (3.9) and (3.6), we have
+∥˜a − ˆa∥ℓ1 = ∥(G−1 − ˆG−1) ˆw∥ℓ1 ≤ δ∥ ˆw∥ℓ1 ≤ mδ∥ ˆw∥ ≤ m(Λs + Λε1)δ.
+Combining these two estimates, we find that
+∥a∗ − ˆa∥ℓ1 ≤ mMΛε1 + m(Λs + Λε1)δ.
+We now insert this bound into (3.13) to obtain
+∥u∗
+H − ˆuH∥H1(Ω) ≤ mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ).
+Thus, for u∗ := u0 + u∗
+H and using (3.3), we have
+∥u∗ − ˆu∥H1(Ω)
+≤
+∥u0 − ˆu0∥H1(Ω) + ∥u∗
+H − ˆuH∥H1(Ω)
+≤
+ε1 + mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ) ≤ ε,
+(3.14)
+Since u = u0 + uH, we have
+∥u − u∗∥H1(Ω) = ∥uH − u∗
+H∥H1(Ω) ≤ R(KH
+w′)H1(Ω) = R(Kw)H1(Ω),
+and the statement of the theorem follows from this inequality and (3.14).
+Remark 3.3. Note that in numerical computations the quantity ˆ
+M is available while M is unknown. Thus
+in practice, in order to achieve the prescribed accuracy ε, we can first impose that ε2 < (2m ˆ
+MΛ)−1 and
+derive the inequalities, see (3.8),
+M ≤
+ˆ
+M
+1 − m ˆ
+MΛε2
+≤ 2 ˆ
+M,
+∥G−1 − ˆG−1∥1 ≤
+ˆ
+M 2
+1 − m ˆ
+MΛε2
+mΛε2 ≤ 2 ˆ
+M 2mΛε2 =: ˆδ,
+where the last inequality is proven in a similar fashion to (3.9). If we then follow the proof of Theorem 3.2,
+the requirement in (3.14) can be substituted by
+ε1 + 2m ˆ
+MΛsε2 + (C0 + ε2)(2m ˆ
+MΛε1 + m(Λs + Λε1)ˆδ) ≤ ε,
+and thus all participating quantities are computable.
+Remark 3.4. The result in Theorem 3.2 can easily be extended to the case of noisy data, that is, to the case
+when the observations
+˜w = w + η,
+where η is a noise vector of norm ∥η∥ ≤ κ. Indeed, the application of the algorithm to this noisy data leads
+to finding in Step 1 the vector ˆw := w + η − λ(ˆu0) that satisfies
+∥w′ − ˆw∥ ≤ Λε1 + κ,
+and
+∥ ˆw∥ ≤ Λs + ε1Λ + κ,
+where w′ = w − λ(u0). Inspection of the above proof shows that under the same assumption as in Theorem
+3.2, one has the recovery bound
+∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε + Cκ,
+for every
+u ∈ Kw,
+where C := (M + δ)m(C0 + ε2).
+9
+
+Remark 3.5. For simplicity, we did not introduce in the above analysis the possible errors in the application
+of the λi to the approximations ˆu0 and ˆφj, and in the numerical solution to the system ˆGˆa = ˆw, which would
+simply result in similar conditions involving the extra tolerance parameters.
+3.4. Point evaluation data. We now want to extend the numerical algorithm and its analysis to the case
+when the data functionals λj, j = 1, . . . , m, are point evaluations
+λj(h) := h(xj),
+xj ∈ Ω,
+j = 1, . . . , m.
+Of course these functionals are not defined for general functions h from H1(Ω). However, we can formulate
+the recovery problem whenever the functionals λj are well defined on K. We now discuss settings when this
+is possible.
+Recall that any u ∈ K can be written as u = u0 + uH, where u0 is the solution to (1.1) with right side
+f and g = 0 and uH ∈ Hs(Ω). Point evaluation is well defined for the harmonic functions uH ∈ Hs(Ω),
+provided the points are in Ω. In addition, they are well defined for points on the boundary Γ if the space
+Hs(Ω) continuously embeds into C(Ω). For d = 2, this is the case when s > 1/2 and when d = 3, this is the
+case when s > 1.
+Concerning u0, we will need some additional assumption to guarantee that point evaluation of u0 makes
+sense at the data sites xj, j = 1, . . . , m. For example, it is enough to assume that u0 is globally continuous
+or at least in a neighborhood of each of these points. This can be guaranteed by assuming an appropriate
+regularity of f. In this section, we assume that one of these settings holds. We then write
+w′
+j := uH(xj) = wj − u0(xj),
+j = 1, . . . , m,
+and follow the algorithm of the previous section with the following simple modifications:
+Modified Step 1: We numerically find an approximation ˆu0 to u0, which in addition to
+∥u0 − ˆu0∥H1(Ω) ≤ ε1,
+satisfies the requirement
+(3.15)
+max
+i=1,...,m |u0(xi) − ˆu0(xi)| ≤ ε1.
+To find such a ˆu0 we use standard or adaptive FEM methods. Given that ˆu0 has been constructed, we define
+ˆwj := wj − ˆu0(xj), j = 1, . . . , m, and thus, using (3.15), we have ∥w′ − ˆw∥ ≤ ε1.
+Modified Step 2: For each j = 1, . . . , m, we numerically compute an approximation ˆφj to φj, which in
+addition to
+∥φj − ˆφj∥H1(Ω) ≤ ε2,
+j = 1, . . . , m,
+satisfies the condition
+(3.16)
+max
+i=1,...,m |φj(xi) − ˆφj(xi)| ≤ ε2,
+i, j = 1, . . . , m.
+Condition (3.16) ensures that in Step 3 we can choose the entries ˆgi,j of the matrix ˆG as
+ˆgi,j = ˆφj(xi),
+i, j = 1, . . . , m.
+The Steps 3 and 4 of our algorithm remain the same as in the previous section.
+Theorem 3.6. With the above modifications, Theorem 3.2 holds with the exact same statement in this point
+evaluation setting.
+Proof. The proof is the same as that of Theorem 3.2.
+10
+
+4. Finite element approximations of the Riesz representers
+The computation of an approximation ˆu0 to u0, required in Step 1 of the algorithm, can be carried out by
+standard finite element Galerkin schemes. Depending on our knowledge on f one can resort to known a priori
+estimates for ε1, or may employ standard a posteriori estimates to ensure that the underlying discretization
+provides a desired target accuracy. Therefore, in the remainder of this section, we focus on a numerical
+implementation of Step 2 of the proposed algorithm.
+Our proposed numerical algorithm for Step 2 is to use finite element methods to generate the approx-
+imations ˆφj of the Riesz representers φj. Note that each of the functions φj is harmonic on Ω but we do
+not require that the sought after numerical approximation ˆφj is itself harmonic but only that it provides
+an accurate H1(Ω) approximation to φj. This allows us to use finite element approximations which are
+themselves not harmonic. However, the ˆφj will necessarily have to be close to being harmonic since they
+approximate a harmonic function in the H1(Ω) norm.
+Our numerical approach to constructing a ˆφj, discussed in §4.1, is to use discretely harmonic finite
+elements. Here, ˆφj is a discrete harmonic extension of a finite element approximation to the trace ψj = T (φj)
+computed by solving a Galerkin problem.
+In order to reduce computational cost (see Remark 4.2), we
+incorporate discrete harmonicity as constraints and introduce in §4.2 an equivalent saddle point formulation
+that has the same solution ˆφj, and which is the one that we practically employ in the numerical experiments
+given in §5. We give in §4.4 an a priori analysis with error bounds for ∥φj − ˆφj∥H1 in terms of the finite
+element mesh size, in the case where the measurement functionals are continuous on H1(Ω). These error
+bounds can in turn be used to ensure the prescribed accuracy ε2 in Step 2. We finally discuss in §4.5 the
+extensions to the point value case where pointwise error bounds on |ˆφj(xi) − ˆφj(xi)| are also needed.
+In order to simplify notation, we describe these procedures for finding an approximation ˆφ to the Riesz
+representer φ ∈ Hs = Hs(Ω) of a given linear functional ν on Hs. This numerical procedure is then applied
+with ν = λj, to find the numerical approximations ˆφj to the Riesz representer φj.
+For simplicity, throughout this section, we work under the assumption that Ω is a polygonal domain of R2
+or polyhedral domain of R3. This allows us to define finite element spaces based on triangular or simplicial
+partitions of Ω that in turn induce similar partitions on the boundary. We assume that 1
+2 < s < 3
+2, which
+is the relevant range for such domains, as explained in §2. Our analysis can be extended to more general
+domains with smooth or piecewise smooth boundaries, for example by using isoparametric elements near the
+boundary, however at the price of considerably higher technicalities.
+4.1. A Galerkin formulation. Let s > 1/2 be fixed and assume that ν is any linear form continuous on
+Hs(Ω) with norm
+(4.1)
+Cs := max{ν(v) : ∥v∥Hs(Ω) = 1}
+In view of the the definition of the Hs norm, the representer φ ∈ Hs(Ω) of ν for the corresponding inner
+product can be defined as
+φ = Eψ,
+where E is the harmonic extension operator of (4.3) below and where ψ ∈ Hs(Γ) is the solution to the
+following variational problem:
+(4.2)
+⟨ψ, η⟩Hs(Γ) = µ(η) := ν(Eη),
+η ∈ Hs(Γ).
+Note that this problem admits a unique solution and we have
+∥ψ∥Hs(Γ) = ∥φ∥Hs(Ω) = Cs.
+Recall that
+(4.3)
+Eg := argmin{∥∇v∥L2(Ω) : vΓ = g}.
+The function Eg is characterized by T (Eg) = g and
+�
+Ω
+∇Eg · ∇v = 0,
+v ∈ H1
+0(Ω).
+11
+
+From the left inequality in (1.3), one has
+(4.4)
+∥Eg∥H1(Ω) ≤ CE∥g∥H1/2(Γ),
+g ∈ H1/2(Γ),
+where CE can be taken to be the inverse of the constant c0 in (1.3).
+Therefore, one approach to discretizing this problem is the following: consider finite element spaces Vh
+associated to a family of meshes {Th}h>0 of Ω, where as usual h denotes the maximum meshsize. We define
+Th to be the space obtained by restriction of Vh on the boundary Γ, that is,
+Th = T (Vh)
+Since we have assumed that Ω is a polygonal or polyhedral domain, the space Th is a standard finite element
+space for the boundary mesh. Having also assumed that s < 3/2, when using standard H1 conforming finite
+elements such as Pk-Lagrange finite elements, we are ensured that Th ⊂ Hs(Γ). We denote by
+Wh := {vh ∈ Vh : T (vh) = 0},
+the finite element space with homogeneous boundary conditions.
+We define the discrete harmonic extension operator Eh associated to Vh as follows : for gh ∈ Th,
+Ehgh := argmin{∥∇vh∥L2(Ω) : vh ∈ Vh, T (vh) = gh}.
+Note that Ehgh is not harmonic. Similar to E, the function Ehgh is characterized by T (Ehgh) = gh and
+�
+Ω
+∇Ehgh · ∇vh = 0,
+vh ∈ Wh.
+Then, we define the approximation φh ∈ Vh to φ as
+φh = Ehψh,
+where ψh ∈ Th is the solution to the following variational problem:
+(4.5)
+⟨ψh, gh⟩Hs(Γ) = µh(gh) := ν(Ehgh),
+gh ∈ Th.
+Here we are assuming that, in addition to be defined on Hs(Ω), the functional ν is also well defined on the
+space Vh. We shall further consider separately two instances where this is the case : (i) ν is a continuous
+functional on H1(Ω) and (ii) ν is a point evaluation functional.
+Note that (4.5) is not the straightforward Galerkin approximation of (4.2), since µh differs from µ. This
+complicates somewhat the further conducted convergence analysis. The numerical method we employ for
+computing φh is to numerically solve an equivalent saddle point problem described below.
+We apply the strategy (4.5) to ν := λj for each j and thereby obtain the corresponding approximations
+ˆφj := φh ∈ Vh. Since Step 2 requires that we guarantee the error ∥φj − ˆφj∥H1 ≤ ε2, our main goal in
+this section is to establish a quantitative convergence bound for ∥φ − φh∥H1. We also need to establish a
+pointwise convergence bound for |φ(x) − φh(x)| when considering the modified version of Step 2 in the case
+that the measurements are point values.
+Similar to E, it will be important in our analysis to control the stability of Eh in the sense of a bound
+(4.6)
+∥Ehgh∥H1(Ω) ≤ DE∥gh∥H1/2(Γ),
+gh ∈ Th,
+with a constant DE that is independent of h. However, such a uniform bound is not readily inherited from
+the stability of E. As observed in [6], its validity is known to depend on the existence of uniformly H1-stable
+linear projections onto Vh preserving the homogeneous boundary condition, that is, projectors Ph onto Vh
+that satisfy
+(4.7)
+Ph(H1
+0(Ω)) = Wh
+and
+∥Phv∥H1(Ω) ≤ B∥v∥H1(Ω),
+v ∈ H1(Ω),
+for some B independent of h. One straightforward consequence of this is that if v ∈ H1(Ω) with v|Γ ∈ Th
+then Ph(v)|Γ = v|Γ.
+We next show that the existence of such projectors is sufficient to guarantee the stability of Eh. For this,
+suppose (4.7) holds and gh ∈ Th. Then PhEgh ∈ Vh and the trace of PhEgh is equal to gh. It follows that
+∥Ehgh − PhEgh∥H1(Ω)
+≤
+CP ∥∇Ehgh − ∇PhEgh∥L2(Ω)
+≤
+CP ∥∇Ehgh∥L2(Ω) + CP ∥∇PhEgh∥L2(Ω),
+≤
+2CP ∥PhEgh∥H1(Ω),
+12
+
+where CP is the Poincar´e constant for Ω. Here, the last inequality follows from the minimizing property of
+Ehgh. Thus, by triangle inequality, one has
+∥Ehgh∥H1(Ω) ≤ (1 + 2CP )∥PhEgh∥H1(Ω) ≤ (1 + 2CP )B∥Egh∥H1(Ω) ≤ (1 + 2CP )BCE∥gh∥H1/2(Γ),
+which is (4.6) with DE = (1 + 2CP )BCE.
+The requirement of uniformly stable projectors Ph with the property (4.7) is satisfied by projectors of
+Scott-Zhang type [26] when the family of meshes {Th}h>0 is shape regular, that is, when all elements T
+have a uniformly bounded ratio between their diameters h(T ) and the diameter ρ(T ) of their inner circle.
+In other words, the shape parameter
+(4.8)
+σ = σ({Th}h>0) := sup
+h>0
+max
+T ∈Th
+h(T )
+ρ(T ),
+is finite. In all that follows in the present paper, we work under such an assumption on the meshes Th.
+Therefore, (4.6) holds when Vh is subordinate to such partitions.
+4.2. A saddle point formulation. Before attacking the convergence analysis, we need to stress an impor-
+tant computational variant of the above described Galerkin method, that leads to the same solution φh. It is
+based on imposing harmonicity via a Lagrange multiplier. For this purpose, we introduce the Hilbert space
+Xs(Ω) that consists of all v ∈ H1(Ω) such that vΓ ∈ Hs(Γ), and equip it with the norm
+∥v∥Xs(Ω) :=
+�
+∥vΓ∥2
+Hs(Γ) + ∥∇v∥2
+L2(Ω)
+�1/2
+.
+Then, the Riesz representer φ is equivalently determined as the solution of the saddle point problem: find
+(φ, π) ∈ Xs(Ω) × H1
+0(Ω) such that
+a(φ, v) + b(v, π)
+=
+ν(v),
+v ∈ Xs(Ω)
+b(φ, z)
+=
+0,
+z ∈ H1
+0(Ω),
+where the bilinear forms are given by
+a(φ, v) := ⟨φΓ, vΓ⟩Hs(Γ)
+and
+b(v, π) := ⟨∇v, ∇π⟩L2(Ω).
+Clearly the second equation in (4.9) means that φ is harmonic and testing the first equation with a v ∈ Hs(Ω)
+shows that φ is the Riesz representer of µ.
+This saddle point formulation is well-posed: the bilinear forms a and b obviously satisfies the continuity
+properties
+a(φ, v) ≤ ∥φΓ∥Hs(Γ)∥vΓ∥Hs(Γ) ≤ ∥φ∥Xs(Ω)∥v∥Xs(Ω),
+φ, v ∈ Xs(Ω),
+and for the standard norm ∥v∥H1
+0 (Ω) = ∥∇v∥L2(Ω),
+b(v, π) ≤ ∥∇v∥L2(Ω)∥∇π∥L2(Ω) ≤ ∥v∥Xs(Ω)∥π∥H1
+0(Ω),
+v ∈ Xs(Ω), π ∈ H1
+0(Ω).
+In addition, for all v ∈ Hs(Ω), one has
+∥v∥2
+Xs(Ω) ≤ ∥vΓ∥2
+Hs(Γ) + ∥v∥2
+H1(Ω) ≤ ∥vΓ∥2
+Hs(Γ) + C2
+E∥v∥2
+H1/2(Γ) ≤ (1 + C2
+E)a(v, v),
+which shows that a is coercive on the null space of b in Xs(Ω). Finally, the bilinear form b satisfies the
+inf-sup condition
+inf
+π∈H1
+0 (Ω)
+sup
+v∈Xs(Ω)
+b(v, π)
+∥v∥Xs(Ω)∥π∥H1
+0(Ω)
+≥
+inf
+π∈H1
+0 (Ω)
+b(π, π)
+∥π∥Xs(Ω)∥π∥H1
+0(Ω)
+= 1.
+Therefore the standard LBB theory ensures existence and uniqueness of the solution pair (φ, π).
+We now discretize the saddle point problem by searching for (φh, πh) ∈ Vh × Wh such that
+a(φh, vh) + b(vh, πh)
+= ν(vh),
+vh ∈ Vh
+b(φh, zh)
+= 0,
+zh ∈ Wh.
+Remark 4.1. The equivalence with the previous derivation of φh by the Galerkin approach is easily checked:
+the second equation tells us that the solution φh is discretely harmonic, and therefore equal to Ehψh for some
+ψh ∈ Th. Then taking vh of the form Ehgh for gh ∈ Th gives us exactly the Galerkin formulation (4.5).
+13
+
+This discrete saddle point problem is uniformly well-posed when we equip the space Wh with the H1
+0
+norm, and the space Vh with the Xs norm. The continuity of a and b, and the inf-sup condition for b follow
+by the exact same arguments applied to the finite element spaces, with the same constants. On the other
+hand, we need to check the uniform ellipticity of a in the space VH
+h ⊂ Vh of discretely harmonic functions,
+which can be defined as
+VH
+h := {vh ∈ Vh : b(vh, zh) = 0, zh ∈ Wh},
+or equivalently as the image of Th by the operator Eh. For all vh ∈ Vh,H and gh = T (uh), we write
+∥vh∥2
+Xs(Ω) ≤ ∥gh∥2
+Hs(Γ) + ∥vh∥2
+H1(Ω) ≤ ∥gh∥2
+Hs(Γ) + D2
+E∥gh∥2
+H1/2(Γ) ≤ (1 + D2
+E)a(vh, vh),
+where we have used the discrete stability of Eh.
+Remark 4.2. In practice, we use this discrete saddle point formulation for the computation of φh rather
+than the equivalent Galerkin formulation (4.5) for the following reason. Let Nh := dim Vh, Mh := dim Wh,
+and Ph := dim Th = Nh − Mh.
+Computing the right hand side load vector in (4.2) requires computing
+discretely harmonic extensions of Ph basis functions, which means solving Ph linear systems of dimension
+Mh. In addition one has to solve the sparse linear system (4.5) of size Ph followed by another system of size
+Mh to compute φh = Ehψh. Using optimal iterative solvers of linear complexity the minimum amount of
+work needed to compute one representer scales then like
+PhMh ∼ N
+1+ d−1
+d
+h
+.
+while solving the saddle point problem requires the order of Nh + Mh ∼ Nh operations.
+On the other
+hand the characterization of φh through (4.5) appears to be more convenient when deriving error bounds for
+∥φ − φh∥H1(Ω). This is the objective of the next sections.
+4.3. Preparatory results. In the derivation of error bounds for ∥φ − φh∥H1(Ω), we will need several ingre-
+dients.
+The first is the following lemma that quantifies the perturbation induced by using Eh in place of E.
+Lemma 4.3. For any gh ∈ Th, one has
+(4.9)
+∥(E − Eh)gh∥H1(Ω) ≤ C2hr−1∥gh∥Hs(Γ).
+where C2 depends on r and s, the shape-parameter σ, and on the geometry of Ω.
+Proof. From the properties of E and Eh, one has
+⟨∇(E − Eh)gh, ∇vh⟩ = 0,
+vh ∈ Wh
+This orthogonality property shows that
+∥∇(Egh − Ehgh)∥L2(Ω) ≤ ∥∇(Egh − Ehgh − vh)∥L2(Ω),
+vh ∈ Wh,
+and therefore
+∥∇(Egh − Ehgh)∥L2(Ω) ≤
+min
+vh∈Vh,T (vh)=gh ∥∇(Egh − vh)∥L2(Ω) ≤ ∥∇(Egh − PhEgh)∥L2(Ω),
+where Ph is the stable projector that preserves homogeneous boundary condition, see (4.7). It follows that
+∥∇(Egh − Ehgh)∥L2(Ω) ≤ (1 + B) min
+vh∈Vh ∥Egh − vh∥H1(Ω),
+where B is the uniform H1-stability bound on Ph. By standard finite element approximation estimates and
+(2.4), we have
+min
+vh∈Vh ∥Egh − vh∥H1(Ω) ≤ Chr−1∥Egh∥Hr(Ω) ≤ CC1hr−1∥gh∥Hs(Γ),
+where the constant C depends on r and on the shape parameter σ. The estimate (4.9) follows by Poincar´e
+inequality since Egh − Ehgh ∈ H1
+0(Ω).
+The second ingredient concerns the regularity of the solution to the variational problem
+(4.10)
+⟨κ, v⟩Hs(Γ) = γ(v),
+v ∈ Hs(Γ).
+14
+
+For a general linear functional γ ∈ H−s(Γ), that is, continuous on Hs(Γ), we are only ensured that the
+solution κ is bounded in Hs(Γ), with ∥κ∥Hs(Γ) = ∥γ∥H−s(Γ). However, if γ has some extra regularity, this
+then translates into additional regularity of κ.
+As a simple example, consider the case where γ is in addition
+continuous on L2(Γ), that is
+(4.11)
+γ(v) = ⟨g, v⟩L2(Γ),
+for some g ∈ L2(Γ), and assume that we work with s = 1 and a polygonal domain. Then the variational
+problem has a solution κ ∈ H1(Γ) and in addition κ ∈ H2(E) for each edge E with weak second derivative
+given by
+−κ′′ = g − κ ∈ L2(Γ).
+In turn, standard finite element approximation estimates yield
+min
+κh∈Th ∥κ − κh∥H1(Γ) ≤ Ch∥g∥L2(Γ),
+with a constant C that depends on the shape parameter σ.
+Of course, gain of regularity theorems for elliptic problems are known in various contexts. However, we
+have not found a general treatment of gain of regularity that addresses the setting of this paper. In going
+forward, we do not wish to systematically explore this gain in regularity and approximability for more general
+values of s and smoothness of γ since this would significantly enlarge the scope of this paper. Instead, we
+state it as the following general assumption.
+Assumption R: for s >
+1
+2 and δ > 0, there exists r(s, δ) > 0 such that if γ ∈ H−s+δ(Γ) for some
+δ > 0, then the solution κ to (4.10) satisfies
+(4.12)
+min
+κh∈Th ∥κ − κh∥Hs(Γ) ≤ Chr(s,δ)∥γ∥H−s+δ(Γ),
+with a constant C that depends on s, δ, and on the shape parameter σ.
+The above example shows that r(1, 1) = 1 for a polygonal domain. We expect that this assumption always
+holds for the range 1
+2 < s < 3
+2 that is considered here.
+4.4. An a priori error estimate for ∥φ − φh∥H1(Ω). In this section, we work under the assumption that
+the linear functional ν is continuous on H1(Ω) with norm
+Cν := max{ν(v) : ∥v∥H1(Ω) = 1}.
+Let us first check that this assumption implies a uniform a priori bound on ∥ψh∥Hs(Γ). Indeed, we may write
+∥ψh∥2
+Hs(Γ) = ⟨ψh, ψh⟩Hs(Γ) = ν(Ehψh) ≤ CνDE∥ψh∥H1/2(Γ) ≤ CνDE∥ψh∥Hs(Γ),
+where the first inequality used (4.6). Therefore,
+(4.13)
+∥ψh∥Hs(Γ) ≤ CνDE.
+We have seen in §2 that the function φ belongs to the standard Sobolev space Hr(Ω) for r defined in
+(2.3). We use this r throughout this section. From (2.4), there exists a constant C1 such that
+(4.14)
+∥Ew∥Hr(Ω) ≤ C1∥w∥Hs(Γ),
+w ∈ Hs(Γ).
+As noted in §2, the amount of smoothness r depends both on s and on the geometry of Ω. What is important
+for us is that since s > 1/2, we have shown in (2) that r > 1. For example, for smooth domains it is r = s+ 1
+2.
+The fact that φ ∈ Hr(Ω) hints that the finite element approximation φh to φ should converge with a certain
+rate.
+This is indeed the case as given in the following result.
+Theorem 4.4. Under Assumption R, we have
+(4.15)
+∥φ − φh∥H1(Ω) ≤ CCνht,
+where t = min{r − 1, r(s, s + 1
+2) + r(s, s − 1
+2)}. The constant C depends on s and on the geometry of Ω, and
+on the family of meshes through the shape parameter σ.
+15
+
+Proof. We use the decomposition
+(4.16)
+φ − φh = Eψ − Ehψh = E(ψ − ψh) + (E − Eh)ψh,
+The second term can be estimated with the help of Lemma 4.3 applied to gh = ψh which gives
+∥(E − Eh)ψh∥H1(Ω) ≤ C2hr−1∥ψh∥Hs(Γ) ≤ C2DECνhr−1,
+from the a priori estimate (4.13) for ψh. We thus have obtained a bound in O(hr−1) for the H1 norm of the
+second term in (4.16).
+For the first term, we know that
+∥E(ψ − ψh)∥H1(Ω) ≤ CE∥ψ − ψh∥H1/2(Γ),
+and so we are led to estimate ψ − ψh in the H1/2(Γ) norm. For this purpose, we introduce the intermediate
+solution ψh ∈ Th to the problem
+⟨ψh, gh⟩Hs(Γ) = µ(gh) = ν(Egh),
+gh ∈ Th,
+and we use the decomposition
+(4.17)
+ψ − ψh = (ψ − ψh) + (ψh − ψh).
+We estimate the second term in (4.17) by noting that for any gh ∈ Th,
+⟨ψh − ψh, gh⟩Hs(Γ) = ν((E − Eh)gh) ≤ Cν∥(E − Eh)gh∥H1(Ω) ≤ CνC2hr−1∥gh∥Hs(Γ),
+where we have again used Lemma 4.3. Taking gh = ψh − ψh we obtain a bound O(hr−1) for its Hs(Γ) norm,
+and in turn for its H1/2(Γ) norm.
+It remains to estimate ∥ψ − ψh∥H1/2(Γ). Note that ψh is exactly the Galerkin approximation of ψ since
+we use the same linear form µ in both problems. In fact, we have
+⟨ψ − ψh, gh⟩Hs(Γ) = 0,
+gh ∈ Th,
+that is ψh is the Hs-orthogonal projection of ψ onto Th and therefore
+∥ψ − ψh∥Hs(Γ) = min
+κh∈Th ∥ψ − κh∥Hs(Γ).
+Since the linear form µ satisfies
+|µ(g)| = |ν(Eg)| ≤ Cν∥Eg∥H1(Ω) ≤ CνCE∥g∥H1/2(Γ),
+and thus belongs to H−1/2(Γ), we may apply the estimate (4.12) to γ = ν, κ = ψ, δ = s − 1
+2 > 0, to reach
+(4.18)
+∥ψ − ψh∥H1/2(Γ) ≤ ∥ψ − ψh∥Hs(Γ) ≤ CCνCEhr(s,s− 1
+2 ).
+This proves the theorem for the value t = min{r − 1, r(s, s − 1
+2)} > 0.
+We finally improve the value of t by using a standard Aubin-Nitsche duality argument as follows. We now
+take κ to be the solution of (4.10) with
+γ(v) = ⟨ψ − ψh, v⟩H1/2(Γ),
+v ∈ H1/2(Γ),
+where ⟨., .⟩H1/2(Γ) stands for the H1/2 scalar product associated with the norm ∥.∥H1/2(Γ). We then write
+∥ψ − ψh∥2
+H1/2(Γ) = ⟨ψ − ψh, ψ − ψh⟩H1/2(Γ) = ⟨κ, ψ − ψh⟩Hs(Γ) = ⟨κ − κh, ψ − ψh⟩Hs(Γ),
+where the last equality comes from Galerkin orthogonality. It follows that
+∥ψ − ψh∥2
+H1/2(Γ) ≤ ∥κ − κh∥Hs(Γ)∥ψ − ψh∥Hs(Γ) ≤ Chr(s,s+ 1
+2 )∥ψ − ψh∥H1/2(Γ)∥ψ − ψh∥Hs(Γ),
+where we have again used (4.12) now with δ = s+ 1
+2. Using the already established estimate (4.18), it follows
+that
+∥ψ − ψh∥H1/2(Γ) ≤ CCECνh˜t,
+with ˜t := r(s, s + 1
+2) + r(s, s − 1
+2). With all such estimates, the desired convergence bound follows with
+t := min{r − 1, ˜t}.
+16
+
+Remark 4.5. In the case of a polygonal domain and s = 1 which is further considered in our numerical
+experiments, we know that r = 3
+2 and r(1, 1) = 1 so that ˜t ≥ r(1, 3
+2) ≥ 1. In turn the convergence bound is
+established with t = r − 1 = 1
+2, a rate that we observe in practice, see §5.
+4.5. The case of point value evaluations. We discuss now the case where
+ν(v) = δz(v) = v(z),
+for some z ∈ Ω. In order to guarantee that point evaluation is a continuous functional on Hs, we assume
+that
+s > d − 1
+2
+,
+that is s > 1
+2 for d = 2, and s > 1 for d = 3. We want to find the Riesz representer of such a point evaluation
+functional on Hs. Note that our assumption on s ensures the continuous embeddings
+Hs(Γ) ⊂ C(Γ),
+as well as
+Hs(Ω) ⊂ Hr(Ω) ⊂ C(Ω),
+since in view of (2.3)
+r = min
+�
+s + 1
+2, r∗�
+> d
+2.
+where in the inequality we recall that r∗ > 3
+2 for polygonal domains.
+The point evaluation functional ν is thus continuous on Hs(Ω) with norm Cs bounded independently of
+the position of z. Of course, the Galerkin scheme analyzed above for ν ∈ H1(Ω)∗ continues to make sense
+since ν is well defined on the space Vh.
+As explained in §3.4, the prescriptions in Step 2 of the recovery algorithm need to be strengthened in
+the point evaluation setting. Thus, we are interested in bounding the pointwise error |φ(x) − φh(x)| at the
+measurement points, in addition to the H1-error ∥φ − φh∥H1(Ω). In what follows, we establish a modified
+version of Theorem 4.4 in the point value setting that gives a convergence rate for ∥φ − φh∥H1(Ω), and
+in addition for ∥φ − φh∥L∞(Ω) ensuring the pointwise error control. We stress that the numerical method
+remains unchanged, that is, φh is defined in the exact same way as previously. The new ingredients that
+are needed in our investigation are two classical results on the behavior of the finite element method with
+respect to the L∞ norm.
+The first one is the so-called weak discrete maximum principle which states that there exists a constant
+Cmax such that, for all h > 0,
+(4.19)
+∥Ehgh∥L∞(Ω) ≤ Cmax∥gh∥L∞(Γ),
+gh ∈ Th.
+This result was first established in [4] with constant Cmax = 1 for piecewise linear Lagrange finite elements
+under acuteness assumptions on the angles of the simplices. The above version with Cmax ≥ 1 is established in
+[25] for Lagrange finite elements of any degree on 2d polygonal domains, under the more general assumption
+that the meshes {Th}h>0 are quasi-uniform (in addition to shape regularity, all elements of Th have diameters
+of order h). A similar result is established in [13] on 3d convex polyhedrons.
+The second ingredient we need is a stability property in the L∞ norm of the Galerkin projection Rh :
+H1
+0(Ω) → Wh where Rhv, v ∈ H1
+0(Ω), is defined by
+�
+Ω
+∇Rhv · ∇vh =
+�
+Ω
+∇v · ∇vh,
+vh ∈ Wh.
+Specifically, this result states that there exists a constant Cgal and exponent a ≥ 0 such that, for all h > 0,
+(4.20)
+∥Rhv∥L∞(Ω) ≤ Cgal(1 + | ln(h)|)a∥v∥L∞(Ω),
+v ∈ L∞(Ω) ∩ H1
+0(Ω),
+that is, the Ritz projection is stable and quasi-optimal, uniformly in h, up to a logarithmic factor. This
+result is established in [25] for Lagrange finite elements on 2d polygonal domains and quasi-uniform partitions,
+with a = 1 in the case of piecewise linear elements and a = 0 for higher order elements. A similar result is
+established in [13] with a = 0 for convex polygons and polyhedrons. In going further, we assume that the
+choice of finite element meshes ensures the validity of (4.19) and (4.20).
+17
+
+We begin our analysis with the observation that under the additional mesh assumptions, Lemma 4.3 can
+be adapted to obtain an estimate on ∥(E − Eh)gh∥L∞(Ω).
+Lemma 4.6. For any gh ∈ Th, one has
+(4.21)
+∥(E − Eh)gh∥L∞(Ω) ≤ C3(1 + | ln(h)|)a)hr− d
+2 ∥gh∥Hs(Γ),
+where C3 depends on (r, s), the geometry of Ω, and the family of meshes through Cgal.
+Proof. For any vh ∈ Vh such that T (vh) = gh, we write
+∥(E − Eh)gh∥L∞(Ω) ≤ ∥Egh − vh∥L∞(Ω) + ∥Ehgh − vh∥L∞(Ω).
+It is readily seen that Ehgh − vh = Rh(Ehgh − vh) = Rh(Egh − vh). Indeed RhEhgh − RhEgh ∈ Wh and
+�
+Ω
+∇(Rh(Ehgh − Egh)) · ∇vh =
+�
+Ω
+∇(Ehgh − Egh) · ∇vh = 0 for all vh ∈ Wh. Therefore, by (4.20), we obtain
+∥(E − Eh)gh∥L∞(Ω) ≤ (1 + Cgal(1 + | ln(h)|)a)
+min
+vh∈Vh,T (vh)=gh ∥Egh − vh∥L∞(Ω).
+On the other hand, we are ensured that Egh belongs to Hr(Ω) where r >
+d
+2, and therefore has H¨older
+smoothness of order r − d
+2 > 0 with
+∥Egh∥Cr− d
+2 (Ω) ≤ Ce∥Egh∥Hr(Ω) ≤ CeC1∥gh∥Hs(Γ),
+where Ce is the relevant continuous embedding constant. By standard finite element approximation theory,
+min
+vh∈Vh,T (vh)=gh ∥Egh − vh∥L∞(Ω) ≤ Chr− d
+2 ∥Egh∥Cr− d
+2 (Ω),
+where C depends on r and the shape-parameter σ and therefore we obtain (4.21).
+We are now in position to give an adaptation of Theorem 4.4 to the point value setting.
+Theorem 4.7. Under Assumption R, for any t1 < min{r − d
+2, r(s, s + 1
+2) + r(s, s − 1
+2)}, one has
+(4.22)
+∥φ − φh∥H1(Ω) ≤ Cht1,
+and for any t2 < min{r − d
+2, 2r(s, s − d−1
+2 )}, one has
+(4.23)
+∥φ − φh∥L∞(Ω) ≤ Cht2.
+The constant C depends in both cases on s, t1 and t2, on the geometry of Ω, as well as on the family of
+meshes through the constants Cmax and Cgal, and the shape parameter σ.
+Proof. We estimate ∥φ − φh∥H1(Ω) by adapting certain steps in the proof of Theorem 4.4. The first change
+lies in the a priori estimate of the Hs(Γ) norm of ψh that was previously given by (4.13) which is not valid
+anymore since Cν = ∞. Instead, we write
+∥ψh∥2
+Hs(Γ) = ⟨ψh, ψh⟩Hs(Γ) = ν(Ehψh) ≤ ∥Ehψh∥L∞(Ω) ≤ Cmax∥ψh∥L∞(Γ) ≤ CmaxBs∥ψh∥Hs(Γ),
+where we have used (4.19) and where Bs is the continuous embedding constant between Hs(Γ) and L∞(Γ).
+In turn, we find that
+(4.24)
+∥ψh∥Hs(Γ) ≤ CmaxBs,
+which results in the slightly modified estimate
+∥(E − Eh)ψh∥H1(Ω) ≤ C2CmaxBshr−1,
+for the second term of (4.16).
+For the first term E(ψ − ψh), we proceed in a similar manner to the proof of Theorem 4.4. Namely, we
+estimate the H1/2(Γ) norms of two summands in (4.17). The estimate of ∥ψh − ψh∥H1/2(Γ) is modified as
+follows. We note that for any gh ∈ Th,
+⟨ψh − ψh, gh⟩Hs(Γ) = ν((E − Eh)gh) ≤ ∥(E − Eh)gh∥L∞(Ω) ≤ C3(1 + | ln(h)|)a)hr− d
+2 ∥gh∥Hs(Γ),
+18
+
+where we have now used Lemma 4.6. Taking gh = ψh − ψh we obtain a bound of order O(hr− d
+2 ) up to
+logarithmic factors for its Hs norm, and in turn for its H1/2 norm. The estimate of ∥ψ − ψh∥H1/2(Γ) is
+left unchanged and of order O(h˜t). Combining these various estimates, we have established (4.22) for any
+t1 < min{r − d
+2, ˜t}, with ˜t := r(s, s + 1
+2) + r(s, s − 1
+2).
+We next estimate ∥φ − φh∥L∞(Ω) by the following adaptation of the proof of Theorem 4.4. For the first
+term (E − Eh)ψh of (4.16) we use Lemma 4.6 combined with the estimate (4.24) of ψh which give us
+∥(E − Eh)ψh∥L∞(Ω) ≤ CmaxBsC3(1 + | ln(h)|)a)hr− d
+2 .
+For the second term E(ψ − ψh), we use the continuous maximum principle to obtain
+∥E(ψ − ψh)∥L∞(Ω) ≤ ∥ψ − ψh∥L∞(Γ) ≤ ∥ψh − ψh∥L∞(Γ) + ∥ψ − ψh∥L∞(Γ)
+For the first summand, we write
+∥ψh − ψh∥L∞(Γ) ≤ Ce∥ψh ��� ψh∥Hs(Γ),
+where Ce is the relevant continuous embedding constant, and we have already observed that ∥ψh − ψh∥Hs(Γ)
+satisfies a bound in O(hr− d
+2 ) up to logarithmic factors. For the second summand, we may write
+∥ψ − ψh∥L∞(Γ) ≤ Ce∥ψ − ψh∥Hs(Γ),
+where Ce is the relevant continuous embedding constant. Since ν belongs to H−s+δ(Γ) for all δ < s − d−1
+2 ,
+we can apply the estimate (4.12) to reach a convergence bound
+∥ψ − ψh∥Hs(Γ) ≤ Chr(s,δ),
+where C depends on the closeness of δ to s − d−1
+2 , and on the family of meshes through the shape parameter
+σ. Combining these estimates then gives (4.23) for any t2 < min{r − d
+2, ˜t} where ˜t = r(s, s − d−1
+2 ), since δ
+can be picked arbitrarily close to s − d−1
+2 .
+We can improve the range of t2 as follows: pick any s such that d−1
+2
+< s < s and write
+∥ψ − ψh∥L∞(Γ) ≤ Ce∥ψ − ψh∥Hs(Γ),
+where Ce is the relevant continuous embedding constant. We then apply a similar Aubin-Nitsche argument
+to derive an estimate
+∥ψ − ψh∥Hs(Γ) ≤ Chr(s,δ)+r(s,s−s).
+Combining these estimates gives (4.23) for any t2 < min{r − d
+2, t}, where t := 2r(s, s − d−1
+2 ) since s can be
+picked arbitrarily close to d−1
+2
+and δ arbitrarily close to s − d−1
+2 .
+5. Numerical Illustrations
+In this section, we implement some examples of our numerical method. For this, we have to specify the
+domain Ω, the functionals λj, and a function u ∈ H1(Ω) which gives rise to the data vector w = λ(u).
+While our numerical method can be applied to general choices for these quantities, in our illustrations we
+make these choices so that the computations are not too involved but yet allow us the flexibility to illustrate
+certain features of our algorithm. The specific choices we make for our numerical example are the following.
+The domain: In order to simplify the presentation, we restrict ourselves when Ω = (0, 1)2 but point out
+again that the algorithm can be extended to more general domains.
+The function u: For the function u we choose the harmonic function u = uH where
+(5.1)
+uH(x, y) = ex cos(y),
+(x, y) ∈ Ω := (0, 1)2.
+This choice means that u0 = 0 and therefore allows us not to deal with the computation of ˆu0. This choice
+corresponds to the right side f = 0. Note that the trace of uH on the boundary Γ is piecewise smooth and
+continuous. Therefore, we have T (uH) ∈ H1(Γ). We take s = 1 as our assumption on the value of s. This
+means that we shall seek Riesz representor for the functionals given below when viewed as acting on H1(Ω).
+19
+
+5.1. The case of linear functionals defined on H1(Ω). In this section, we consider numerical experiments
+for linear functionals defined on H1(Ω). In our illustrative example, we relabel these functionals by double
+indices associated with a regular square grid. More precisely,
+(5.2)
+λi,j(v) :=
+1
+√
+2πr2
+�
+Ω
+v(z)e− 1
+2
+|z−zi,j|2
+r2
+dz, v ∈ H1(Ω),
+i, j = 1, ..., √m.
+Here, we assume that m is a square integer and r = 0.1 in our simulations. The centers zi,j ∈ Ω are uniformly
+distributed
+(5.3)
+zi,j :=
+1
+√m + 1(i, j),
+i, j = 1, ..., √m.
+Recall that our numerical algorithm as described in Section 3.2 is based on finite element methods.
+Specifically, we us the finite element spaces
+Vh :=
+�
+vh ∈ C0(Ω) : vh|T ∈ Q1,
+T ∈ Th
+�
+,
+where Th are subdivisions of Ω made of squares of equal side length h and Q1 denotes the space of polynomials
+of degree at most 1 in each direction. In order to study the effect of the mesh-size we specifically consider
+h = hn := 2−n,
+n = 4, . . . , 9,
+that is, bilinear elements on uniformly refined meshes with mesh-size 2−n.
+We display in Table 1 the results of our numerical recovery algorithm. The entries in the table are the
+recovery errors
+e(m, n) := ∥uH − ˆuH∥H1(Ω),
+where ˆuH ∈ Vhn is the recovery for the particular values of m and n.
+n
+m
+4
+9
+16
+25
+36
+4
+0.7
+0.28
+0.2
+141.73
+49.43
+5
+0.7
+0.28
+0.18
+16.0
+16.31
+6
+0.7
+0.28
+0.18
+0.2
+1.79
+7
+0.7
+0.28
+0.18
+0.16
+0.11
+8
+0.7
+0.28
+0.18
+0.09
+0.06
+9
+0.7
+0.28
+0.18
+0.09
+0.06
+Table 1. Recovery error e(m, n) for different amounts of Gaussian measurements m and
+finite element refinements n.
+We have proven in this paper that our numerical recovery algorithm is near optimal with constant C that
+can be made arbitrarily close to one by choosing n sufficiently large. This means that the error e(m, n)
+satisfies e(m, n) ≤ CR(KH
+w )H1(Ω) for n sufficiently large.
+Increasing the number m of measurements is
+expected to decrease this Chebyshev radius. While one is tempted to think that the entries in each column
+of the table provides an upper bound for the Chebyshev radius of KH
+w for these measurements, this is not
+guaranteed since we are only measuring the error for one function from Kw, namely uH, and not all possible
+functions from Kw. However, the entries in any given column provide a lower bound for the Chebyshev
+radius of KH
+w provided n is sufficiently large.
+Increasing the number m of measurements requires a finer resolution, i.e., increasing n, of the finite
+element discretization until the perturbation ε in Theorem 3.2 is sufficiently small. This is indeed confirmed
+by the results in Table 1 where stagnating error bounds (in each fixed column) indicate the corresponding
+tip-over point. We notice in particular that for small values of n, the error becomes very large as m grows.
+This is explained by the fact that the Gramian matrix G becomes severely ill-conditioned, and in turn
+the prescriptions on ∥G − ˆG∥1 cannot be fulfilled when using finite element approximation of the Riesz
+representers on too coarse meshes. An overall convergence of the recovery error to zero can, of course, only
+take place when both m and n increase.
+20
+
+5.2. The case of point value measurements. In this section, we describe our numerical experiments in
+the case where the linear functionals λi,j are point evaluations at points from Ω. Recall that while the λi,j are
+not defined for general functions in H1(Ω) they are defined for functions in the model class KH := U(Hs(Ω))
+provided s is sufficiently large (s > 1/2 for d = 2 and s > 1 for d = 3). This means that the optimal recovery
+problem is well posed in such a case. We have given in §3.4 sufficient conditions on a numerical algorithm to
+give near optimal recovery and then we have gone on to show in §4.5 that our proposed numerical algorithm
+based on discrete harmonics converges to a near optimal recovery with any constant C > 1 provided that
+the finite element spaces are discretized fine enough.
+In the numerical experiments of this section, we again take Ω = (0, 1)2, s = 1, and the data to be the
+point values of the harmonic function uH defined in (5.1). We choose the evaluation points to be the zi,j of
+(5.3). We now provide in Table 2 the recovery error e(m, n). The observed behavior is similar to the case of
+Gaussian averages; see Table 1.
+n
+m
+4
+9
+16
+25
+36
+4
+0.70
+0.28
+0.19
+14.43
+15.49
+5
+0.70
+0.28
+0.18
+32.56
+8.02
+6
+0.70
+0.28
+0.18
+1.51
+2.27
+7
+0.70
+0.28
+0.18
+0.53
+0.89
+8
+0.70
+0.28
+0.18
+0.20
+0.14
+9
+0.70
+0.28
+0.18
+0.14
+0.11
+Table 2. Recovery error e(m, n) for different amounts of point evaluation measurements
+m and refinements n.
+5.3. Additional comments on the approximation of Riesz representers. Finally, we provide a little
+more information on the computations that may be of interest to the reader. We work in the same setting
+as in the previous sections. Let us begin with the rate of convergence of our numerical approximations to
+the Riesz representers.
+We first consider the computation of the Riesz representer for the Gaussian measurement functional
+centered at z = zi,j := (0.75, 0.5). Let φn ∈ Vhn be the approximation to the Riesz representer φ produced
+by the finite element computation. Figure 5.1 shows the error ∥φn − φ9∥H1(Ω), n = 2, . . . , 8. This graph
+indicates an error decay Ch1/2
+n
+which matches the rate guaranteed by Theorem 4.4, see also Remark 4.5.
+Next consider the computation of the Riesz representer for point evaluation at the same z. Figure 5.1
+reports the numerical computations of error in both the H1(Ω) and L∞(Ω) norms. Again, the graph indicates
+an error decay Ch1/2
+n
+for the H1(Ω) norm which matches the rate guaranteed by Theorem 4.7 and a decay
+rate closer to Chn for the L∞(Ω) norm (Theorem 4.7 only guarantees Ch1/2
+n ).
+6. Optimal data sites: Gelfand widths and sampling numbers
+In this section, we make some comments on the number of measurements m that are needed to guarantee
+a prescribed error in the recovery of u. Bounds on m are known to be governed by the Gelfand width for
+the case of general linear functionals and by sampling numbers when the functionals are required to be point
+evaluations. We explain what is known about these quantities for our specific model classes. As we shall see
+these issues are not completely settled for the model classes studied in this paper. The problem of finding
+the best choice of functionals, respectively point evaluations, is in need of further research.
+We have seen that the accuracy of the optimal recovery of u ∈ Kw is given by the Chebyshev radius
+R(Kw) := R(Kw)H1(Ω) or equivalently R(KH
+w ) := R(KH
+w )H1(Ω) for the harmonic component. The worst
+case recovery error R(K) over the class K is defined by
+(6.1)
+R(K)H1(Ω) := sup
+w∈Rm R(Kw)H1(Ω),
+Notice that this worst case error fixes the measurement functionals but allows the measurements w to come
+from any function in K. Both the individual error R(Kw) and the worst case error R(K) are very dependent
+21
+
+0.000010
+0.000100
+0.001000
+0.010000
+0.100000
+1.000000
+100
+1000
+dim(Vhn)
+Gaussian: H1 error
+Point evaluation: L∞ error
+Point evaluation: H1 error
+order 1
+2
+Figure 5.1. Approximation errors for the Riesz representers of the Gaussian and point
+evaluation functionals.
+on the choice of the data functionals λj. For example, in the case that these functionals are point evaluations
+at points z1, . . . , zm ∈ ¯Ω, then R(Kw) and R(K) will depend very much on the positioning of these points
+in ¯Ω.
+In the case of general linear functionals, one may fix m and then search for the λ1, . . . , λm that minimize
+the worst case recovery error over the class K. This minimal worst case error is called the Gelfand width of
+K. If we restrict the linear functionals to be given by point evaluation, we could correspondingly search for
+the sampling points x1, . . . , xm minimizing the worst case recovery error. This minimal error is called the
+deterministic sampling number of K.
+The goal of this section is not to provide new results on Gelfand widths and sampling numbers, since we
+regard this as a separate issue in need of a systematic study, but to discuss what is known about them in
+our setting and refer the reader to the relevant papers. Let us recall that R(Kw) is equivalent to R(KH
+w )H1
+and so we restrict our discussion in what follows to sampling of harmonic functions.
+6.1. Optimal choice of functionals. Suppose we fix the number m of observation to be allowed and ask
+what is the optimal choice for the λj, j = 1, . . . , m, and what is the optimal error of recovery for this choice.
+The answer to the second question is given by the Gelfand width of K. Given a compact set K of a Banach
+space X, we define the Gelfand width of K in X by
+(6.2)
+dm(K)X :=
+inf
+λ1,...,λm R(K)X
+where the infimum is taken over the linear functionals defined on X. Let us mention that this definition
+differs from that employed in the classical literature [21] where dm(K)X is defined as the infimum over all
+spaces F of codimension n of max{∥v∥X : v ∈ K ∩F}. The two definitions are equivalent in the case where
+K is a centrally symmetric set such that K + K ⊂ CK for some constant C ≥ 1.
+Any set of functionals which attains the infimum in (6.2) would be optimal. The Gelfand width is often
+used as a benchmark for performance since it says that no matter how the m functionals λ1, . . . , λm are
+chosen, the error of recovery of u ∈ K cannot be better than dm(K)X.
+When X is a Hilbert space and K is the ball of a Hilbert space Y with compact embedding in X, it is
+known that the Gelfand width coincides with the Kolmogorov width, that is
+dm(K)X = dm(K)X :=
+inf
+dim(E)=m dist(K, E)X =
+inf
+dim(E)=m max{∥v − PEv∥X : v ∈ K},
+where the infimum is taken over all linear spaces E of dimension m. This is precisely our setting as discussed
+in §3: taking X = H1 := H1(Ω) and K as in (1.4), we have
+(6.3)
+dm(K)H1(Ω) = dm(KH)H1(Ω) = dm(KH)H1(Ω) ∼ dm(KB)H1/2(Γ) = dm(KB)H1/2(Γ),
+22
+
+where the equivalence follows from (1.3). Upper and lower bounds for the Gelfand width of KB in L2(Γ)
+are explicitely given in [20].
+We can estimate the rate of decay of the Kolmogorov and Gelfand width of KB in H1/2(Γ) by the following
+general argument: as explained in §2.1, for the admissible range of smoothness, the Sobolev spaces Hs(Γ)
+have an intrinsic description by locally mapping the boundary onto domains of Rd−1. More precisely, in [17]
+and [10], the Hs(Γ) norm of g is defined as
+(6.4)
+∥g∥Hs(Γ) :=
+� J
+�
+j=1
+∥gj∥2
+Hs(Rj)
+�1/2
+,
+where the Rj are open bounded rectangles of Rd−1 that are mapped by transforms γj into portions Γj that
+constitute a covering of Γ, and gj = g ◦ γj are the local pullbacks.
+From this it readily follows that the Gelfand and Kolmogorov m-width of the unit ball of Hs(Γ) in the
+norm Ht(Γ), with 0 ≤ t < s behaves similar to that of the unit ball of Hs(R) in the norm Ht(R) where R is
+a bounded rectangle of Rd−1. The latter is known to behave like m− s−t
+d−1 . Therefore, for KH = U(Hs) with
+s > 1
+2 in the admissible range allowed by the boundary smoothness, one has
+(6.5)
+cm− s−1/2
+d−1 ≤ dm(KH)H1(Ω) ≤ Cm− s−1/2
+d−1 ,
+m ≥ 1,
+where c and C are positive constants depending only on Ω and s.
+Remark 6.1. We have already observed in §2 that the space Hs(Ω) is continuously embedded in the Sobolev
+space Hr(Ω) with r := max{s+ 1
+2, r∗} and in particular r = s+ 1
+2 for smooth domains. However the Gelfand
+and Kolmogorov widths of the unit ball of Hr(Ω) in H1(Ω) have the slower decay rate m− r−1
+d
+= m− s−1/2
+d
+compared to (6.5) for Hs(Ω).
+This improvement reflects the fact that the functions from Hs(Ω) have d
+variables but are in fact determined by functions of d − 1 variables. The reduction in dimension from d to
+d − 1 is related to the fact that in our formulation of our problem we have complete knowledge of f.
+6.2. Optimal choice of sampling points. We turn to the particular setting where the λj are point
+evaluations functionals,
+λj(v) = v(xj),
+at m points xj ∈ Ω. Similar to the Gelfand width, the deterministic sampling numbers are defined as
+(6.6)
+ρm(K)X :=
+inf
+x1,...,xm R(K)X,
+A variant of this is to measure the worst case expected recovery error when the m points are chosen at
+random according to a probabilty distribution and search for the distribution that minimizes this error,
+leading to the randomized sampling number of K. Obviously, one has
+(6.7)
+ρm(K)X ≥ dm(K)X.
+In the majority of the literature, deterministic and randomized sampling numbers are studied with error
+measured in the L2(Ω) norm. In this setting, concrete strategies for optimal deterministic and randomized
+point design have been given when K is the unit ball of a reproducing kernel Hilbert space H defined on Ω.
+In particular, the recent results in [16, 12, 18, 5] show that under the assumption
+�
+m>0
+|dm(K)L2(Ω)|2 < ∞,
+then, for all t > 1
+2,
+sup
+m≥1
+mtdm(K)L2(Ω) < ∞ =⇒ sup
+m≥1
+mtρm(K)L2(Ω) < ∞.
+In words, under the above assumptions, optimal recovery in L2(Ω) has the same algebraic convergence rate
+when using optimally chosen point values compared to an optimal choice of general linear functionals.
+While similar general results have not been established for Gelfand width and sampling numbers in the
+H1 norm, we argue that they hold in our particular setting where H = Hs(Ω). For simplicity, as in §4,
+we consider a domain that is either a polygon when d = 2 or polyhedron when d = 3, and thus consider
+the range
+d−1
+2
+< s <
+3
+2 where the restriction from below ensures that Hs(Ω) ⊂ C(Ω).
+Recalling the
+23
+
+finite element spaces Vh and their traces Th on the boundary, based on quasi-uniform meshes {Th}h>0, we
+consider for a given h > 0 the measurement points x1, . . . , xm that are the mesh vertices located on Γ. By
+the quasi-uniformity property the number m = m(h) of these points satisfies
+ch1−d ≤ m ≤ Ch1−d,
+for some c, C > 0 independent of h. If v ∈ Hs(Ω), its trace vΓ belongs to Hs(Γ). Then, denoting by Ih
+the piecewise linear interpolant on the boundary, standard finite element approximation theory ensures the
+estimate
+∥vΓ − IhvΓ∥H1/2(Γ) ≤ Chs− 1
+2 ∥vΓ∥Hs(Γ) = Chs− 1
+2 ∥v∥Hs(Ω),
+for some C that only depends on s. Therefore, introducing ˜v := EIhv, one has
+∥v − ˜v∥H1(Ω) ≤ CE∥vΓ − IhvΓ∥H1/2(Γ) ≤ CDEm− s−1/2
+d−1 ∥v∥Hs(Ω).
+Since ˜v only depends on the value of v at the points x1, . . . , xm, we have thus proved an upper bound of
+order m− s−1/2
+d−1
+for ρm(KH)H1(Ω), and in turn for ρm(K)H1(Ω). In view of (6.7) and (6.5), a lower bound of
+the same order must hold. In summary, similar to the Gelfand widths, the sampling numbers satisfy
+(6.8)
+˜cm− s−1/2
+d−1 ≤ ρm(K)H1(Ω) ≤ ˜Cm− s−1/2
+d−1 ,
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+Guergana
+Petrova,
+Department
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+A&M
+University,
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+25
+

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