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STABILITY AND TRANSITIONS OF THE SECOND GRADE POISEUILLE FLOW
11 Sep 2015
Saadet Ozer
Taylan Sengul
STABILITY AND TRANSITIONS OF THE SECOND GRADE POISEUILLE FLOW
11 Sep 2015
In this study we consider the stability and transitions for the Poiseuille flow of a second grade fluid which is a model for non-Newtonian fluids. We restrict our attention to flows in an infinite pipe with circular cross section that are independent of the axial coordinate.We show that unlike the Newtonian (ǫ = 0) case, in the second grade model (ǫ = 0 case), the time independent base flow exhibits transitions as the Reynolds number R exceeds the critical threshold Rc ≈ 4.124ǫ −1/4 where ǫ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects.At R = Rc, we find that generically the transition is either continuous or catastrophic and a small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates. The time period of the bifurcated solution tends to infinity as R tends to Rc. Our numerical calculations suggest that for low ǫ values, the system prefers a catastrophic transition where the bifurcation is subcritical.We also find that there is a Reynolds number R E with R E < Rc such that for R < R E , the base flow is globally stable and attracts any initial disturbance with at least exponential speed. We show that R E ≈ 12.87 at ǫ = 0 and R E approaches Rc quickly as ǫ increases.
Introduction
Certain natural materials manifest some fluid characteristics that can not be represented by well-known linear viscous fluid models. Such fluids are generally called non-Newtonian fluids. There are several models that have been proposed to predict the non-Newtonian behavior of various type of materials. One class of fluids which has gained considerable attention in recent years is the fluids of grade n [11,8,13,12,7,17,6]. A great deal of information for these types of fluids can be found in [4]. Among these fluids, one special subclass associated with second order truncations is the so called second-grade fluids. The constitutive equation of a second grade fluid is given by the following relation for incompressible fluids:
t = −pI + µA 1 + α 1 A 2 + α 2 A 2 1 ,
where t is the stress tensor, p is the pressure, µ is the classical viscosity, α 1 and α 2 are the material coefficients. A 1 and A 2 are the first two Rivlin-Ericksen tensors defined by
A 1 = ∇v + ∇v T , A 2 =Ȧ 1 + A 1 ∇v + ∇v T A 1 ,
where v is the velocity field and the overdot represents the material derivative with respect to time. This type of constitutive relation was first proposed in [2]. The following conditions: α 1 + α 2 = 0, µ ≥ 0, α 1 ≥ 0, must be satisfied for the second-grade fluid to be entirely consistent with classical thermodynamics and the free energy function achieves its minimum in equilibrium [5]. Equation of motion for an incompressible second grade Rivlin Ericksen fluid is represented as:
ρ(v t + w × v + ∇ |v| 2 2 ) = −∇p + µ∆v + α[∆v t + ∆w × v + ∇(v · ∆v + 1 4 |A 1 | 2 )], ∇ · v = 0.
where ρ is the density, α = α 1 = −α 2 , represents the second order material constant. Subscript t denotes the partial derivative with respect to time, w is the usual vorticity vector defined by w = ∇ × v. We next define the non-dimensional variables:
v * = v U , p * = p ρ U 2 , t * = tU L , x * = x L ,
where U and L are characteristic velocity and length, respectively. By letting ǫ represent the second order non-dimensional material constant which measures the relative strength of second order viscous effects compared to inertial effects and defining the Reynolds number, R = ρU L µ , ǫ = α ρL 2 , the equation of motion, with asterisks omitted, can be expressed as:
(1) ∇p = 1 R ∆v + ǫ(∆w × v + ∆v t ) − v t − w × v,
where the characteristic pressurep is defined as:
p = p + |v| 2 2 − ǫ(v∆v + 1 4 |A 1 | 2 ).
Taking curl of both sides of (1) we can simply write the equation of motion as:
(2) ∇ × [ 1 R ∆v + ǫ(∆w × v + ∆v t ) − v t − w × v] = 0,
which is the field equation of incompressible unsteady second grade Rivlin-Ericksen fluid independent of the choice of any particular coordinate system. Now we restrict our interest to flows in a cylindrical tube and assume that the velocity is dependent only on two cross-sectional variables x, y and the time t. The incompressiblity of the fluid allows us to introduce a stream function ψ such that v = (ψ y , −ψ x , w) where ψ = ψ(t, x, y) and also w = w(t, x, y).
We further take the cross section of the cylinder to be a disk with unit radius and consider the no-slip boundary conditions. Then the equations (2) admit the following steady state solution
w 0 = 1 2 (1 − x 2 − y 2 ), ψ 0 = 0.
Here the characteristic velocity has been chosen as U = pL 2 4µ . First considering the deviation w ′ = w − w 0 and ψ ′ = ψ − ψ 0 and then introducing the polar coordinates w ′′ (t, r, θ) = w ′ (t, r cos θ, r sin θ) and ψ ′′ (t, r, θ) = ψ ′ (t, r cos θ, r sin θ) and ignoring the primes, the equations become
(3) ∂ ∂t (1 − ǫ∆)w = 1 R ∆w + Rψ θ + J(ψ, (1 − ǫ∆)w), ∂ ∂t ∆(ǫ∆ − 1)ψ = − 1 R ∆ 2 ψ + ǫR∆w θ + J((1 − ǫ∆)∆ψ, ψ) + ǫJ(∆w, w),
in the interior of the unit disk Ω where J is the advection operator
J(f, g) = 1 r (f r g θ − f θ g r ).
The field equations are supplemented with no-slip boundary conditions for the velocity field (4) w = ψ = ∂ψ ∂r = 0 at r = 1.
In this paper, our main aim is to investigate the stability and transitions of (3) subject to (4). We first prove that the system undergoes a dynamic transition at the critical Reynolds number R c ≈ 4.124ǫ −1/4 . As R crosses R c the steady flow loses its stability, and a transition occurs. If we denote the azimuthal wavenumber of an eigenmode by m, then two modes, called critical modes hereafter, with m = 3 and radial wavenumber 1, become critical at R = R c . Using the language of dynamic transition theory [10], we can show that the transition is either Type-I(continuous) or Type-II(catastrophic). In Type-I transitions, the amplitudes of the transition states stay close to the base flow after the transition. Type-II transitions, on the other hand, are associated with more complex dynamical behavior, leading to metastable states or a local attractor far away from the base flow.
We show that the type of transition preferred in system (3) is determined by the real part of a complex parameter A which only depends on ǫ. In the generic case of nonzero imaginary part of A, there are two possible transition scenarios depending on the sign of the real part of A: continuous or catastrophic. In the continuous transition scenario, a stable, small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates on R > R c . The time period of the bifurcated solution tends to infinity as R approaches R c , a phenomenon known as infinite period bifurcation [9]. The dual scenario is the catastrophic transition where the bifurcation is subcritical on R < R c and a repeller bifurcates. In the non-generic case where the imaginary part of A vanishes, the limit cycle degenerates to a circle of steady states.
The transition number A depends on the system parameter ǫ in a non-trivial way, hence it is not possible to find an analytical expression of A as a function of ǫ. So, A must be computed numerically for a given ǫ. Physically, the transition number can be considered as a measure of net mechanical energy transferred from all modes back to the critical modes which in turn modify the base flow. We show that A is determined by the nonlinear interactions of the critical modes (m = 3) with all the modes having m = 0 and m = 6. Moreover, our numerical computations suggest that for low ǫ fluids (ǫ < 1), just a single nonlinear interaction, namely the one with m = 0 and radial wavenumber 1 mode, dominates all the rest contributions to A. Our numerical experiments with low ǫ, i.e. ǫ < 1, suggest that the real part of A is always positive indicating a catastrophic transition at R = R c .
We also determine the Reynolds number threshold R E > 0 below which the Poiseuille flow is globally stable, attracting all initial conditions with at least exponential convergence in the H 1 0 (Ω) norm for the velocity. We find that R E ≈ 12.87 when ǫ = 0. The gap between R E and R c shrinks to zero quickly as ǫ is increased.
The paper is organized as follows: In Section 3, the linearized stability of the system is studied and the principle of exchange of stabilities is investigated. In Section 4, transition theorem of the system is presented with its proof given in Section 5. Section 6 is devoted to the energy stability of the system. In Section 7, we give a detailed numerical analysis. Finally in Section 8 the conclusions and possible extensions of this study are discussed in detail.
The model and the functional setting
Throughout Re z, Im z, z will denote the real part, imaginary part and the complex conjugate of a complex number z. Let
X 1 = H 1 0 (Ω) × H 2 0 (Ω), X = L 2 (Ω) × L 2 (Ω),
where Ω is the unit disk in R 2 , H 1 0 (Ω) and H 2 0 (Ω) denote the usual Sobolev spaces and L 2 (Ω) is the space of Lebesgue integrable functions.
For φ i = wi(r,θ) ψi(r,θ) ∈ X, i = 1, 2, the inner product on X is defined by
(5) φ 1 , φ 2 = 2π 0 1 0 (w 1 w 2 + ψ 1 ψ 2 )rdrdθ,
with the norm on X defined as φ 2 = φ, φ . We define the linear operators M : X 1 → X and N : X 1 → X as
(6) M = I − ǫ∆ 0 0 ∆(ǫ∆ − I) , N = 1 R ∆ R∂ θ ǫR∆∂ θ − 1 R ∆ 2 ,
and the nonlinear operator H : X 1 → X as
H(φ) = J(ψ, (1 − ǫ∆)w) J((1 − ǫ∆)∆w, ψ) + ǫJ(∆w, w) , for φ = w ψ ∈ X 1 .
We will use H to denote both the nonlinear operator as well as the bilinear form
(7) H(φ I , φ J ) = J(ψ I , (1 − ǫ∆)w J ) J((1 − ǫ∆)∆w I , ψ J ) + ǫJ(∆w I , w J ) , for φ I = [w I , ψ I ] T , φ J = [w J , ψ J ] T .
Also we will use the symmetrization of this bilinear form
(8) H s (φ I , φ J ) = H(φ I , φ J ) + H(φ J , φ I ).
Then the equations (3) and (4) can be written in the following abstract form
(9) M φ t = N φ + H(φ), φ ∈ X 1 ,
with initial condition
φ(0) = φ 0 ∈ X 1 .
Linear Stability
To determine the transitions of (9), the first step is to study the eigenvalue problem N φ = βM φ of the linearized operator. This is equivalent to the problem (10)
1 R ∆w + Rψ θ = β(1 − ǫ∆)w, 1 R ∆ 2 ψ − ǫR∆w θ = β(1 − ǫ∆)∆ψ,
subject to the boundary conditions (4). An interesting feature of the eigenvalue problem is the following.
Lemma 1. Any eigenvalue β of (10) with boundary conditions (4) is real.
Proof. Multiplying the first equation of (10) by ∆w and the second equation by ψ, integrating over the domain Ω, we obtain after integration by parts
(11) ( 1 R + βǫ) ∆w 2 + R Ω ψ θ ∆wdrdθ = −β ∇w 2 , and (12) − ( 1 R + βǫ) ∆ψ 2 + ǫR Ω ∆w θ ψdrdθ = β ∇ψ 2 . Let A 1 = ǫ ∆w 2 + ∆ψ 2 , A 2 = ǫ ∇w 2 + ∇ψ 2 A 3 = 2ǫ Ω ∆w θ ψdrdθ. Now consider −ǫ×(11) + (12) which is (13) − ( 1 R + βǫ)A 1 + RRe(A 3 ) = βA 2 ,
after integrating by parts. Taking the imaginary part of (13) gives
Im(β)(ǫA 1 + A 2 ) = 0. Since (ǫA 1 + A 2 ) ≥ 0, we must have Im(β) = 0.
Now we turn to the problem of determining explicit expressions of the solutions of the eigenvalue problem of the linearized operator. Thanks to the periodicity in the θ variable, for m ∈ Z and j ∈ Z + , we denote the eigenvectors of (10) by (14) φ m,j (r, θ) = e imθ ϕ m,j (r),
ϕ m,j (r) = w m,j (r) ψ m,j (r)
with corresponding eigenvalues β m,j . Let us set the eigenvalues β m,j for m = 0 to be ordered so that β m,1 ≥ β m2 ≥ · · · for each m ∈ Z.
Plugging the ansatz (14) into (10) and omitting j we obtain two ODE's in the r-variable.
(15) 1 R + βǫ ∆ m w m + imRψ m = βw m , − 1 R + βǫ ∆ 2 m ψ m + iǫmR∆ m w m = −β∆ m ψ m ,
with boundary conditions
(16) w m (1) = ψ m (1) = ψ ′ m (1) = 0 where ∆ m = d 2 dr 2 + 1 r d dr − m 2 r 2 .
When m = 0, the equations (15) become decoupled and we easily find that there are two sets of eigenpairs given by
w 1 0,j (r) = J 0 (α 0,j r), ψ 1 0,j (r) = 0, β 1 0,j = −α 2 0,j R(1 + ǫα 2 0,j ) , w 2 0,j (r) = 0, ψ 2 0,j (r) = J 0 (α 1,j r) − J 0 (α 1,j ), β 2 0,j = −α 2 1,j R(1 + ǫα 2 1,j ) ,
where α k,j is the jth zero of the kth Bessel function J k . In particular, β 0,j < 0 for all ǫ and for all R.
In the m = 0 case, as the eigenvalues are real by Lemma 1, the multiplicity of each eigenvalue β = β m,j = β −m,j ∈ R is generically two with corresponding eigenvectors φ m,j and φ −m,j = φ m,j .
Solving the first equation of (15) for ψ m and plugging it into the second equation of (15) yields a sixth order equation
(17) (λ + ∆ m )(µ + ∆ m )∆ m w m = 0 where (18) λ = √ ǫmR − β m,j 1 R + ǫβ m,j , µ = − √ ǫmR − β m,j 1 R + ǫβ m,j .
It is easy to check that λ = 0 or µ = 0 yield only trivial solutions to equations (16) and (17). So we will assume λ = 0, and µ = 0. When m > 0, the general solution of (17) is
w m = c 1 r m + c 2 J m ( √ λr) + c 3 J m ( √ µr) + c 4 r −m + c 5 Y m ( √ λr) + c 6 Y m ( √ µr),
where J m and Y m are the Bessel functions of the first and the second kind respectively.
The boundedness of the solution and its derivatives at r = 0 implies that c 4 = c 5 = c 6 = 0 and we get the eigensolutions
(19) w m,j (r) = c 1 r m + c 2 J m ( √ λr) + c 3 J m ( √ µr) if m > 0 w −m,j , if m < 0 ψ m,j (r) = d 1 r m + d 2 J m ( √ λr) + d 3 J m ( √ µr) if m > 0 ψ −m,j , if m < 0 with d 1 = −iβ m,j c 1 mR , d 2 = −i √ ǫc 2 , d 3 = i √ ǫc 3 .
The eigenvalues and two of the three coefficients c 1 , c 2 , c 3 in (19) are determined by the boundary conditions (16) which form a linear system for the coefficients c k . This system has a nontrivial solution only when the dispersion relation
(20) 1 J m ( √ λ) J m ( √ µ) β √ ǫmRJ m ( √ λ) − √ ǫmRJ m ( √ µ) βm √ ǫmR √ λJ ′ m ( √ λ) − √ ǫmR √ µJ ′ m ( √ µ) = 0, is satisfied. Using the identity J ′ m (z) = m z J m (z) − J m+1 (z)
we can show that (20) is equivalent to
(21) √ λJ m ( √ λ)J m+1 ( √ µ) + √ µJ m ( √ µ)J m+1 ( √ λ) = 0,
where J m is the Bessel function of the first kind of order m.
To compute the critical Reynolds number R c , we set β = 0 in (21) which, after some manipulation, yields
(22) I m ( √ λ)J ′ m ( √ λ) − J m ( √ λ)I ′ m ( √ λ) = 0.
Once (22) is solved for λ, the corresponding Reynolds number is obtained by the relation λ = √ ǫmR 2 (from (18) when β = 0). We note here that this is the exact same equation as the one obtained in [12]. For each m, the equation (22)
/m) 1/2 for m = 1, . . . , 5. infinitely many solutions {λ m,j } ∞ j=1 where λ m,j increases with j and λ m,j → ∞ as j → ∞. Letting R m,j = ǫ −1/4 (λ m,j /m) 1/2 , we have β m,j = 0 when R = R m,j .
We define the critical Reynolds number
R c = min m,j R m,j = min m R m,1 = ǫ −1/4 min m∈Z+ λ m,1 m , so that β m,j ≤ 0 if R ≤ R c .
There has been recent progress on the properties of zeros of (22). In [1], the estimate
2 4 (m + 1)(m + 2)(m + 3) (m + 4)(m + 5) √ 5m + 15 < λ 2 m,1 < 2 4 (m + 1)(m + 2)(m + 3)(m + 4)(m + 5) 5m + 17 ,(23)
on λ m,1 is obtained. Using (23), we can show that the upper bound for λ m,1 /m for m = 3 is less than its lower bound for all m ≥ 7 which implies that λ 3,1 /3 < λ m,1 /m for all m ≥ 7. Thus R c is minimized for some m smaller than 7 and hence can be found by brute force. Looking at Table 1, we find that the expression above is indeed minimized when m = 3 and obtain the relation (24), i.e. R c = R 3,1 ≈ 4.124ǫ −1/4 .
Defining the left hand side of (21) as ω(β, R), the equation (21) becomes ω(β, R) = 0. By the implicit function theorem, this defines β 3,1 (R) for R near R c with β 3,1 (R c ) = 0. With the aid of symbolic computation, we can compute
dβ 3,1 dR | R=Rc = ∂ω/∂R ∂ω/∂β | R=Rc,β=β3,1 = 0.12 √ ǫ 0.02 + ǫ > 0.
Thus we have proved the Principal of Exchange of Stabilities which we state below.
Theorem 1. For ǫ = 0, let (24) R c = ǫ −1/4 λ 3,1 3 ≈ 4.124ǫ −1/4 . Then (25) β 3,1 (R) = β −3,1 (R) < 0 if R < R c = 0 if R = R c > 0 if R > R c β m,j (R c ) < 0, if (m, j) = (±3, 1).
Note that Theorem 1 is in contrast to the ǫ = 0 case where the basic flow is linearly stable for all Reynolds numbers. This can be seen easily by noting that when ǫ = 0, the inner product of the second equation in (10) with ψ yields
(26) ∆ψ 2 = −βR ∇ψ 2 .
With the Dirichlet boundary conditions, from (26), it follows easily that β < 0 if ψ = 0.
For the proof and presentation of Theorem 2, we also need to solve the eigenvalue problem of the adjoint linear operator which yields adjoint modes orthogonal to the eigenmodes of the linear operator. Adjoint problem is obtained by taking the inner product of (9) by φ * and moving the derivatives via integration by parts onto φ * by making use of the boundary conditions. This yields the following adjoint problem
N * φ * = β * M * φ * , where M * = M = I − ǫ∆ 0 0 ∆(ǫ∆ − I) , N * = 1 R ∆ −ǫR∆∂ θ R∂ θ − 1 R ∆ 2 , φ * = w * ψ * ,
and w * , φ * satisfies the same boundary conditions (4) as w and φ. We denote the adjoint eigenvectors by φ * m,j = w * m,j ψ * m,j T and we also have the adjoint eigenvalues β * m,j = β m,j . The reason we introduce the adjoint eigenmodes is to make use of the following orthogonality relation
(27) φ m,i , M φ * n,j = 0 if (m, i) = (n, j).
Dynamic Transitions
Let us briefly recall here the classification of dynamic transitions and refer to [10] for a detailed rigorous discussion. For ǫ = 0, as the critical Reynolds number R c is crossed, the principle of exchange of stabilities (25) dictates that the nonlinear system always undergoes a dynamic transition, leading to one of the three type of transitions, Type-I(continuous), II(catastrophic) or III(random). On R > R c , the transition states stay close to the base state for a Type-I transition and leave a local neighborhood of the base state for a Type-II transition. For Type-III transitions, a local neighborhood of the base state is divided into two open regions with a Type-I transition in one region, and a Type-II transition in the other region. Type-II and Type-III transitions are associated with more complex dynamics behavior.
Below we prove that for (3), only two scenarios are possible. In the first scenario, the system exhibits a Type-I (or continuous) transition and a stable attractor will bifurcate on R > R c which attracts all sufficiently small disturbances to the Poiseuille flow. We prove that this attractor is homeomorphic to the circle S 1 and is generically a periodic orbit. The Figure 1 shows the stream function of the bifurcated time-periodic solution. The dual scenario is that the system exhibits a Type-II (or catastrophic) transition.
The type of transition at R = R c depends on the transition number
(28) A = ∞ j=1 A 0,j + A 6,j ,
where A m,j represent the nonlinear interaction of the critical modes with the mode with azimuthal wavenumber m and radial wavenumber j. The formulas for A m,j are (29) To recall the meaning of various terms in (29), β n,j is the jth eigenvalue of the nth azimuthal mode with corresponding eigenvector φ n,j and adjoint eigenvector φ * n,j . ·, · denotes the inner product (5), H denotes the bilinear form (7), H s is its symmetrization (8), M is the linear operator defined by (6).
A 0,j = 1 φ 3,1 , M φ * 3,1 Φ 0,j H s (φ 3,1 , φ 0,j ), φ * 3,1 A 6,j = 1 φ 3,1 , M φ * 3,1 Φ 6,j H s (φ 3,1 , φ 6,j ), φ * 3,1 Φ 0,j = 1 −β 0,j φ 0,j , M φ * 0,j H s (φ 3,1 , φ 3,1 ), φ * 0,j , Φ 6,j = 1 −β 6,j φ 6,j , M φ * 6,j H(φ 3,1 , φ 3,1 ), φ * 6,j ,
Theorem 2. If ǫ = 0 then the following statements hold true.
(1) If Re(A) < 0 then the transition at R = R c is Type-I and an attractor Σ R bifurcates on R > R c which is homeomorphic to S 1 . If Im(A) = 0 then Σ R is a cycle of steady states. If Im(A) = 0 then Σ R is the orbit of a stable limit cycle given by Here w 3,1 and ψ 3,1 are the vertical velocity and stream function of the eigenmode of the linear operator with corresponding eigenvalue β 3,1 . (2) If Re(A) > 0 then the transition at R = R c is Type-II and a repeller Σ R bifurcates on R < R c . If Im(A) = 0 then Σ R is a cycle of steady states. If Im(A) = 0 then Σ R is the orbit of of an unstable limit cycle given by (30) with β 3,1 replaced by −β 3,1 .
Remark. In the generic case of Im(A) = 0, Theorem 2 guarantees the existence of a stable (unstable) bifurcated periodic solution on R > R c (R < R c ). By (31), the period of the bifurcated solution approaches to infinity as R ↓ R c (R ↑ R c ).
Proof Of Theorem 2
As is standard in the dynamic transition approach, the proof of Theorem 2 depends on the reduction of the field equations (9) on to the center manifold.
Let us denote the (real) eigenfunctions and adjoint eigenfunctions corresponding to the critical eigenvalue β 3,1 by e 1 (r, θ) = Re(φ 3,1 (r, θ)) e 2 (r, θ) = Im(φ 3,1 (r, θ)) e * 1 (r, θ) = Re(φ * 3,1 (r, θ)) e * 2 (r, θ) = Im(φ * 3,1 (r, θ)) By the spectral theorem, the spaces X 1 and X can be decomposed into the direct sum
X 1 = E 1 ⊕ E 2 , X = E 1 ⊕ E 2 , where E 1 = span{e 1 , e 2 },E 2 = {u ∈ X 1 | u, e * i = 0 i = 1, 2}, E 2 = closure of E 2 in X.
Since M : X 1 → X is an invertible operator, we can define L = M −1 N and G = M −1 H. Now the abstract equation (9) can be written as
(32) dφ dt = Lφ + G(φ).
The linear operator L in (32) can be decomposed into
L = J ⊕ L, J = L | E1 : E 1 → E 1 , L = L | E2 : E 2 → E 2 ,
Since the eigenvalues are real, we have Le k = β 3,1 e k for k = 1, 2. Hence we have J = β 3,1 (R)I 2 , where I 2 is the 2 × 2 identity matrix. We know that when J is diagonal, we have the following approximation of the center manifold function Φ : E 1 → E 2 near R ≈ R c ; see [10].
(33) − LΦ(x) = P 2 G k (x) + o(k),
The meaning of the terms in the above formula (33) are as follows.
a) o(k) = o( x k ) + O(|β 3,1 (R)| x k ) as R → R c , x → 0, b) P 2 : X → E 2 is the canonical projection, c) x is the projection of the solution onto E 1 ,(34)
x(t, r, θ) = x 1 (t)e 1 (r, θ) + x 2 (t)e 2 (r, θ) d) G k denotes the lowest term of the Taylor expansion of G(u) around u = 0. In our case G is bilinear and thus k = 2 in and G = G k . It is easier to carry out the reduction using complex variables. So we write (34) as
(35) x(t, r, θ) = z(t)φ 3,1 (r, θ) + z(t)φ 3,1 (r, θ)
where z(t) = 1 2 (x 1 (t) − ix 2 (t)). Let us expand the center manifold function by (36) Φ = (n,j) =(±3,1) Φ n,j (t)φ n,j (r, θ)
Plugging the above expansion into the center manifold approximation formula (33), taking inner product with M φ * n,j and using the orthogonality (27) we have
(37) Φ n,j = 1 −β n,j φ n,j , M φ * n,j H(x), φ * n,j + o(2).
Since H is bilinear,
H(x) = H(zφ 3,1 + zφ 3,1 ) = z 2 H(φ 3,1 , φ 3,1 ) + zzH s (φ 3,1 , φ 3,1 ) + z 2 H(φ 3,1 , φ 3,1 ), with the operator H s defined by (8). Thanks to the orthogonality 2π 0 e inθ e −imθ dθ = 2πδ nm , we have
(39) H(φ m1,i1 , φ m2,i2 ), φ * m3,i3 = 0, if m 1 + m 2 = m 3 With φ 3,1 = φ −3,1 , this implies (40) H(x), φ * n,j = 0 if n / ∈ {0, −6, 6}.
According to (40), (36) and (37),
(41) Φ(t) = ∞ j=1 Φ 0,j (t)φ 0,j + Φ 6,j (t)φ 6,j + Φ −6,j (t)φ −6,j + o(2)
That is the center manifold is o(2) in eigendirections whose azimuthal wavenumber is not 0, 6 or −6. The equation (38) implies that (42) and (37), we get the coefficients of the center manifold in (41)
(42) H(x), φ * 0,j = zz H s (φ 3,1 , φ 3,1 ), φ * 0,j , H(x), φ * 6,j = z 2 H(φ 3,1 , φ 3,1 ), φ * 6,j , H(x), φ * −6,j = H(x), φ * 6,j . By(43) Φ 0,j = zzΦ 0,j + o(2), Φ 6,j = z 2Φ 6,j + o(2), Φ −6,j = Φ 6,j ,
whereΦ 0,j andΦ 6,j are given by (29).
As the dynamics of the system is enslaved to the center manifold for small initial data and for Reynolds numbers close to the critical Reynolds number R c , it is sufficient to investigate the dynamics of the main equation (9) on the center manifold. For this reason we take
φ(t) = x(t) + Φ(t), in (9) to obtain (44) dz dt M φ 3,1 + dz dt M φ 3,1 = zN φ 3,1 + zN φ 3,1 + H(x + Φ).
To project the above equation onto the center-unstable space E 1 , we take inner product of (44) with φ * 3,1 and use
M φ 3,1 , φ * 3,1 = 0, and N φ 3,1 = β 3,1 M φ 3,1 , N φ 3,1 = β 3,1 M φ 3,1 ,
to get the following reduced equation of (9).
(45) dz dt = β 3,1 (R)z + 1 φ 3,1 , M φ * 3,1 H(x + Φ), φ * 3,1 .
The reduced equation (45) describes the transitions of the full nonlinear system for R near R c and small initial data. At this stage, the nonlinear term in (45) is too complicated to explicitly describe the transition. Thus we need to determine the lowest order expansion in z of the nonlinear term H(x + Φ), φ * 3,1 . By the bilinearity of H,
(46) H(x + Φ), φ * 3,1 = H(x), φ * 3,1 + H s (x, Φ), φ * 3,1 + H(Φ), φ * 3,1 .
The first term in (46) vanish by (40) and the last term in (46) is o(3) as H(Φ) = o(3) since Φ = O(2) and H is bilinear. Thus (46) becomes
(47) H(x + Φ), φ * 3,1 = H s (x, Φ), φ * 3,1 + o(3)
. Using the expression (35) for x, we can rewrite (47) as
(48) H(x + Φ), φ * 3,1 = z H s (φ 3,1 , Φ), φ * 3,1 + z H s (φ 3,1 , Φ), φ * 3,1 + o(3)
. Now we use the expansion (41) of Φ in (48) and the orthogonality relations
H s (φ 3,1 , φ n,j ), φ * 3,1 = 0 if n = 0,
and H s (φ 3,1 , φ n,j ), φ * 3,1 = 0 if n = 6, which follow from (39) to arrive at (49)
H(x + Φ), φ * 3,1 = ∞ j=1 zΦ 0,j H s (φ 3,1 , φ 0,j ), φ * 3,1 + zΦ 6,j H s (φ 3,1 , φ 6,j ), φ * 3,1 + o(3).
Defining the coefficient A by (28) and making use of (43) and (49), we write down the approximate equation of (45) as
(50) dz dt = β 3,1 (R)z + A|z| 2 z + o(3).
To finalize the proof, there remains to analyze the stability of the zero solution of (50) for small initial data. In polar coordinates z(t) = |z|e iγ , (50) is equivalent to
(51) d|z| dt = β 3,1 (R)|z| + Re(A)|z| 3 + o(|z| 3 ), dγ dt = Im(A)|z| 2 + o(|z| 3 ).
For R > R c as β 3,1 > 0, it is clear from (51) that z = 0 is unstable if Re(A) > 0 and is locally stable if Re(A) < 0. In the latter case, the bifurcated solution is
z(t) = −β 3,1 (R) Re(A) exp −i Im(A) Re(A) β 3,1 (R)t .
Thus to determine the stability of the bifurcated state as R crosses the critical Reynolds number R c , we need to compute the sign of the real part of A.
The details of the assertions in the proof of Theorem 2 follow from the attractor bifurcation theorem in [10]. That finishes the proof.
Energy Stability
In this section we study the energy stability of the equations (3) which is related to at least exponential decay of solutions to the base flow. We refer to [16] for a multitude of applications of this theory.
For f , g, h in H 1 0 (Ω), the following two properties of J follows from integrating by parts
d dt ( w 2 + ǫ ∇w 2 ) = − 1 R ∇w 2 + R ψ θ , w + ǫ J(∆w, ψ), w ,(54)1 2 d dt ( ∇ψ 2 + ǫ ∆ψ 2 ) = − 1 R ∆ψ 2 + ǫR ∆w θ , ψ + ǫ J(∆w, w), ψ .
Adding equations (53) and (54) and using (52) once again, we arrive at
(55) 1 2 d dt E(t) = − 1 R I 1 (t) + RI 2 (t),
where E = w 2 + ǫ ∇w 2 + ∇ψ 2 + ǫ ∆ψ 2
I 1 = ∇w 2 + ∆ψ 2 I 2 = ψ θ , w − ǫ∆w . Letting (56) 1 R 2 E = max X1\{0} I 2 I 1 ,
we have by (55)
(57) d dt E ≤ −2R( 1 R 2 − 1 R 2 E )I 1 .
Since I 1 ≥ 0 and I 2 = 0 whenever ψ θ = 0, R E must be nonnegative.
Since w ∈ H 1 0 (Ω) and ∇ψ ∈ H 1 0 (Ω), by the Poincaré inequality, |∇w| 2 ≥ η 1 |w| 2 and |∆ψ| 2 ≥ η 1 |∇ψ| 2 , where η 1 ≈ 5.78 is the first eigenvalue of negative Laplacian on Ω. Thus we have (58)
I 1 ≥ η 1 1 + ǫη 1 E Now let c R = 2Rη 1 1 + ǫη 1 ( 1 R 2 − 1 R 2 E ),
and suppose that R < R E . Then c R > 0 and by (57) and (58),
d dt E(t) ≤ −c R E(t).
Hence the Gronwall's inequality implies
E(t) ≤ e −cRt E(0).
In particular, for R ≤ R E , c R > 0 and any initial disturbance in X 1 will decay to zero implying the unconditional stability of the basic steady state solution.
Using the variational methods to maximize the quantity in (56), we find the resulting Euler-Lagrange equations as
(59) ∆w + R 2 2 (1 − ǫ∆)ψ θ = 0, ∆ 2 ψ + R 2 2 (1 − ǫ∆)w θ = 0.
Considering (59) as an eigenvalue problem with R playing the role of the eigenvalue, R E is just the smallest positive eigenvalue. To solve (59), we plug the ansatz w = e imθ w m (r) and ψ = e imθ ψ m (r) into (59) which yields
(60) ∆ m w m + i mR 2 2 (1 − ǫ∆ m )ψ m = 0 ∆ 2 m ψ m + i mR 2 2 (1 − ǫ∆ m )w m = 0, where ∆ m = d 2 dr 2 + 1 r d dr − m 2 r 2 .
Taking ∆ m of the second equation above and using the first equation, we obtain
(61) p(∆ m )ψ m = 0 where p(ξ) = ξ 3 + m 2 R 4 4 (1 − ǫξ) 2 .
Let ξ 1 , ξ 2 and ξ 3 be the three roots of p. As the discriminant of p is negative, one root is real and the others are complex conjugate. The factorization of the operator in (61) gives
(62) (∆ m − ξ 1 )(∆ m − ξ 2 )(∆ m − ξ 3 )ψ m = 0.
The general solution of (62) is
ψ m = 3 k=1 c k I m ( ξ k r) +c k K m ( ξ k r),
where I m and K m are the modified Bessel functions. The boundedness of the solution at r = 0 necessitatesc k = 0 for k = 1, 2, 3. Thus
ψ m = 3 k=1 c k I m ( ξ k r).ξ −1 1 I m ( √ ξ 1 ) ξ −1 2 I m ( √ ξ 2 ) ξ −1 3 I m ( √ ξ 3 ) I m ( √ ξ 1 ) I m ( √ ξ 2 ) I m ( √ ξ 3 ) √ ξ 1 I ′ m ( √ ξ 1 ) √ ξ 2 I ′ m ( √ ξ 2 ) √ ξ 3 I ′ m ( √ ξ 3 ) = 0.
For fixed m and ǫ, the equation (63) has infinitely many solutions R = R m,j (ǫ), j ∈ Z + . Letting
(64) R m = min j∈Z+ R m,j ,
the critical Reynolds number is given by
(65) R E = min m∈Z+ R m .
We present the numerical computations of R E in the next section.
A N ,
where A N is the series in (28) truncated at N , i.e. A N = N j=1 A 0,j + A 6,j and A m,j represents the nonlinear interaction of the critical modes and the mode with azimuthal wavenumber m and radial wavenumber j given by (29). A symbolic computation software is used to compute A N . We present our numerical computations of A N in Figure 2 for ǫ = 1, 10 −1 , 10 −2 , 10 −3 and 1 ≤ N ≤ 10. The imaginary part of A is nonzero and we are only interested in the sign of the real part of A to determine the type of transition according to Theorem 2. To simplify the presentation, we scale all A N 's so that |Re(A 1 )| = 1. The plots in Figure 2 suggest that the convergence of the truncations A N → A is rapid for small ǫ but a higher order truncation (larger N ) is necessary to accurately resolve A for larger ǫ. For ǫ < 10 −1 , even A 1 is a good approximation to determine the sign of A. For example, the relative error for approximating A with A 1 is approximately %2 for ǫ = 10 −3 and increases to approximately %18 for ǫ = 1.
We also measure the relative strength of the nonlinear interactions, i.e. the ratio
(66) B N = N j=1 Re(A 6,j ) N j=1 Re(A 0,j ) ,
in Figure 3. It is seen from Figure 3 that the contribution from the modes with m = 0 dominates when ǫ is low. But as ǫ increases, the contribution from modes with m = 6 start to become significant. For example, for ǫ = 10 −3 , B N approaches 8 × 10 −5 , for ǫ = 10 −2 , B N approaches −2 × 10 −3 and for ǫ = 10 −1 , B N approaches −8 × 10 −3 . In particular, for low ǫ, we have A ≈ A 1 ≈ A 0,1 .
More significantly, our numerical results presented in Figure 2 show that the real part of A is positive for ǫ = 10 −3 , 10 −2 , 10 −1 , 1, meaning that the transition is catastrophic by Theorem 2. Thus the system moves to a flow regime away from the base Poiseuille flow and the system exhibits complex dynamical behavior for R > R c . 7.2. Determination of Energy Stability Threshold R E . With a standard numerics package, R m (ǫ) in (64) can be computed for given m and ǫ. Then by (65), R E is computed by taking minimum in (65) over all (computed) R m . In Table 2, it is shown that for ǫ = 10 −4 , and ǫ = 10 −3 , R E is obtained for m = 1 while for ǫ = 10 −2 and ǫ = 2 × 10 −2 , R E is obtained for m = 2. In Figure 5, we plot R m (m = 1, 2, 3) for 0 < ǫ ≤ 5 × 10 −2 . We see that the curves R 1 and R 2 intersect approximately at ǫ = 0.009 while R 2 and R 3 intersect approximately at ǫ = 0.024. As ǫ is increased, the value of m for which R E is minimized also increases. For higher values of m, the roots of the determinant in (63) becomes increasingly hard to find. In Table 2, the last column gives the value of R c , the linear instability threshold, computed by (24). Note that the interval [R E , R c ] consists of Reynolds numbers for which the base flow is either not globally stable or globally stable but not not exponentially attracting. We plot the R E and R c data from Table 2 in Figure 4 which shows that this interval shrinks rapidly as ǫ is increased. Table 2. R m denotes the first positive root of (63) and R E is the minimum of R m taken over all m. R c is the linear instability threshold.
ǫ R 1 R 2 R 3 R 4 R 5 R E R
Concluding Remarks
In this work, we considered both the energy stability and transitions of the Poiseuille flow of a second grade fluid in an infinite circular pipe with the restriction that the flow is independent of the axial variable z.
We show that unlike the Newtonian (ǫ = 0) case, in the second grade model (ǫ = 0 case), the time independent base flow exhibits transitions as the Reynolds number R exceeds the critical threshold R c ≈ 4.124ǫ −1/4 where ǫ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects. At R = R c we prove that a transition takes place and that the type of the transition depends on a complex number A. In particular depending on A, generically, either a continuous transition to a periodic solution or a catastrophic transition occurs where the bifurcation is subcritical. The time period of the periodic solution approaches to infinity as R approaches R c , a phenomenon known as infinite period bifurcation.
We show that the number A = ∞ j=1 A 0,j + A 6,j where A m,j denotes the nonlinear interaction of the two critical modes with the mode having azimuthal wavenumber m and radial wavenumber j. Our numerical results suggest that for low ǫ (ǫ < 1), A is approximated well by A 0,1 . That is, a single nonlinear interaction between the critical modes and the mode with azimuthal wavenumber 0 and radial wavenumber 1 dominates all the rest of interactions and hence determines the transition.
Our numerical results suggest that a catastrophic transition is preferred for low ǫ values (ǫ < 1). This means, an unstable time periodic solution bifurcates on R < R c . On R > R c , the system has either metastable states or a local attractor far away from the base flow and a more complex dynamics emerges.
We also show that for R < R E with R E < R c , the Poiseuille flow is at least exponentially globally stable in the H 1 0 (Ω) norm for the velocity. We find that R E ≈ 12.87 when ǫ = 0 and the gap between R E and R c diminishes quickly as ǫ is increased.
There are several directions in which this work can further be extended. First, in this work we consider a pipe with a circular cross section of unit radius and find that the first two critical modes have azimuthal wavenumber equal to 3. Increasing (or decreasing) the radius of the cross section will also increase (decrease) the azimuthal wavenumber of the critical modes. The analysis in this case would be similar to our presentation except in the case where four critical modes with azimuthal wave numbers m and m + 1 become critical. In the case of four critical modes, more complex patterns will emerge due to the cross interaction of the critical modes [15,3]. An analysis in the light of [14] is required to determine transitions in this case of higher multiplicity criticality.
Second, for the Reynolds number region between R E and R c , there may be regions where the base flow is either globally stable but not exponentially attractive or regions where the domain of attraction of the base flow is not the whole space. A conditional energy stability analysis is required to resolve these Reynolds number regimes [16].
Third, the results we proved in this work for the second grade fluids can also be extended to fluids of higher grades and to other types of shear flows.
Fourth, in this work we restricted attention to 2D flows. In the expense of complicating computations and results, a similar analysis could be considered for 3D flows which depend also on the axial variable z.
Figure 1 .
1The time periodic stream function ψ per given by (30) which rotates in time, clockwise if T > 0 and counterclockwise if T < 0.
if f , g and h are linearly dependent. Taking inner product of the first equation in (3) with w and the second equation in (3) with ψ and using the property (52)
Now applying the operator 1 − ǫ∆ m to the second equation in (60) and using the first equation of (60), we obtain p(∆ m )w m = 0, i.e. the same equation (61), this time for w m . Hencew m = 3 k=1 d k I m ( ξ k r).The first equation in (60) gives the relation d k = −i mR 2 2 (ǫ − ξ −1 k )c k between c k 's and d k 's. Now the boundary conditions w m (1) = ψ m (1) = ψ ′ m (1) = 0 constitute a homogeneous system of three linear equations for the coefficients c k 's. The existence of nontrivial solutions is then equivalent to the vanishing determinant of this system which after some manipulation becomes (63)
Figure 2 .
2The plots of Re(A N ) for different ǫ values. All A N 's are scaled so that |Re(A 1 )| = 1.
Figure 3 .
3B N , defined by (66), measures the relative strength of the nonlinear interactions of the critical modes with m = 6 modes to the m = 0 modes.
Figure 4 .
4R E and R c curves in the ǫ − R plane.
Figure 5 .
5The plot of R m vs ǫ for m = 1, 2, 3.
Table 1. The smallest positive root λ m,1 of (22) and (λ m,1has
m
1
2
3
4
5
6
λ m,1
21.260 34.877 51.030 69.665 90.739 114.21
(λ m,1 /m) 1/2 4.610 4.175 4.124 4.173 4.260 4.36
c 0 12.87 13.49 14.84 16.37 17.95 12.87 ∞ 10 −4 12.86 13.47 14.81 16.32 17.88 12.86 42.4 10 −3 12.77 13.31 14.54 15.90 17.29 12.77 23.84 10 −2 11.99 11.95 12.44 12.95 13.39 11.95 13.40 2 × 10 −2 11.26 10.83 10.91 11.03 11.13 10.83 11.27
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| zyda_arxiv-0030000 |
Plankton-FL: Exploration of Federated Learning for Privacy-Preserving Training of Deep Neural Networks for Phytoplankton Classification
Daniel Zhang [email protected]
Vikram Voleti
Mila
Alexander Wong
Jason Deglint [email protected]
University of Waterloo
University of Montreal
University of Waterloo
Blue Lion Labs
University of Waterloo
Blue Lion Labs
Plankton-FL: Exploration of Federated Learning for Privacy-Preserving Training of Deep Neural Networks for Phytoplankton Classification
10.1038/s41746-020-00323-1
Creating high-performance generalizable deep neural networks for phytoplankton monitoring requires utilizing large-scale data coming from diverse global water sources. A major challenge to training such networks lies in data privacy, where data collected at different facilities are often restricted from being transferred to a centralized location. A promising approach to overcome this challenge is federated learning, where training is done at site level on local data, and only the model parameters are exchanged over the network to generate a global model. In this study, we explore the feasibility of leveraging federated learning for privacy-preserving training of deep neural networks for phytoplankton classification. More specifically, we simulate two different federated learning frameworks, federated learning (FL) and mutually exclusive FL (ME-FL), and compare their performance to a traditional centralized learning (CL) framework. Experimental results from this study demonstrate the feasibility and potential of federated learning for phytoplankton monitoring.
Introduction
The uncontrollable growth of particular phytoplankton and algae species can cause the formation of harmful algae blooms (HABs). If not properly monitored and controlled, HABs can have severe, negative impacts on various industries, natural ecosystems, and the environment [1]. HABs are a growing concern as research has shown that climate change has led to an increase in the frequency and severity of HABs [2]. A very important step in the monitoring and controlling of HAB formation is the identification of phytoplankton and algae species. Unfortunately, this process is largely manual and thus is highly time-consuming and prone to human error. As such, effective methods for automating the species identification process are highly desired.
Recent advances in machine learning, in particular deep learning, have shown considerable promise for monitoring and assessment of phytoplankton and algae [3,4]. However, a significant bottleneck to training such models is the need for large-scale data coming from different water sources across different countries in order to create high-performance, generalizable models. Since the data collected at the different facilities are often restricted from being transferred to a centralized location for training due to data privacy concerns, this makes it infeasible to leverage traditional, centralized learning frameworks for building such models.
A particularly promising direction for tackling this data privacy challenge lies in federated learning (FL), which involves training local models at individual local nodes on the premises (prem) of each local data source and communicating only the parameters and updates of these local models to a server for generating a global model to reap the benefits from the different local data without having seen any of the individual data sources [5]. FL has demonstrated considerable success in the domains of mobile computing [5,6] and healthcare [7], and thus can hold considerable potential for the application of phytoplankton monitoring and assessment.
In this study, we explore the feasibility of leveraging federated learning to train deep, convolutional neural networks for the purpose of image-driven phytoplankton classification, which we will refer to as Plankton-FL. Our main contributions in this study are as follows:
(1) we simulate and study two federated learning frameworks as potential realizations of Plankton-FL: (centralized) federated learning (FL) and mutually exclusive FL (ME-FL), (2) we evaluate the performance of both Plankton-FL frameworks, and (3) we compare them to a traditional, centralized learning framework (CL). Figure 1 provides a visual representation of each of the three environments. Both privacy-preserving federated learning frameworks are evaluated against CL for the training of deep neural networks for phytoplankton classification.
Methodology
Background
Federated learning has been shown to be very effective for training deep neural networks on decentralized data while ensuring data privacy [5]. Specifically, when there is sensitive data from various sources, federated learning can be leveraged. A typical federated learning framework consists of 2 components: a global model and K clients. Each client contains its own local model, and they are trained iteratively and independently on their respective data. It is assumed that all of the data available is partitioned into K clients, P k . The local models are then used to update the global model [5]. This process is repeated for N rounds in order for the global model to generalize. The objective for federated learning is calculated using equation 1.
f (w) = K ∑ k=1 n k n F k (w) where F k (w) = 1 n k ∑ i∈P k f i (w)(1)
In equation 1, f i (w) denotes the loss function (x i , y i ; w) for an observation (x i , y i ) and model parameters w. Also, n k and n denote the |P k | and the total number of observations, respectively.
The centralized federated learning algorithm which governs the communication between the global and local models is known as FederatedAveraging (FedAvg) and was introduced by McMahan et al. [5]. FedAvg is the iterative process of training all the local models, taking the average of all the updated weights from the local models, and then using it to update the global model. As described by McMahan et al., pseudo-code is provided in Algorithm 1.
Algorithm 1
McMahan et al. [5]'s implementation of the FedAvg algorithm. C is the set of all clients; B is the local model batch size; LE is the number of local epochs; η is the learning rate; w are the weights, and is the loss function 1: Global/Server: 2: C ← Set of all available clients 3: initialize w 0 4: for each round, r = 1,2, ... do 5: for each client, k ∈ C, do 6: w k r+1 ← Local-Training(k, w r ) 7: end for 8: FedAvg: w r+1 ← ∑ K k=1 n k n w k r+1 9:
SYNC Global→Local: w k r+1 ← w r+1 ∀k 10: end for 11: 12: Local-Training(k, w): In this paper, we used the FedAvg method, as described in Algorithm 1, when training our two instances of Plankton-FL. To test the feasibility and potential of federated learning, three different experiments were simulated. Specifically, a centralized learning baseline (CL), a (centralized) federated learning framework (FL), and a mutually exclusive, federated learning framework (ME-FL). Figure 1 provides a visual representation of each of the three experiments.
for batch b ∈ B do 16: w ← w − η * ∇ w (w; b)
Centralized Learning (CL)
For the CL experiment, we have two data sources that we consolidate into a single server. From there we train a centralized model, which can then be deployed back to the edge devices for assessment and monitoring. The model was trained for a maximum of 75 epochs, with an early stopping criteria: A minimum of 50 epochs and a δ between test accuracies of 0.000001.
Federated Learning (FL)
In the FL experiment, all of the training data was combined, randomly shuffled, and distributed to clients. Each client trained their own local model, on-prem, and only communicated their parameters back to the global server. FL was run for 10 iterations, where each iteration number corresponded to the number of clients. Namely, for the first iteration, there was only one client containing all of the training data, identical to CL, and with each increasing iteration another client was added (i.e. second iteration utilized two clients, etc.). Although, in reality, no single client will contain all of the data, for the purpose of comparing to CL, it made sense to start with a single client.
Mutually Exclusive FL (ME-FL)
Unlike FL, in ME-FL, instead of combining all of the data together, shuffling, and distributing them, the clients only contained data from a single source, making them mutually exclusive. Again, each of the clients trained their own model, on-prem, and only communicated their parameters back to the global server. ME-FL was run for 9 iterations, starting from 2 clients up to 10. Iterations start from 2 clients due to the nature of the experiment; since each client only has data from a single source, it would not make sense to only have a single client. This modified setup ensures that we always have data from both sources.
Experimental Setup
Dataset
The dataset was provided by Blue Lion Labs and was collected from two mutually exclusive sources, Halifax and Waterloo. It contained 301 distinct microscope specimen photos, each at a resolution of 3208 x 2200 pixels. The phytoplankton contained in
# of clients
Model Architecture
For the purpose of this exploration, we took the majority class present in each image as the label to do image classification. This ensures that all models receive the same amount of information.
Given the task, we built a custom convolutional neural network with four convolutional layers, three max-pooling layers, two dense layers, and an output layer. Across all intermediate layers, the ReLU activation function was used, and at the output, a softmax activation was used to predict the probabilities of each class. For all the convolutional layers, a kernel of size 3x3 and a stride of 1x1 was used and for all of the pooling layers, a pool and stride size of 2x2 was used. Additionally, dropout was used with a rate of 0.25 after the convolutional layers and a rate of 0.5 after the first dense layer.
Model Training and Evaluation
When training, the images were resized to a resolution of 128 x 128 pixels and further augmented, using a horizontal flip, vertical flip, rotation, and color jitter, to create a larger data set of 2107 images. Across all experiments, the model architectures were held the same, a batch size of 8 was used, and the data was split into 80% training and 20% test. We also tune the learning rate across all experiments via a grid search over three different learning rates (LR) of 0.001, 0.0001, and 0.0005. Both of the federated learning experiments were run for 75 rounds and each local model was trained for 1 epoch. Furthermore, given it is a multi-label image classification task, the metric considered is prediction accuracy and the loss function is categorical cross-entropy. comparing CL and FL, we observe that for a single client FL outperforms CL. However, this is expected because FL with a single client is the exact same setup as CL; we expect the test accuracies to be very close in magnitude, and it is entirely possible that FL can outperform CL in this scenario. For all other number of clients, FL has a progressively worse test accuracy and is continuously outperformed by CL. In addition, across all number of clients, we observe that ME-FL consistently gets outperformed by CL and FL, which further demonstrates the impracticality of this method. Figure 2 provides a visual comparison of CL, FL, and ME-FL across all epochs. We specifically look at the results for 10 clients of both FL and ME-FL, as in reality there are often large numbers of clients. From the figure, CL and FL both appear to learn, whereas ME-FL does not appear to learn at all. Comparing CL and FL, we observe that CL converges much faster than FL, which tells us that CL learns faster than FL. Overall, across all experiments, generally, we observed that CL performed the best, FL performed the second best, and ME-FL had the worst performance and this same trend is observed across each learning rate.
Results & Discussion
Comparison of Performance Across Experiments
Downward Trend in FL Across Number of Clients
From table 1 and figure 2, we observe that FL performs relatively well, which prompted an investigation into its properties. Figure 3 displays the global model test accuracies for FL, for all numbers of clients, across the three learning rates. We observe a downward trend in the test accuracies as the number of clients increases. With an increasing number of clients, the global model needs to process and learn more information (i.e. the global model has to aggregate weights from more sources). With more information to process, learning is slowed down, yielding a worse generalization.
Causation of Poor ME-FL Performance
The largest contributor as to why ME-FL had a subpar performance relative to FL was because ME-FL was trained on individual, mutually exclusive clients. In our FL experiments we utilize a homogeneous model architecture, that is, all clients and the global server have the same model architecture. The nature of FL yields an independent and identically distributed (IID) distribution of labels across clients. However, in ME-FL, the distribution across clients is non-IID. As discussed in other literature, homogeneous federated learning performs poorly on non-IID data distributions [6]. Given this limitation, heterogeneous federated learning [8,9] is an alternative approach that should be explored. In this method, clients are allowed to differ in network architecture, allowing for more flexibility. Research has been done to explore applications of heterogeneous federated learning to mutually exclusive data [10] and it has typically been the preferred approach over homogeneous federated learning.
Conclusion & Future Works
This work demonstrates the feasibility and potential of Plankton-FL for the privacy-preserving building of high-performance, generalizable models for phytoplankton assessment without the need to exchange data. We simulated two different federated learning frameworks and compared their performance to a traditional, centralized learning framework. Although centralized learning yields the best performance, it does not address privacy concerns. Federated learning preserves privacy but fails to generalize when clients are mutually exclusive. We find that when clients share class labels with one another, federated learning both generalizes well and provides a privacy-preserving alternative to centralized learning.
Given the outcomes of this paper, the immediate future work includes (1) implementing this framework for object detection to build off the current work of image classification, (2) utilizing a heterogeneous federated learning framework and conducting the same experiments to assess the relative performance to homogeneous federated learning, and (3) explore novel federated learning-related methods. For example, another method that can be utilized is git re-basin, which aims to train individual models on disjoint datasets and merge them together [11]. Finally, careful consideration must be taken on how federated learning frameworks will be deployed in the field to ensure data privacy between clients. This will help provide a secure and accurate method for identifying different species of phytoplankton and help alleviate the manual workload.
Fig. 1 :
1Traditional centralized learning (CL) (top) and the two federated learning frameworks as realizations of Plankton-FL: federated learning (FL) (middle) and mutually exclusive FL (ME-FL) (bottom).
on client k into B batches 14: for each local epoch, l = 1...LE do 15:
Table 1
1provides a numerical comparison across all experiments
utilizing a learning rate of 0.0001. Note that, CL, FL, and ME-FL
were trained for 51, 75, and 75 epochs, respectively. Firstly, when
AcknowledgmentsThis work was funded by the Waterloo AI Institute and Mitacs. The dataset was provided by Blue Lion Labs, and the computing resources were provided by the Vision and Image Processing (VIP) Lab at the University of Waterloo and Blue Lion Labs.
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| zyda_arxiv-0036000 |
Distributionally Robust Optimization using Cost-Aware Ambiguity Sets
Mathijs Schuurmans
Panagiotis Patrinos
Distributionally Robust Optimization using Cost-Aware Ambiguity Sets
We present a novel framework for distributionally robust optimization (DRO), called cost-aware DRO (CADRO).The key idea of CADRO is to exploit the cost structure in the design of the ambiguity set to reduce conservatism. Particularly, the set specifically constrains the worst-case distribution along the direction in which the expected cost of an approximate solution increases most rapidly. We prove that CADRO provides both a high-confidence upper bound and a consistent estimator of the out-of-sample expected cost, and show empirically that it produces solutions that are substantially less conservative than existing DRO methods, while providing the same guarantees.
I. INTRODUCTION
We consider the stochastic programming problem
minimize x∈X IE[ (x, ξ)](1)
with X ⊆ IR n a nonempty, closed set of feasible decision variables, ξ ∈ Ξ a random variable following probability measure P, and : IR n × Ξ → IR a known cost function. This problem is foundational in many fields, including operations research [1], machine learning [2], and control (e.g., stochastic model predictive control) [3]. Provided that the underlying probability measure P is known exactly, this problem can effectively be solved using traditional stochastic optimization methods [1], [4]. In reality, however, only a data-driven estimateP of P is typically available, which may be subject to misestimations-known as ambiguity. Perhaps the most obvious method for handling this issue is to disregard this ambiguity and instead apply a sample average approximation (SAA) (also known as empirical risk minimization (ERM) in the machine learning literature), where (1) is solved usingP as a plug-in replacement for P. However, this is known to produce overly optimistic estimates of the optimal cost [4,Prop. 8.1], potentially resulting in unexpectedly high realizations of the cost when deploying the obtained optimizers on new, unseen samples. This downward bias of SAA is closely related to the issue of overfitting, and commonly refered to as the optimizer's curse [5], [6].
Several methods have been devised over the years to combat this undesirable behavior. Classical techniques such as regularization and cross-validation are commonly used in machine learning [2], although typically, they are used as heuristics, providing few rigorous guarantees, in particular for M. Schuurmans small sample sizes. Alternatively, the suboptimality gap of the SAA solution may be statistically estimated by reserving a fraction of the dataset for independent replications [7]. However, these results are typically based on asymptotic arguments, and are therefore not valid in the low-sample regime. Furthermore, although this type of approach may be used to validate the SAA solution, it does not attempt to improve it, by taking into account possible estimation errors. More recently, distributionally robust optimization (DRO) has garnered considerable attention, as it provides a principled way of obtaining a high-confidence upper bound on the true out-of-sample cost [6], [8], [9]. In particular, its capabilities to provide rigorous performance and safety guarantees has made it an attractive technique for data-driven and learning-based control [10]- [12]. DRO refers to a broad class of methods in which a variant of (1) is solved where P is replaced with a worst-case distribution within a statistically estimated set of distributions, called an ambiguity set.
As the theory essentially requires only that the ambiguity set contains the true distribution with a prescribed level of confidence, a substantial amount of freedom is left in the design of the geometry of these sets. As a result, many different classes of ambiguity sets have been proposed in the literature, e.g., Wasserstein ambiguity sets [9], divergencebased ambiguity sets [6], [12], [13] and moment-based ambiguity sets [8], [14]; See [15], [16] for recent surveys.
Despite the large variety of existing classes of ambiguity sets, a common characteristic is that their design is considered separately from the optimization problem in question. Although this simplifies the analysis in some cases, it may also induce a significant level of conservatism; In reality, we are only interested in excluding distributions from the ambiguity set which actively contribute to increasing the worst-case cost. Requiring that the true distribution deviates little from the data-driven estimate in all directions may therefore be unnecessarily restrictive. This intuition motivates the introduction of a new DRO methodology, which is aimed at designing the geometry of the ambiguity sets with the original problem (1) in mind. The main idea is that by only excluding those distributions that maximally affect the worstcase cost, higher levels of confidence can be attained without introducing additional conservatism to the cost estimate.
Contributions: (i) We propose a novel class of ambiguity sets for DRO, taking into account the structure of the underlying optimization problem; (ii) We prove that the DRO cost is both a high-confidence upper bound and a consistent estimate of the optimal cost of the original stochastic program (1); (iii) We demonstrate empirically that the provided ambiguity set outperforms existing alternatives.
Notation: We denote [n] = {1, . . . , n}, for n ∈ IN. |S| denotes the cardinality of a (finite) set S. e i ∈ IR n is the ith standard basis vector in IR n . Its dimension n will be clear from context. We denote the level sets of a function f : IR n → IR as lev ≤α f := {x ∈ IR n | f (x) ≤ α}. We write 'a.s.' to signify that a random event occurs almost surely, i.e., with probability 1. We denote the largest and smallest entries of a vector v ∈ IR n as v max := max i∈[n] v i and v min = min i∈[n] v i , respectively, and define its range
as rg(v) := v max − v min . δ X is the indicator of a set X: δ X (x) = 0 if x ∈ X, +∞ otherwise.
II. PROBLEM STATEMENT
We will assume that the random variable ξ is finitely supported, so that without loss of generality, we may write Ξ = {1, . . . , d}. This allows us to define the probability mass vector p = (P[ξ = i]) d i=1 , and enumerate the cost
realizations i = (· , i), i ∈ [d].
Furthermore, it will be convenient to introduce the mapping L : IR n → IR d as L(x) = ( 1 (x), . . . , d (x)). We will pose the following (mostly standard) regularity assumption on the cost function.
Assumption II.1 (Problem regularity). For all i ∈ [d] (i) i is continuous on X; (ii) i := i + δ X is level-bounded;
Since any continuous function is lower semicontinuous (lsc), Assumption II.1 combined with the closedness of X implies inf-compactness, which ensures attainment of the minimum [17,Thm. 1.9]. Continuity of i is used mainly in Lemma A.5 to establish continuity of the solution mapping V -defined below, see (2). However, a similar result can be obtained by replacing condition (i) by lower semicontinuity and uniform level-boundedness on X. However, for ease of exposition, we will not cover this modification explicitly.
Let p ∈ ∆ d := {p ∈ IR d + | d i=1 p i = 1}
denote the true-but-unknown probability mass vector, and define V : IR n ×∆ d → IR : (x, p) → p, L(x) , to obtain the parametric optimization problem with optimal cost and solution set
V (p) = min x∈X V (x, p) and X (p) = argmin x∈X V (x, p). (2)
The solution of (1) is retrieved by solving (2) with p = p .
Assume we have access to a datasetΞ := {ξ 1 , . . . , ξ m } ∈ Ξ m collected i.i.d. from p . In order to avoid the aforementioned downward bias of SAA, our goal is to obtain a data-driven decisionx m along with an estimateV m such that
P[V (x m , p ) ≤V m ] ≥ 1 − β,(3)
where β ∈ (0, 1) is a user-specified confidence level. We address this problem by means of distributionally robust optimization, where instead of (2), one solves the surrogate problemV
m = min x∈X max p∈Am V (x, p).(DRO)
Here, A m ⊆ ∆ d is a (typically data-dependent, and thus, random) set of probability distributions that is designed to contain the true distribution p with probability 1−β, ensuring that (3) holds. Trivially, (3) is satisfied with β = 0 by taking A m ≡ ∆ d . This recovers a robust optimization method, i.e., min x∈X max i∈[d] i (x). Although it satisfies (3), this robust approach tends to be overly conservative as it neglects all available statistical data. The aim of distributionally robust optimization is to additionally ensure thatV m is a consistent estimator, i.e.,
lim m→∞V m = V (p ), a.s.(4)
We will say that a class of ambiguity sets is admissible if the solutionV m of the resulting DRO problem (DRO) satisfies (3) and (4). Our objective is to develop a methodology for constructing admissible ambiguity sets that take into account the structure of (DRO) and in doing so, provide tighter estimates of the cost, while maintaining (3) with a given confidence level β.
III. COST-AWARE DRO
In this section, we describe the proposed DRO framework, which we will refer to as cost-aware DRO (CADRO). The overall method is summarized in Alg. 1. Here, > 0 is determined to satisfy (5) and α = max p∈A TV L(x), p . Since A TV ⊂ A, A satisfies (5) with a higher confidence level 1 − β, but nevertheless, we have max p∈A V (x, p) = max p∈A TV V (x, p).
A. Motivation
We start by providing some intuitive motivation. Consider the problem (DRO). In order to provide a guarantee of the form (3), it obviously suffices to design A m such that
P[p ∈ A m ] ≥ 1 − β.(5)
However, this condition alone still leaves a considerable amount of freedom to the designer. A common approach is to select A m to be a ball (expressed in some statistical metric/divergence) around an empirical estimatep of the distribution. Depending on the choice of metric/divergence (e.g., total variation [18], Kullback-Leibler [6], Wasserstein [9], . . . ), several possible variants may be obtained. Using concentration inequalities, one can then select the appropriate radius of this ball, such that (5) is satisfied. A drawback of this approach, however, is that the construction of A m is decoupled from the original problem (1). Indeed, given that A m takes the form of a ball, (5) essentially requires the deviation ofp from p to be small along every direction. If one could instead enlarge the ambiguity set without increasing the worst-case cost, then (5) could be guaranteed for smaller values of β without introducing additional conservatism. This idea is illustrated in Fig. 1.
Conversely, for a fixed confidence level β, one could thus construct a smaller upper boundV m , by restricting the choice of p only in a judiciously selected direction. Particularly, we may set A m = {p ∈ ∆ d | L(x), p ≤ α m } for some candidate solution x ∈ X, where α m is the smallest (potentially data-dependent) quantity satisfying (5). This directly yields an upper bound on the estimateV m . Namely, for x ∈ X (p ), we have with probability 1 − β,
V (x , p ) (a) ≤ V (x m , p ) ≤ max p∈Am V (x m , p) =V m = min x∈X max p∈Am V (x, p) (b) ≤ max p∈Am V (x, p) = α m .
Here, inequalities (a) and (b) become equalities whenx m = x = x. Thus, a reasonable aim would be to select x to be a good approximation of x . We will return to the matter of selecting x in §III-C. First, however, we will assume x to be given and focus on establishing the coverage condition (5).
B. Ambiguity set parameterization and coverage
Motivated by the previous discussion, we propose a family of ambiguity sets parameterized as follows. Let v ∈ IR d be a fixed vector (we will discuss the choice of v in §III-C). Given a sampleΞ = {ξ 1 , . . . , ξ m } of size |Ξ| = m drawn i.i.d. from p , we consider ambiguity sets of the form
AΞ(v) := {p ∈ ∆ d | p, v ≤ αΞ(v)},(6)
where α :
Ξ m × IR d (Ξ, v) → αΞ(v) ∈
IR is a datadriven estimator for p , v , selected to satisfy the following assumption, which implies that (5) holds for A m = AΞ(v).
Assumption III.1. P[ p , v ≤ αΞ(v)] ≥ 1 − β, ∀v ∈ IR d .
Note that the task of selecting α to satisfy Assumption III.1 is equivalent to finding a high-confidence upper bound on the mean of the scalar random variable v, e ξ , ξ ∼ p . It is straightforward to derive such bounds by bounding the deviation of a random variable from its empirical mean using classical concentration inequalities like Hoeffding's inequality .
Proposition III.2 (Hoeffding bound). Fix v ∈ IR d and let Ξ with |Ξ| = m, be an i.i.d. sample from p ∈ ∆ d , with empirical distributionpΞ = 1 m ξ∈Ξ e ξ . Consider the bound αΞ(v) = v,pΞ + r m rg(v).(7)
This bound satisfies Assumption III.1, if r m satisfies
r m = min 1, log( 1 /β) 2m .(8)
Proof. Define
y k := p −e ξ k , v , so that 1 m m k=1 y k = p − p m , v . Since v is fixed, y k , k ∈ [m] are i.i.d., and we have IE[y k ] = 0 and (by Lemma A.1), |y k | ≤ rg(v), ∀k ∈ IN.
This establishes the (vacuous) case r m = 1 in (8). For the nontrivial case, we apply Hoeffding's inequality [19, eq. 2.11]
P 1 m m k=1 y k > t ≤ exp −2mt 2 rg(v) 2 .(9)
Setting t = r m rg(v), equating the right-hand side of (9) to the desired confidence level β, and solving for r m yields the desired result.
Although attractive for its simplicity, this type of bounds has the drawback that it applies a constant offset (depending only on the sample size, not the data) to the empirical mean, which may be conservative, especially for small samples. Considerably sharper bounds can be obtained through a more direct approach. In particular, we will focus our attention on the following result due to Anderson [20], which is a special case of the framework presented in [21]. We provide an experimental comparison between the bounds in Appendix B.
Proposition III.3 (Ordered mean bound [21]). Let
η k := v, e ξ k , k ∈ [m], so that IE[η k ] = v, p . Let η (1) ≤ η (2) ≤ · · · ≤ η (m) ≤ η denote the sorted sequence, with ties broken arbitrarily, where η := max i∈[d] v i . Then, there exists a γ ∈ (0, 1) such that Assumption III.1 holds for αΞ(v) = κ m −γ η (κ) + m i=κ+1 η (i) m +γη, κ = mγ .(γ = log( 1 /β) 2m , for sufficiently large m.(11)
This asymptotic expression will be useful when establishing theoretical guarantees in Section IV.
C. Selection of v
The proposed ambiguity set (6) depends on a vector v. As discussed in §III-A, we would ideally take v = L(x ) with x ∈ X (p ). However, since this ideal is obviously out of reach, we instead look for suitable approximations. In particular, we propose to use the available datasetΞ in part to select v to approximate L(x ), and in part to calibrate the mean bound α.
To this end, we will partition the available datasetΞ into a training set and a calibration set. Let τ : IN → IN be a user-specified function determining the size of the training set, which satisfies τ (m) ≤ cm for some c ∈ (0, 1); and (12a)
τ (m) → ∞ as m → ∞.(12b)
Correspondingly, let {Ξ T ,Ξ C } be a partition ofΞ, i.e.,Ξ T ∩ Ξ C = ∅ andΞ T ∪Ξ C =Ξ. Given that |Ξ| = m, we ensure that |Ξ T | = τ (m) and thus |Ξ C | = m := m − τ (m). Note that by construction, m ≥ (1 − c)m, with c ∈ (0, 1), and thus, both |Ξ T | → ∞ and |Ξ C | → ∞ as m → ∞. Due to the statistical independence of the elements inΞ, it is inconsequential how exactly the individual data points are divided intoΞ T andΞ C . Therefore, without loss of generality, we may takê
Ξ T = {ξ 1 , . . . , ξ τ (m) } andΞ C = {ξ τ (m)+1 , . . . , ξ m }.
With an independent datasetΞ T at our disposal, we may use it to design a mapping v τ (m) : Ξ τ (m) → IR d , whose output will be a data-driven estimate of L(x ). For ease of notation, we will omit the explicit dependence on the data, i.e., we write v τ (m) instead of v τ (m) (Ξ T ). We propose the following construction. Letp τ (m) = 1
τ (m) τ (m) k=1 e ξ k denote the empirical distribution ofΞ T and set v τ (m) = L(x τ (m) ), with x τ (m) ∈ argmin x∈X V (x,p τ (m) ).(13)
Remark III.4. We underline that although (13) is a natural choice, several alternatives for the training vector could in principle be considered. To guide this choice, Lemma IV.2 provides sufficient conditions on the combination of α and v τ (m) to ensure consistency of the method.
Given v τ (m) as in (13), we will from hereon use the following shorthand notation whenever convenient:
A m := AΞ C (v τ (m) ), α m := αΞ C (v τ (m) ),(14)
with AΞ C (v τ (m) ) as in (6). We correspondingly obtain the cost estimateV m according to (DRO).
D. Selection of τ
Given the conditions in (12), there is still some flexibility in the choice of τ (m), which defines a trade-off between the quality of v τ (m) as an approximator of L(x ) and the size of the ambiguity set A m .
An obvious choice is to reserve a fixed fraction of the available data for the training set, i.e., set τ (m) /m equal to some constant. However, for low sample counts m, the mean bound α m will typically be large and thus A m will not be substantially smaller than the unit simplex ∆ d , regardless of v τ (m) . As a result, the obtained solution will also be rather insensitive to v τ (m) . In this regime, it is therefore preferable to reduce the conservativeness of α m quickly by using small values of τ (m) /m (i.e., large values of m = m − τ (m)).
Conversely, for large sample sizes, α m is typically a good approximation of p , v τ (m) and the solution to (DRO) will be more strongly biased to align with v τ (m) . Thus, the marginal benefit of improving the quality of v τ (m) takes priority over reducing α m , and large fractions τ (m) /m become preferable. Based on this reasoning, we propose the heuristic τ (m) = µν m(m+1) µm+ν , µ, ν ∈ (0, 1).
Note that µ and ν are the limits of τ (m) /m as m → 0 and m → ∞, respectively. Eq. (15) then interpolates between these extremes, depending on the total amount of data available. We have found µ = 0.01, ν = 0.8 to be suitable choices for several test problems.
E. Tractable reformulation
The proposed ambiguity set takes the form of a polytope, and thus, standard reformulations based on conic ambiguity sets apply directly [23]. Nevertheless, as we will now show, a tractable reformulation of (DRO) specialized to the ambiguity set (6) may be obtained, which requires fewer auxiliary variables and constraints .
Proposition III.5 (Tractable reformulation of (DRO)). Fix parametersp ∈ ∆, v ∈ IR d , and α ∈ IR and let A = {p ∈ ∆ d | p, v ≤ α} be an ambiguity set of the form (6).
Denoting V A := min x∈X max p∈A V (x, p), we have V A = min x∈X,λ≥0 λα + max i∈[d] { i (x) − λv i }.(16)
Proof. Let g(z) := max p∈∆ d { p, z | p, v ≤ α}, where z ∈ IR d and α are constants with respect to p. By strong duality of linear programming [24],
g(z) = min λ≥0 max p∈∆ d p, z − λ( p, v − α) = min λ≥0 λα + max p∈∆ d p, z − λv
Noting that max p∈∆ d y = max i∈[d] y i , ∀y ∈ IR d and that V A = min x∈X g(L(x)), we obtain (16).
If the functions { i } i∈ [d] are convex, then (16) is a convex optimization problem, which can be solved efficiently using off-the-shelf solvers. In particular, if they are convex, piecewise affine functions, then it reduces to a linear program (LP). For instance, introducing a scalar epigraph variable, one may further rewrite (16) as min x∈X,λ≥0,z∈IR
{λα + z | L(x) − λv ≤ z1},(17)
which avoids the non-smoothness of the pointwise maximum in (16) at the cost of a scalar auxiliary variable. Even for general (possibly nonconvex) choices of i , (16) is a standard nonlinear program, which can be handled by existing solvers. We conclude the section by summarizing the described steps in Alg. 1.
Algorithm 1 CADRO
IV. THEORETICAL PROPERTIES
We will now show that the proposed scheme possesses the required theoretical properties, namely to provide (i) an upper bound to the out-of-sample cost, with high probability (cf. (3)); and (ii) a consistent estimate of the true optimal cost (cf. (4)). Let us start with the first guarantee, which follows almost directly by construction.
Theorem IV.1 (Out-of-sample guarantee). Fix m > 0, and letV m ,x m be generated by Alg. 1. Then,
P[V (x m , p ) ≤V m ] ≥ 1 − β.(18)
Proof. If p ∈ A m , then
V m (x) := max p∈Am V (x, p) ≥ V (x, p ), ∀x ∈ X.(19)
Sincex m ∈ argmin x∈X V m (x), (19)
implies that V (x m , p ) ≤ V m (x m ) =V m , where the last equality holds by definition (DRO). Consequently, p ∈ A m =⇒ V (x m , p ) ≤V m , and thus P[V (x m , p ) ≤V m ] ≥ P[p ∈ A m ].
Since v τ (m) is constructed independently fromΞ C , Assumption III.1 ensures that (5) holds with respect to A m = AΞ C (v τ (m) ), establishing the claim. We now turn our attention to the matter of consistency. That is, we will show that under suitable conditions on the mean bound α and the training vector v in (6),V m converges almost surely to the true optimal value, as the sample size m grows to infinity. We will then conclude the section by demonstrating that for the choices proposed in §III-B and III-C, the aforementioned conditions hold.
. If v τ (m) = L(x τ (m) ), with x τ (m) , α m = αΞ C (v τ (m) ) chosen to ensure (i) p m , v τ (m) ≤ αΞ C (v τ (m) ), a.s.; (ii) lim sup m→∞ αΞ C (v τ (m) ) ≤ V (p ),x ∈ X, p m , L(x) ≤ V m (x) ≤ α m + ε m (x) ∞ .
Minimizing with respect to x yields that for all m,
V SAA m ≤V m ≤ α m ,(20)
whereV SAA m := V (p m ) (cf. (2)). By the law of large numbers,p m → p , a.s. Furthermore, under Assumption II.1, Lemma A.5 states that the optimal value mapping V (p) is continuous, which implies that alsoV SAA m → V (p ), a.s. The claim then follows directly from condition (ii).
Informally, Lemma IV.2 requires that the mean bound is bounded from below by the empirical mean, and from above by a consistent estimator of the optimal cost. The latter excludes choices such as the robust minimizer x τ (m) ∈ argmin max i∈[d] i (x) in the construction of v τ (m) . However, besides (13), one could consider alternatives, such as a separate DRO scheme to select v τ (m) . A more extensive study of such alternatives, however, is left for future work. We now conclude the section by showing that (13) (13), then,V m → V (p ), a.s.
Proof. It suffices to show that conditions (i) and (ii) of Lemma IV.2 are satisfied by αΞ C (v τ (m) ). Condition (i): Consider αΞ C (v) as in (10) for an arbitrary v ∈ IR d , and let (η (i) ) i∈[m ] denote ( v, e ξ ) ξ∈ΞC , sorted in increasing order, then, we may write
p m , v = 1 m m i=1 η (i) ,(21)
and thus,
αΞ C (v) − p m , v = κ m − γ η (κ) − κ i=1 η (i) m + γη, (a) ≥ κ m − γ η (κ) − κ m η (k) + γη, = γ(η − η (k) ) (γ≥0) ≥ 0, ∀v ∈ IR d ,
where (a) follows from the fact that η (i) are sorted.
Condition (ii): By Lemma A.4, there exists a constant
v ≥ v τ (m) ∞ , ∀m > 0, a.
s. . Therefore, using (10) and (21),
α m − p m , v τ (m) ≤ ( κ m − γ)v + κ m v + γv = 2v( κ m ) (b) ≤ 2v(γ + 1 m ),(22)
for all m > 0, where (b) follows from κ = m γ ≤ m γ +1. By construction (see (12) and below), we have that both τ (m) → ∞ and m → ∞. Thus, using (11), γ + 1 m = log( 1 /β) 2m + 1 m → 0. Combined with (22), this yields that
lim sup m→∞ αΞ C (v τ (m) ) − v τ (m) ,pΞ C ≤ 0.(23)
Finally, by the law of large numbers,p m → p andp τ (m) → p , a.s. Thus (under Assumption II.1), Corollary A.6 ensures that lim m→∞ p m , L(x τ (m) ) = V (p ), which, combined with (23) yields the required result.
Theorem IV.4 (Consistency -Hoeffding bound). LetV m be generated by Alg. 1, for m > 0. If α m = αΞ C (v τ (m) ) is selected according to Proposition III.2, with v τ (m) as in (13) then,V m → V (p ), a.s.
Proof. We show that condition (i) and condition (ii) of Lemma IV.2 are satisfied.
Condition (i): Trivial, noting that r m > 0 by (8). Condition (ii): By the law of large numbers, we have that p m → p , a.s., and thus, by Corollary A.6, p m , v τ (m) → V (p ). Furthermore, by Lemma A.4, there exists a constant v such that rg(v τ (m) ) ≤ 2v for all m ∈ IN. Thus, for r m given by (8), we have
lim sup m→∞ αΞ C (v τ (m) ) ≤ lim sup m→∞ p m , v τ (m) + r m v = V (p ).
This concludes the proof.
V. ILLUSTRATIVE EXAMPLE
As an illustrative example, we consider the following facility location problem, adapted from [25,Sec. 8.7.3]. Consider a bicycle sharing service setting out to determine locations x (i) ∈ X i ⊆ IR 2 , i ∈ [n x ], at which to build stalls where bikes can be taken out or returned. We will assume that X i are given (polyhedral) sets, representing areas within the city suitable for constructing a new bike stall. Let z (k) ∈ IR 2 , k ∈ [d], be given points of interest (public buildings, tourist attractions, parks, etc.). Suppose that a person located in the vicinity of some point z (k) decides to rent a bike. Depending on the availability at the locations x (i) , this person may be required to traverse a distance
k (x) = max i∈[nx] x (i) − z (k) 2 , where x = (x (i) ) i∈[nx] .
With this choice of cost, (16) can be cast as a second order cone program. Thus, if the demand is distributed over (z (k) ) k∈ [d] according to the probability mass vector p ∈ ∆ d , then the average cost to be minimized over X = X 1 ×· · ·×X d is given by V (x, p ) as in (2). We will solve a randomly generated instance of the problem, illustrated in Fig. 2. As p is unknown, one has to collect data, e.g., by means of counting passersby at the locations z (k) . As this may be a costly operation, it is important that the acquired data is used efficiently. Furthermore, in order to ensure that the potentially large up-front investment is justified, we are required to provide a certificate stating that, with high confidence, the quality of the solution will be no worse than what is predicted. Thus, given our collected sample of size m, our aim is to compute estimatesV m , satisfying (3).
X1 X2 X3 z (k) argmin x∈X V (x, p ⋆ ) argmin x∈X max k∈[d] ℓ k (x) CADRO (m = 20)
We compare the following data-driven methods.
CADRO Solves (DRO) according to Alg. 1, setting τ (m) as in (15), with µ = 0.01, ν = 0.8. [28], ensuring that (5) is satisfied. SAA Using the same data partition {Ξ T ,Ξ C } as CADRO, we useΞ T to compute x m = x τ (m) as in (13), and we useΞ C to obtain a high-confidence upper boundV m = αΞ C (L(x τ (m) )), utilizing Proposition III.3.
D-DRO Solves (DRO), with an ambiguity set of the form
A m = {p ∈ ∆ d | D(p m , p) ≤ r D m }, with D ∈ {TV,
Note that D-DRO does not require an independent data sample in order to satisfy (3).
Remark V.1. Other methods could be used to validate SAA (e.g., cross-validation [2], replications [7]), but these methods only guarantee the required confidence level asymptotically. In order to obtain a fair comparison, we instead use the same mean bound, namely (10) for both CADRO and SAA, so both methods provide the same theoretical guarantees. Moreover, we note that a different data partition could be used for SAA. However, preliminary experiments have indicated that significantly increasing or decreasing τ (m) resulted in deteriorated bounds on the cost.
We set n x = 3, d = 50, β = 0.01, and apply each method for 100 independently drawn datasets of size m. In Fig. 3, we plot the estimated costsV m and the achieved out-of-sample cost V (x m , p ), for increasing values of m. We observe that CADRO provides a sharper cost estimateV m than the other approaches. In particular, the classical DRO formulations require relatively large amounts of data before obtaining a non-vacuous upper bound on the cost. The right-hand panel in Fig. 3 shows that additionally, CADRO returns solutions which exhibit superior out-of-sample performance than the compared approaches, illustrating that it does not rely on conservative solutions to obtain better upper bounds.
VI. CONCLUSION AND FUTURE WORK
We proposed a DRO formulation, named cost-aware DRO (CADRO), in which the ambiguity set is designed to only restrict errors in the distribution that are predicted to have significant effects on the worst-case expected cost. We proved out-of-sample performance bounds and consistency of the resulting DRO scheme, and demonstrated empirically that this approach may be used to robustify against poor distribution estimates at small sample sizes, while remaining considerably less conservative than existing DRO formulations. In future work, we aim to extend the work to continuous distributions.
| p − e i , v | ≤ rg(v), for all i ∈ [d], v ∈ IR d and p ∈ ∆ d . Proof. For any i ∈ [d], v ∈ IR d and p ∈ ∆ d , | p − e i , v | ≤ max{max i∈[d] p, v − v i , max i∈[d] v i − p, v } = max{ p, v − v min , v max − p, v } (a) ≤ v max − v min = rg(v),
where (a) follows from the fact that max Then, for all x ∈ X, we have
p∈∆ p, v = v max and max p∈∆ − p, v = max i∈[d] {−v i } = −v min .V (x) ≤ αΞ(v) + L(x) − v ∞ , a.s. Proof. Define ε(x) = L(x) − v for x ∈ X. We have V (x) = max p∈AΞ(v) p, v + p, ε(x) (6) ≤ αΞ(v) + max p∈AΞ(v) p, ε(x) .
The claim directly follows because AΞ(v) ⊆ ∆ d and max p∈∆ d p, z = max i {z i }, ∀z ∈ IR d [29,Ex. 4.10].
Lemma A.3 (Uniform level-boundedness). If Assump- tion II.1(ii) holds, then V (x, p) = p, L(x) + δ X×∆ d (x, p) is level-bounded in x locally uniformly in p. Proof. Since V (x, p) is a convex combination of i (x), i ∈ [d], V (x, p) ≤ α implies that ∃i ∈ [d] : i (x) ≤ α. Therefore, lev ≤α V ( · , p) ⊆ i∈[d] lev ≤α i =: U α , for all p ∈ ∆ d . By Assumption II.1(ii), lev ≤α i is bounded for all i ∈ [d].
Since the union of a finite number of bounded sets is bounded, U α is bounded. Furthermore, for p / ∈ ∆ d , V (x, p) = ∞, and thus lev ≤α V ( · , p) = ∅ ⊆ U α , ∀p / ∈ ∆ d Thus, lev ≤α V (· , p) ⊆ U α for all p ∈ IR d .
Lemma A.4 (Uniform boundedness of v τ (m) ). Let v τ (m) be defined as in (13). Then, there exists a v ∈ IR + such that v τ (m) ∞ ≤ v, ∀m ∈ IN, a.s.
Proof. By Assumption II.1, there exists
r := min x∈X max i∈[d] i (x) ≥ min x∈X V (x, p), ∀p ∈ ∆ d ,
so that, by (13),
x τ (m) ∈ lev ≤r V ( · ,p τ (m) ), ∀m ∈ IN.
Since V (x, p) is level-bounded uniformly in p (cf. Lemma A.3), there exists a compact set C ⊆ IR n satisfying (i) the optimal value V (p) defined by (2), is continuous at p relative to ∆ d .
x τ (m) ∈ lev ≤r V ( · ,p τ (m) ) ⊆ C, ∀m ∈ IN.(24)
(ii) For anyp m → p , and for any x m ∈ X (p m ), {x m } m∈IN is bounded and all its cluster points lie in X (p ).
Proof. If L is continuous, then V (x, p) can be written as the composition V ≡ g • F of the lsc function g : IR 2d → IR : (y, z) → y, z + δ ∆ d (p), and F : IR nd → IR 2d : (x, p) → (L(x), p). By [17, Ex. 1.40(a)], this implies that V is lsc, and so is (x, p) → V (x, p) + δ X (x). Moreover, by Lemma A.3, it is level-bounded in x locally uniformly in p. Furthermore, p → V (x, p) is continuous relative to ∆ d for all fixed x ∈ X.
On the other hand, since the sequence {x τ (m) ∈ X} m is bounded, and L is continuous on X, p m , L(x τ (m) ) has at least one cluster point and lim sup m→∞ p m , L(x τ (m) ) < ∞. Assume then, for the sake of contradiction, that there exists a cluster point V = lim sup m→∞ p m , L(x τ (m) ) > V (p ). Sincep m → p , this implies, by continuity of L, that there must exist a limit point x / ∈ X (p ) of {x τ (m) } m , contradicting Lemma A.5. We conclude that lim sup m→∞ p m , L(x τ (m) ) ≤ V (p ).
Combining (25) and (26) completes the proof.
B. Comparison with the Hoeffding bound
We consider another instance of the example set-up from §V, and compare CADRO using the the Hoeffding bound (Proposition III.2) and the ordered mean bound (Proposition III.3) . Figure 4 shows the cost estimateV m and the out-ofsample cost V (x m , p ) for the TV-DRO method and the aforementioned versions of CADRO. We note that the radius of the ambiguity set for TV-DRO is computed using the Bretagnolle-Huber-Carol inequality [27, Prop. A.6.6] with slightly improved constants. As this result is based on the same Hoeffding-type inequality as Proposition III.2, The apparent performance gains of CADRO with the Hoeffding bound are thus to be attributed primarily to the geometry of the ambiguity set. However, unlike divergence-based ambiguity sets, which rely on concentration inequalities to bound deviations of the distribution from the empirical mean, (6) does not require the use of concentration inequalities. Rather, any high-confidence upper bound on the mean of a scalar random variable satisfying the conditions of Lemma IV.2 may be used, allowing the use of more sophisticated approaches (e.g., Proposition III.3). This results in the improvements visible in Fig. 4, without requiring alterations to the DRO method itself.
and P. Patrinos are with the Department of Electrical Engineering (ESAT-STADIUS), KU Leuven, Kasteelpark Arenberg 10, 3001 Leuven, Belgium. Email: {mathijs.schuurmans, panos.patrinos}@esat.kuleuven.be This work was supported by the Research Foundation Flanders (FWO) research projects G081222N, G033822N, G0A0920N; European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 953348; Ford KU Leuven Research Alliance Project KUL0075;
Fig. 1 .
1Conceptual motivation for the structure of the ambiguity set(6). The cost contour lines {p ∈ ∆ 3 | L(x), p = α} corresponding to some x ∈ X are shown for increasing values of α (dark to light), together with the sets A TV := ∆ 3 ∩ IB 1 (p, ) and A := {p ∈ ∆ 3 | L(x), p ≤ α}.
Require: i.i.d. datasetΞ = {ξ1, . . . , ξm}; τ (m) (cf. (12)); Confidence parameter β ∈ (0, 1). Ensure: (Vm,xm) satisfy (3)-(4) Cf. §IV ΞT ← {ξ1, . . . , ξ τ (m) },ΞC ← {ξ τ (m)+1 , . . . , ξm} v τ (m) ← evaluate (13) (Vm,xm) ← solve (DRO) with Am = AΞ C (v τ (m) ) Use(16)
Lemma IV.2 (consistency conditions). LetΞ T ,Ξ C be two independent samples from p , with sizes |Ξ T | = τ (m) and |Ξ C | = m := m−τ (m). Letp m := 1 m ξ∈ΞC e ξ denote the empirical distribution of the calibration setΞ C
a.s. ThenV m → V (p ), a.s., whereV m is given by (DRO). Proof. Let V m (x) := max p∈Am p, L(x) . It is clear from condition (i) and (6) thatp m ∈ A m . Let us furthermore define ε m (x) = L(x) − L(x τ (m) ). Then, by Lemma A.2 , we have for all
satisfy the requirements of Lemma IV.2. Theorem IV.3 (Consistency -Ordered mean bound). LetV m be generated by Alg. 1, for m > 0. If α m = αΞ C (v τ (m) ) is selected according to Proposition III.3, with v τ (m) as in
Fig. 2 .
2Illustration of the facility location problem. The colors of the points z (k) represent their probability p k .
Fig. 3 .
3Results of the facility location problem of Section V. (left): The cost estimatesVm satisfying (3) and (4); (right): True out of sample cost V (xm, p ). The points indicate the sample mean, the solid errorbars indicate the empirical 0.95 (upper and lower) quantiles and the semi-transparent errorbars indicate the largest and smallest values over 100 independent runs.
.
Let e i denote the i'th standard basis vector.
Lemma A.2 (Upper bound). Fix v ∈ IR d and consider a sampleΞ . For an ambiguity set AΞ(v), given by (6) with mean bound αΞ(v) , define V (x) := max p∈AΞ(v) V (x, p).
Thus, [17, Thm. 1.17(b),(c)] applies, translating directly to statements (i) and (ii). Corollary A.6. Let {x τ (m) } m∈IN be generated by (13) and let {p m ∈ ∆ d } m∈IN be some sequence withp m → p . Then, lim m→∞ p m , L(x τ (m) ) = V (p ) Proof. By definition of V , we have p m , L(x τ (m) ) ≥ V (p m ), and by Lemma A.5, lim m→∞ V (p m ) → V (p ). Therefore, lim inf m→∞ p m , L(x τ (m) ) ≥ V (p ).
Fig. 4 .
4Results for a problem instance as described in Section V. (left): The cost estimatesVm satisfying (3) and (4); (right): True out of sample cost V (xm, p ). The points indicate the sample mean, the solid errorbars indicate the empirical 0.95 (upper and lower) quantiles and the semi-transparent errorbars indicate the largest and smallest values over 100 independent runs.
10) For finite m, the smallest value of γ ensuring that Proposition III.3 holds, can be computed efficiently by solving a scalar root-finding problem[21, Rem. IV 3]. Furthermore, it can be shown that the result holds for[22, Thm. 11.6.2]
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We use K ij = z (i) − z (j) 2 , i, j ∈ [d]as the transportation cost. 2 This is a slightly improved version of the classical Bretagnolle-Huber-Carol inequality [27, Prop. A.6.6].
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A W Van Der Vaart, J A Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics. New YorkSpringerA. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics. New York: Springer, 2000.
On Choosing and Bounding Probability Metrics. A L Gibbs, F E Su, International Statistical Review. 703A. L. Gibbs and F. E. Su, "On Choosing and Bounding Probability Metrics," International Statistical Review, vol. 70, no. 3, pp. 419-435, 2002.
First-Order Methods in Optimization. A Beck, MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics. A. Beck, First-Order Methods in Optimization. MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Oct. 2017.
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Approximate Killing Fields as an Eigenvalue Problem
12 Aug 2008 (Dated: July 13, 2007)
Christopher Beetle
Department of Physics
Florida Atlantic University
33431Boca Raton, Florida
Approximate Killing Fields as an Eigenvalue Problem
12 Aug 2008 (Dated: July 13, 2007)
Approximate Killing vector fields are expected to help define physically meaningful spins for nonsymmetric black holes in general relativity. However, it is not obvious how such fields should be defined geometrically. This paper relates a definition suggested recently by Cook and Whiting to an older proposal by Matzner, which seems to have been overlooked in the recent literature. It also describes how to calculate approximate Killing fields based on these proposals using an efficient scheme that could be of immediate practical use in numerical relativity.
Spacetime symmetries are essential for defining physically important conserved quantities such as energy and angular momentum in general relativity. For example, when a vacuum spacetime admits a rotational symmetry generated by a Killing vector field ϕ a , then a Komar-type integral [1,2] over an arbitrary 2-sphere in spacetime can reproduce the physically well-defined angular momentum measured at infinity. When spacetime is axi-symmetric, but not vacuum, the difference between these integrals for a pair of different 2-spheres is precisely the ordinary angular momentum computed from the stress-energy of matter in the intervening space. If spacetime is not axisymmetric, however, such formulae become rather ambiguous. They depend not only on the 2-sphere S over which one integrates, but also on the vector field ϕ a used to define the integrand.
These difficulties can be partially avoided under physically favorable conditions. For instance, angular momentum is well-defined at infinity in asymptotically flat spacetimes (see [3] for a recent review), or on an appropriate isolated horizon [4,5] modeling an isolated, nondynamical black hole in a spacetime that may describe interesting dynamics in other regions. Essentially, these techniques identify preferred 2-spheres S (infinity, horizon, etc.) over which to integrate, and thereby reduce the ambiguity in defining the angular momentum. The resulting quasi-local formulae have the general Brown-York [6] form
J[ϕ] := 1 8πG S ϕ a K ab dS b ,(1)
where S is a (perhaps preferred) 2-sphere, K ab is the extrinsic curvature of a spatial slice Σ containing it, ϕ a is a vector field on S, and dS b is the area element of S within Σ. The basic problem remains: the vector field ϕ a is arbitrary unless S has an intrinsic symmetry that can be used to select it. (Now, at least, that symmetry need not extend into the bulk of spacetime.) The horizons of black holes resulting from numerical simulations of astrophysical processes generally have no symmetry of any kind and therefore, seemingly, no preferred vector field ϕ a . The problem is not that such surfaces have no reasonable definition for the angular momentum, but rather that they have infinitely many.
There is one for every vector field ϕ a tangent to the horizon. What is needed is a technique to pick a preferred vector field, and the obvious thing to do is to seek a ϕ a that, in some sense, is as close as possible to a Killing field, even if none is present. This leads intuitively to the idea of an approximate Killing field.
Motivated by the general issues discussed above, several groups have recently proposed elegant definitions of approximate Killing fields on 2-spheres. These include schemes based on Killing transport [7], conformal Killing vectors [8], and most recently a minimization scheme by Cook and Whiting [9]. This paper revives an older approach [10] due to Matzner based on solving an eigenvalue problem. It also suggests a novel adaptation of Matzner's approach to the specific problem of computing a preferred angular momentum for black holes in numerical relativity, and elucidates the intimate relationship between this scheme and that of Cook and Whiting.
Let us begin with a brief review of Matzner's definition [10] of an approximate Killing field on a compact manifold M of dimension n equipped with a Riemannian metric g ab . A continuous symmetry of g ab is generated by a Killing vector field ξ a satisfying L ξ g ab = 2 ∇ (a ξ b) = 0 (2) throughout M , where L ξ denotes the Lie derivative along ξ a and ∇ a is the unique torsion-free connection on M determined by g ab . Taking a divergence, we see that any geometry with a continuous symmetry admits at least one non-trivial solution to the equation
− 2 ∇ b ∇ (b ξ a) = 0.(3)
In principle, even when the geometry is not symmetric, we are still free to seek solutions for this equation. But generically we will not find any. Consider the eigenvalue problem
∆ K ξ a := −2 ∇ b ∇ (b ξ a) = κξ a(4)
on a generic geometry. The operator ∆ K appearing here arises naturally in the transverse decomposition of symmetric tensor fields on Riemannian manifolds [12]. A related operator, denoted ∆ L , arises in the same way from the conformal Killing equation, and plays a similar role in the transverse-traceless decomposition of such fields. Its application to the initial-data problem in general relativity is very well known indeed [13]. Eq. (4) of course admits solutions ξ a only for a certain spectrum of eigenvalues κ, and zero may or may not be among these. Matzner [10] establishes the following four results: the spectrum of eigenvalues κ of Eq. (4) on a compact manifold is (a) discrete, (b) non-negative, (c) corresponds to a complete set of real vector eigenfields ξ a , and (d) contains κ = 0 if and only if the corresponding eigenfield ξ a is a genuine Killing field. That is, the zero eigenspace of Eq. (4) is precisely the finitedimensional vector space of Killing fields of the metric g ab on M . Therefore, Eq. (3) admits no solution if the metric g ab on M has no continuous symmetries, as claimed above. However, we assert that the best approximation to a Killing field on a manifold with no actual symmetry is the unique vector eigenfield ξ a of Eq. 4 with the minimum eigenvalue κ > 0. This is Matzner's definition of an approximate Killing field, and it has several desirable features. It exists generically, reduces to the correct answer when symmetries do exist, and, like a true Killing field on a symmetric manifold, is naturally defined only up to an overall (i.e., constant over M ) scaling.
Like any eigenvalue problem, Eq. (4) admits a variational formulation. Recall the natural L 2 inner product
ζ, ξ := M ζ a ξ a ǫ(5)
on the space of (complex) vector fields over M . Here, ǫ denotes the canonical n-form volume element induced on M by the metric g ab . We minimize the quadratic form
Q K [ξ; κ) := 1 2 ξ, ∆ K ξ − 1 2 κ ξ, ξ − 1 ,(6)
where κ is constant over M and here plays the role of a Lagrange multiplier. Minimizing this functional produces the Euler-Lagrange equations ∆ K ξ a = κξ a and ξ, ξ = 1,
the solutions of which are clearly the vector eigenfields of Eq. (4), normalized to unity in Hilbert space. Many variational problems are solved by initially solving the first, differential equation in Eq. (7) for ξ a as a function of an arbitrary Lagrange multiplier κ, and then using that result in the second, algebraic equation to impose the constraint and determine κ. This does not happen for Eq. (7) because the differential equation is linear in ξ a , and therefore leaves the overall scaling of ξ a undetermined. The second equation serves only to fix that scaling, and cannot also determine κ. The Lagrange multiplier therefore must be fixed when we solve the first equation; for general κ, no solution exists. This is hardly surprising since of course only true eigenvalues κ allow us to solve Eq. (4) for ξ a . However, it does make an approach to Matzner's eigenvalue problem via a variational principle like Eq. (6) rather complicated. There is no algebraic equation to determine the Lagrange multiplier. Indeed, κ is determined in this problem precisely by the condition that it be an eigenvalue of ∆ K , and there is no algebraic equation giving these. Minimizing Q K [ξ; κ) in Eq. (6) by solving the associated Euler-Lagrange equations is neither easier nor harder than solving the eigenvalue problem in Eq. (4).
Cook and Whiting's recent definition [9] of an approximate Killing field uses a variational principle based on a quadratic form closely related to that of Eq. (6). However, it differs in a two important details. First, it focuses on the case where M ≃ S is topologically a 2-sphere, and restricts ξ a to be area-preserving:
L ξ ǫ ab = (∇ c ξ c ) ǫ ab = 0.(8)
The motivation for this restriction arises from the technical details of an eventual application to calculating the angular momentum of a non-symmetric black hole [4]. Second, it is based on a non-standard inner product
ζ, ξ R := ζ, R ξ = S ζ a ξ a R ǫ.(9)
These choices change the form, but not the basic content, of the resulting equations. They still describe a sort of self-adjoint eigenvalue problem.
To restrict to area-preserving vector fields, it is easiest simply to recall that any divergenceless vector fieldξ a on a 2-sphere topology is described by a unique scalar potential Θ such that ξ a = ( * dΘ) a := −ǫ ab ∇ b Θ and S Θ ǫ = 0. (10) Now consider the restricted eigenvalue problem ∆ Kξ a := P ∆ KP ξ a =κξ a ,
whereP denotes the orthogonal projection onto the subspace of area-preserving vector fields within the Hilbert space of Eq. (5). A givenξ a = ( * dΘ) a solves this equation if and only if, for all otherζ a = ( * dΦ) a , we have * dΦ, ∆ K * dΘ =κ * dΦ, * dΘ .
Integrating both sides by parts, and using positivity of the standard L 2 inner product of scalar functions over S, we find that Eq. (11) is equivalent to
∆ ∆ K Θ := ∆ 2 Θ + ∇ a (R ∇ a Θ) =κ ∆ Θ,(13)
where ∆ := −∇ a ∇ a denotes the standard scalar Laplacian. We have shown that the scalar functions Θ solving Eq. (13) generate, via Eq. (10), solutionsξ a of the restricted eigenvalue problem of Eq. (11). Because the projectionP does not typically commute with ∆ K , these ξ a do not generally solve Eq. (4), and the restricted eigenvaluesκ are generally distinct from the eigenvalues κ in the full Hilbert space. In fact, we should generally expect thatκ min > κ min . However, the area-preserving vector eigenfield corresponding to this minimum restricted eigenvalue can also be considered a best approximation to a Killing field, albeit within a restricted class.
To recover the Cook-Whiting approximate Killing field, we must repeat the previous calculation in the inner product of Eq. (9). The operator ∆ K is then no longer self-adjoint, but R −1 times ∆ K is. Accordingly, we seek vector fieldsξ a R = ( * dΘ) a satisfying * dΦ,
R −1 ∆ K * dΘ R =κ R * dΦ, * dΘ R(14)
for allζ a = ( * dΦ) a . Integrating by parts, and once again invoking positivity of the standard inner product of scalar functions, we find that Eq. (14) is equivalent to
∆ ∆ K Θ = −κ R ∇ a (R ∇ a Θ).(15)
Although the notation here differs slightly, this is precisely the Euler-Lagrange equation that Cook and Whiting find [9] by minimizing a quadratic form similar to Eq. (6). Once again, the solutions (ξ a R ,κ R ) of this eigenvalue problem generally differ from the solutions (ξ a , κ) of Eq. (4) and from the solutions (ξ a ,κ) of Eq. (11).
Let us now make two technical comments. First, any constant function Θ = c will give zero on both sides of Eqs. (11) and (15) for all values ofκ orκ R , respectively. These are spurious solutions. They arise only because we have used potentials to describe the subspace of areapreserving vector fields. These solutions are ruled out by the the second condition in Eq. (10), which makes the correspondence betweenξ a and Θ an isomorphism.
Second, the Cook-Whiting inner product in Eq (9) looks a little odd, but it is not immediately clear whether there is anything technically wrong with it. There certainly can be problems. Recall that the scalar curvature in two dimensions varies as
δ 2 R = ∇ b ∇ [a δg b] a − 1 2 2 R δg a a(16)
under a perturbation δg ab of the metric. If this perturbation varies sufficiently rapidly over S, then the first term here can easily dominate the second, as well as the unperturbed, background value 2 R. The result is that a generic spherical geometry, even if perturbatively close to a round sphere in the sense that δg ab has small amplitude, can have regions of negative scalar curvature. (This is intuitively obvious if we imagine "pinching" the surface of a round sphere to create a small, saddle-shaped region of negative curvature.) On such geometries, the "inner product" of Eq. (9) is not positive-definite, and does not define a Hilbert space. However, in the space of all spherical geometries, there should be some finite region of "sufficiently smooth" perturbations of the round sphere for which the total scalar curvature remains everywhere positive. In this region, there is no obvious problem with the Cook-Whiting scheme, but nothing particular to recommend it either. The question could presumably be settled [11] by comparing qualitative features of the approximate Killing fields computed from Eqs. (11) and (15). Matzner's eigenvalue definition of an approximate Killing field is unambiguous, universally applicable, and reproduces the usual Killing fields on a symmetric manifold. But it is not necessarily efficient in practice. Indeed, it would prohibitively expensive to solve any of the eigenvalue problems in Eqs. (4), (11) or (15) on the apparent horizon at every moment of time of a black hole in a numerical simulation. Fortunately, however, there is a simple approximation to speed the process up on a generic geometry. This approximation is based on the Rayleigh-Ritz method [15], and works so long as we only want to find the lowest eigenvalue and the corresponding vector eigenfield.
Consider the Rayleigh-Ritz functional
F [ξ] := ξ, ∆ K ξ ξ, ξ = 2 M ∇ (a ξ b) · ∇ (a ξ b) · ǫ M ξ a ξ a ǫ(17)
on the full Hilbert space of Eq. (5), with the zero vector removed. The local extrema of Eq. (17) occur when ξ a is a vector eigenfield of ∆ K , and the value of F [ξ] at each such extremum is the corresponding eigenvalue κ. Note that the numerator here, which arises via integration by parts of the second-order operator ∆ K in Eq. (4), is precisely one half the square integral of L ξ g ab from Eq. (2). Thus, among all vector fields with fixed L 2 -norm on S, diffeomorphisms along the approximate Killing field modify the metric least in a quantifiable, L 2 sense. It is still not practicable to find the genuine absolute minimum of Eq. (17) on the computer, which of course would yield Matzner's approximate Killing field. But one can approximate that minimum by minimizing F [ξ] within an appropriate space of trial vector fields. This idea is familiar from elementary quantum mechanics, where just such a variational principle is routinely used to approximate the ground-state wave-function of a complicated system. Unless the subspace of trial fields one chooses happens to be orthogonal, or nearly so, to the true minimum ξ a true of F [ξ] in all of Hilbert space, the dominant component of the minimizing trial field ξ a trial should lie along ξ a true in Hilbert space. Most randomlychosen trial spaces will not be orthogonal to ξ a true . This idea allows us to approximate Matzner's approximate Killing field.
There is a natural candidate for the trial space of vector fields in which to minimize Eq. (17) in the specific case M ≃ S of a 2-sphere horizon of a quiescent black hole in numerical relativity. One striking feature of many recent numerical simulations (e.g, [14]) is that the horizons at late times often look fairly regular in the fiducial spacetime coordinates used to do the evolution. Therefore, it is natural to try a space of trial fields ξ a based simply on those coordinates. A specific proposal follows.
Use the fiducial spacetime coordinates in which the numerical evolution occurs to induce spherical coordinates (θ, φ) on the black-hole horizon in some more-or-less natural, but fundamentally ad hoc, way. Then, take the space of scalar trial potentials
Θ(θ, φ) := lmax l=1 l m=−l Θ lmŶ lm (θ, φ),(18)
whereŶ lm (θ, φ) are the ordinary scalar spherical harmonic functions on a round sphere, Θ lm are arbitrary constants, and l max is a chosen cut-off. Each of these potentials generates an area-preserving vector field via Eq. (10), and this will be our trial space [16] within the full Hilbert space of Eq. (5). Therefore, minimize
F [Θ] := * dΘ, ∆ K * dΘ * dΘ, * dΘ = Θ, ∆ ∆ K Θ Θ, ∆ Θ (19) = S 2 g ac g bd − g ab g cd ∇ a ∇ b Θ · ∇ c ∇ d Θ ǫ − S Θ · ∇ a ∇ a Θ · ǫ
within the trial space of potentials given by Eq. (18). Generally, we should expect that the minimizing potential will generate a vector field ξ a trial fairly close to the minimum-eigenvalue area-preserving vector eigenfieldξ a true of Eq. (11). This, in turn, should approximate Matzner's approximate Killing field from Eq. (4). To check the approximation, one could imagine increasing l max until ξ a trial doesn't vary much with the cut-off. Equivalently, one could use a fairly large cut-off-perhaps l max = 5 would be enough-from the start, and check that the amplitudes Θ lm are small for large l. If one prefers to approximate the Cook-Whiting approximate Killing field, one need only insert a factor of the scalar curvature R between the gradients in the denominator of Eq. (19).
There is one significant issue that has not been addressed in this discussion. Even once an approximate Killing field ξ a has been found from the eigenvalue approach, it is still determined only up to overall normalization on S. For a proper rotational Killing field on a symmetric apparent horizon, the correct normalization would demand that the affine length of each Killing orbit should be 2π. It is not immediately clear what convention might be used in the general case, without symmetry, to fix a normalization that goes over to this correct one in the limit of a symmetric manifold. This issue will be discussed more thoroughly, in the context of practical applications, in a forthcoming paper [11].
AcknowledgementsThe author would like to thank Ivan Booth, Manuela Campanelli, Greg Cook, Stephen Fairhurst, Greg Galloway, Carlos Lousto, Charles Torre, Bernard Whiting and Yosef Zlochower for stimulating discussions related to this question. This work has been supported by NSF grants PHY 0400588 and PHY 0555644, and by NASA grant ATP03-0001-0027.
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Isolated and Dynamical Horizons and Their Applications. A Ashtekar, B Krishnan, Living Rev. Relativity. 7A. Ashtekar and B. Krishnan. Isolated and Dynamical Horizons and Their Applications. Living Rev. Relativity 7 (2004), 10. Cited 8 July 2007.
Quasilocal energy and conserved charges derived from the gravitational action. J D Brown, J W York, Jr , Phys. Rev. D. 47J.D. Brown and J.W. York, Jr. Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D 47 (1993) 1407-1419.
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to the space of constant functions on S. However, the key point is that standard properties of the ordinary spherical harmonics show that this space of trial potentials contains no actual constant functions. The space of potentials in Eq. (18) is usually not orthogonal, in the sense of Eq. (10). This is why we have taken lmin = 1 in Eq. (18)The space of potentials in Eq. (18) is usually not orthog- onal, in the sense of Eq. (10), to the space of constant functions on S. However, the key point is that standard properties of the ordinary spherical harmonics show that this space of trial potentials contains no actual constant functions. This is why we have taken lmin = 1 in Eq. (18).
10) maps our space of trial potentials faithfully to a space of trial vector fields with the same dimension, lmax (lmax + 2). Eq Therefore, Therefore, Eq. (10) maps our space of trial potentials faithfully to a space of trial vector fields with the same dimension, lmax (lmax + 2).
| zyda_arxiv-0064000 |
A Wheeler-DeWitt Equation with Time
November 4, 2022
Rotondo Marcello
A Wheeler-DeWitt Equation with Time
November 4, 2022
The equation for canonical gravity produced by Wheeler and De-Witt in the late 1960s still presents difficulties both in terms of its mathematical solution and its physical interpretation. One of these issues is, notoriously, the absence of an explicit time. In this short note, we suggest one simple and straightforward way to avoid this occurrence. We go back to the classical equation that inspired Wheeler and DeWitt (namely, the Hamilton-Jacobi-Einstein equation) and make explicit, before quantization, the presence of a known, classically meaningful notion of time. We do this by allowing Hamilton's principal function to be explicitly dependent on this time locally. This choice results in a Wheeler-DeWitt equation with time. A working solution for the de Sitter minisuperspace is shown. arXiv:2201.00809v4 [gr-qc] 3 Nov 2022
Introduction
One traditional avenue to the quantization of gravity is the geometrodynamical one, represented by the infamous Wheeler-DeWitt (WDW) equation [1,2]. The equation is expected to describe the quantum evolution of the spatial components of the metric tensor of General Relativity (GR), but its solution and interpretation are long-standing problems [3]. In particular, a problem with time occurs when we try to interpret the WDW equation as a Schrödinger-type equation for gravity, because the state it describes appears to be stationary.
To begin with, the absence of time from the WDW equation is a consequence of the fact that the first-class Hamiltonian constraint of GR, of which the WDW equation intends to be the quantization, specifically enforces time diffeomorphism. In other words, it ensures its dynamical laws are valid independently of our choice of time coordinate. When we consider the so-called Hamilton-Jacobi-Einstein (HJE) equation developed by Peres [4], which expresses the constraint on the 00 component of the Einstein field equations in the Hamilton-Jacobi formalism, it is clear that in the classical case, time is absent where it should appear, even though the theory is classical. We know, however, that the HJE equation does not describe a timeless geometry. The reason that the HJE equation is not problematic can be traced back to the fact that a notion of time exists for the evolution of the spatial geometry, as long as the classical notion of trajectory in superspace holds. In that sense, it appears that the actual problem with time is not that it is absent in the Schrödinger-type equation itself, but that we cannot introduce it as we do in the classical case, since space no longer evolves along classical trajectories. It is not clear what becomes of this time beyond the semi-classical level, when gravity does not act as a stage for matter fields, but rather partakes in the quantum dance.
The absence of an external time in the description of GR, which is inherited by its quantization, is sometimes referred to as the "frozen formalism problem", and constitutes only one among other difficulties in the definition of time in classical and quantum physics [5]. In the present work, we will address only this particular aspect of the problem, ignoring its relations to others (a strictly related one being the definition of time-evolving observables for quantum gravity). Two classic reviews on this subject are Isham's [6] and Kuchař's [7]. We invite the reader to read these reviews for detailed references, and to gain a general idea of the large extent of variable approaches. The study of this aspect of the problem of time has certainly evolved significantly since the time of these reviews, with some issues of each approach being successfully addressed, but it ultimately remains open. Faced by the menace represented by the loss of a useful notion of time, three alternative reactions have been adopted by researchers: flight, fight, or freeze (corresponding to Isham's tempus ante quantum, tempus post quantum, and tempus nihil est).
1.
Flight: Time is recognized as a fundamental element of our description of physical phenomena, and attempts are made to define it before quantization, as a functional of the canonical variables. This is a conservative approach that tries to obtain an "external" notion of time as that appearing in Schrödinger's equation.
Fight:
Time is recognized as a fundamental element of our description of physical phenomena, but it is retrieved only after the quantization. This type of approach fights against the interpretative problems presented by the quantum theory to obtain a novel definition of time.
3. Freeze: Timelessness is accepted, time is forsaken as a fundamental notion for the description of quantum gravity, and attempts are made to provide a complete quantum theory otherwise.
The present work adopts a definition of time resulting from the semi-classical approximation of the WDW equation, an approach falling within the second category above. However, we do not limit to the semi-classical regime the definition of time identified by the semiclassical approximation. The point of the present note is in fact to suggest that the "frozen formalism" could be avoided by retaining the use of a classical notion of time suggested by the semi-classical approximation of the theory, even though quantum space does not evolve along classical trajectories. Therefore, our proposal belongs to the first category, in that it carries over to the quantum regime the definition of time justified by the semi-classical approximation. The definition of time that we adopt as a starting point is that naturally resulting from the semi-classical approximation of canonical quantum gravity obtained by expanding the total wave functional in inverse squares of the Planck mass [8,9,10]. Classical GR and the Schrödinger equation for non-gravitational fields straightforwardly recovered this approximation. Time, for the matter fields, is a multi-fingered (i.e., space dependent) functional generated by the classical evolution of background geometry along its trajectory in superspace. The operation has formal analogies with the Wentzel-Kramers-Brillouin (WKB) approximations of quantum mechanics, and the Born-Oppenheimer approximation from molecular physics [11,12,13]. In recent years, special attention was drawn to the problem of re-establishing the unitarity of time evolution in this approach (see, for example, [14,15,16,17]). For a recent work by this author which is related to the present one, see [18].
One key observation that motivated this approach was that this multi-fingered (or "WKB") functional time can be applied to the HJE equation itself, and we can make the presence of that classical time explicit in the HJE equation, which is the timeless form. This alone was the starting point of Wheeler and DeWitt, as the following memoir by one of the authors recalls.
One day in 1965, John Wheeler had a two-hour stopover between flights at the Raleigh-Durham airport in North Carolina. He called Bryce DeWitt, then at the University of North Carolina in Chapel Hill, proposing to meet at the airport during the wait. Bryce showed up with the Hamilton-Jacobi equation of general relativity, published by Asher Peres not long before [...] Bryce mumbled the idea of repeating what Schröedinger did for the hydrogen atom: obtaining a wave equation by replacing the square of derivatives with (i times) a second derivative-a manner for undoing the optical approximation. [...] Wheeler was tremendously excited (he was often enthusiastic) and declared on the spot that the equation for quantum gravity had been found. [19,20] What if DeWitt had presented Wheeler with the HJE equation together with the notion of multi-fingered time? The resulting equation does present the functional time variable just as time appears in the Schrödinger equation, and preserves the correct classical limit for gravity.
In Section 2, we briefly review the definition of this multi-fingered time. In Section 3, we rewrite the HJE equation, allowing an explicit local dependence of Hamilton's principal function on that time, and write the associated WDW equation with time. Finally, in Section 4, we discuss a simple realization in the de Sitter minisuperspace that is of special interest to quantum cosmology.
Classical Time Evolution from the Hamiltonian Constraint
For a straightforward introduction to the emergence of time in the semi-classical approximation of canonical quantum gravity, we refer the reader to the mentioned work by Isham [6], Section 5.4 and references therein. Here, we follow, with some variation, the notation adopted by Kiefer [9]. Consider a generic spacetime with line element
ds 2 = −N 2 dt 2 + N i dtdx i + h ij dx i dx j .(1)
Here, h ij = g µν , µ, ν ∈ {1, 2, 3} is the spatial metric, with Latin indices i, j ∈ (1, 2, 3), N i = g 0i is the shift function, and N = (−g 00 ) −1/2 is the lapse function. In the Hamiltonian formalism of GR, time parametrization invariance is enforced by a first-class Hamiltonian constraint (i.e., a constraint imposed on the Hamiltonian only after the equations of motion are satisfied).
d 3 x (2M ) −1 G AB π A π B + V(h A ) + H φ (φ; h A ) = 0 .(2)
(This constraint can be obtained from the variation of the super-Hamiltonian of GR with respect to the lapse function N in the Arnowitt-Deser-Misner (ADM) formalism [21]. Here, we intend to recall only the elements strictly necessary to follow our discussion.)
In Equation (2), the capital indices, A, B = {ij} represent pairs of Latin indices, and G AB is the DeWitt metric
G AB ≡ G ijkl = 1 2 √ h (h ik h jl + h il h jk − h ij h kl ) ,(3)
underlying superspace, i.e., the space of spatial metrics up to differeomorphism invariance.
The physical scale (we have set = c = 1) of quantum gravity is set by the "geometrodynamical mass" M , which is proportional to the square of the Planck mass m P
M = (m P /2) 2 , m P = (8πG) −1/2 .(4)
The geometrodynamical potential density V is
V = 2M √ h(2Λ − (3) R) ,(5)
where h and (3) R are the determinant and the Ricci scalar of the spatial metric, respectively. The Hamiltonian density operator H φ is taken to describe bosonic matter. The WDW equation results from an attempt to quantize the Hamiltonian constraint (2) straightforwardly, applying it to the "wave functional of the universe", Ψ[h A , φ], thus obtaining
d 3 x (2M ) −1 G AB ∂ A ∂ B + V(h A ) + H φ (h A , φ) Ψ[h A , φ] = 0 ,(6)
where all variables are promoted to the respective operators. For the sake of simplicity, in this section, we have adopted the trivial ordering, and the symbols ∂ A are used to indicate functional derivatives with respect to the metric component indicated by the double index.
The evolution of bosonic quantum fields in classical curved spacetime is obtained by making the ansatz
Ψ[h A , φ] = χ[h A ]ψ[φ; h A ](7)
for the total wave functional, and considering a WKB-like expansion in inverse powers of M [9,10]. In doing so, one aims at wave functionals χ and ψ that describe the "heavy" (i.e., the spatial metric components) and "light" degrees of freedom (i.e., matter), respectively. Notice that ψ depends on the geometry only parametrically, which is indicated by the use of the semicolon. The method consists of substituting the expansion
Ψ[h A , φ] = exp i ∞ n=0 M 1−n S n [h A , φ](8)
in the WDW equation, and equating contributions to equal powers of M .
To order M 2 , one obtains S 0 = S 0 [h A ]: the leading contribution is purely geometrodynamical. To order M 1 , one obtains the vacuum HJE equation
d 3 x (2M ) −1 G AB ∂ A S G ∂ B S G + V = 0 ,(9)
S G = M S 0 being the leading contribution to the phase of the wave functional (8). The HJE Equation (9) appears to be timeless due to the vanishing of the RHS. However, the time evolution of space can still be obtained from the Hamiltonian constraint (2) by expressing the canonical momenta (defined by the Lagrangian as π A = ∂L/∂ḣ A , and identified with π A = ∂ A S G in the Hamilton-Jacobi formalism) in terms of the geometrodynamical velocities. From the Hamiltonian equations of motion in the ADM formalism, these are given bẏ
h A = N π A + 2N (A) ,(10)
N (A) being a shorthand for N (i;j) . The fact that time evolution is retrieved by such substitution, as the Hamiltonian is constrained, is an important point.
At this point, define
ψ 1 [φ; h A ] = exp (iS 1 [φ, h A ]
) and require conservation of the current associated with χ. Then, to order M 0 , one gets the following functional equation for matter
d 3 x iM −1 G AB ∂ A S G ∂ B − H φ ψ 1 [φ; h A ] = 0 .(11)
By using as time the local "multi-fingered" time τ = τ (x) of the parametrization generated by the classical momenta along the classical trajectory in superspace
G AB π A ∂ B x τ (y) = δ(y − x) ,(12)
Equation (11) gives the Schrödinger-type functional equation
d 3 x i • ψ 1 [φ; h A ] − H φ ψ 1 [φ; h A ] = 0 ,(13)
where we employ a circle over the variable to indicate the functional derivative with respect to τ
δ δτ = M −1 G AB π A ∂ B .(14)
Notice that in normal coordinates (N = 1, N 0i = 0), the Equation (13) reduces to the functional Schrödinger equation for bosonic fields. In this case, the passage to the partial derivative with respect to time is granted by the fact that the matter wave functional ψ 1 [φ; h A ] depends explicitly on time only through the background metric, which appears as a set of local parameters, and employing the resulting relation between momenta and velocities (10), (3) reduces to an application of the chain rule.
The Time Evolution of Quantum Space
The functional derivative with respect to τ of a metric component, intended as the functional
h A (x) = d 3 y h A (y) δ(y − x) ,(15)
gives the relation between velocities and momenta,
• h A = M −1 G AB π A .(16)
Putting the relation (16) back into the Hamiltonian equation gives the equations of motion with respect to τ . In other words, we can use τ not only to describe the evolution of quantum fields with respect to the background metric, but also to describe the evolution of the background metric itself. It is a favorable parametrization in that it makes the form of the equations of motion simpler and not explicitly dependent on the coordinate choice (1). Incidentally, notice that, in Peres' HJE equation, the spatial metric components are defined as functions of space alone. How they become a function of time seems to be a problem that is not addressed in the literature. In the present treatment, they become functions of time precisely by defining their dependence on time according to (16).
The main objection in extending the use of multi-fingered time to the quantum evolution is that classical trajectories are lost in that regime. While this is true, we still know what the classical trajectory is, and we can use the "natural" parametrization (i.e., the parametrization in which all equations of motion appear simple as (16)) along it to describe evolution along non-classical trajectories between wave fronts of constant multi-fingered time.
Working with the vacuum model, we can make multi-fingered time explicit in the HJE Equation (9) by rewriting it as
d 3 x ∂ τ S G + (2M ) −1 G AB ∂ A S G ∂ B S G + V = 0 ,(17)
and requiring that
d 3 x∂ τ S G = 0 .(18)
Notice that the derivative with respect to τ is partial. As we previously observed, WKB time (3) only takes into account the dependency on τ through the geometrodynamical degrees of freedom, but S G could, in principle, also be explicitly dependent on it. What we are doing is simply allowing for this possibility. Adding and subtracting the τderivative of S G , and substituting the velocities (16), integration of the HJE Equation (17) tells us that S G is indeed Hamilton's principal function, defined as a functional integral of the Lagrangian.
Notwithstanding the fact that the condition (18) ensures that the Hamiltonian constraint still holds, it allows for the action to remain dependent explicitly on τ locally. We may then try to quantize the HJE Equation (17)à la Schrödinger, and require the global condition (18) to be true only in the classical limit. What we obtain is a WDW equation with time
d 3 x −i ∂Ψ ∂τ + (2M ) −1 G AB ∂ A ∂ B + V Ψ = 0 .(19)
Both the introduction of the coordinate independent functional time derivative and the (classically redundant) condition (18) are necessary to obtain Equation (19). The second condition is somehow reminiscent of another approach [22], where time is recovered by weakening the classical Hamiltonian constraint, required to hold only on average in the quantum regime. In that work, the problem of which time to use to describe the evolution is not addressed, and the condition on time dependence is stricter than ours (see (6) in the referenced paper), resulting in a wave function whose phase does not depend on time even locally.
The de Sitter Minisuperspace
The results of the previous section, i.e., the time evolution Equation (19) combined with the global condition (18), are only formal. Equation (19) still presents the same issues as the "timeless" WDW Equation (6). Besides that, using a spatial-dependent τ to parametrize the spatial geometry is more easily said than done. In the following, we will consider, by way of example, the solution for the spatially flat de Sitter universe, described by the line element
ds 2 = −dt 2 + a(t) 2 δ ij dx i dx j .(20)
Here, a(t) is the scale factor. We will set M = 1/12. As a geometrodynamical variable, rather than the scale factor itself. It will be convenient to adopt
q = (2a) 3 2 3 d 3 x 1 2 ,(21)
which is proportional to the square root of the co-moving spatial volume considered.
The Ricci scalar of the spatial metric vanishes in this model, and the Hilbert-Einstein action Lagrangian, reads simply
L = − 1 2 q 2 + ω 2 q 2 .(22)
Here, we have defined the constant ω = 3Λ/4. The canonical momentum is π q = −q, and the Hamiltonian is
H = − 1 2 π 2 q + 1 2 ω 2 q 2(23)
The Hamiltonian constraint then gives the Friedmann equation
q q 2 = ω 2 .(24)
This allows us to obtain the classical time evolution of spatial volume
q(t) = q(0) exp (ωt) .(25)
The HJE Equation (9) reduces to − 1 2
∂S ∂q
2 + 1 2 ω 2 q 2 = 0 .(26)
Classically, Hamilton's principal function depends on time only implicitly, and is of the form
S = ∓ ω 2 q 2 − q 2 i .(27)
Using Equation (25), one can check that the (one-fingered) time associated with this action indeed coincides with forward coordinate time when we choose the negative sign.
Moving onto the quantization, notice that with our choice of variable, the DeWitt metric, that in terms of the scale factor reads G aa = −2a, is now
G qq = −(3q) 2 3 .(28)
This simplifies the measure
−G (a) da → dq(29)
and, adopting the Laplace-Beltrami operator ordering for the kinetic operator, we have
−G (a) −1 ∂ a −G (a) G aa ∂ a → ∂ 2 q .(30)
Then, in this minisuperspace, the WDW Equation (19) reads
iΨ = 1 2 ∂ 2 q Ψ + 1 2 ω 2 q 2 Ψ ,(31)
which is essentially the one-dimensional Schrödinger equation for the so-called inverted harmonic oscillator. See [23] for a recent review, and [24] for an application to the study of a scalar field in slow-roll inflation. The difference in our case is that the variable is constrained to the positive axis only, and time appears with the opposite sign. As in [24], we require the Gaussian ansatz
ψ(q, t) = A(t) exp −B(t)q 2 .(32)
The equality in (31) of the terms of null and second order in q imposes −iȦ = A B
and
−iḂ = 1 2 ω 2 + 2B 2 .(34)
From the evolution Equation (34), we have
B(t) = ω 2 tan (φ + iωt) .(35)
Here φ is the real part of the constant of integration. The imaginary part of the constant is merged into the choice of initial time, instead, so that the width of the state is minimized at t = 0. The value of φ determines the greater (<π/4) or lesser (>π/4) peaking of the distribution in q, rather than the conjugate momentum at time t = 0.
Substituting B(t) in Equation (33), for a normalized solution, we obtain
A(t) = 2 π ω sin(2φ) 1/4 (cos(φ + iωt)) − 1 2 .(36)
The expectation value for q is
q = (2πω sin(2φ)) − 1 2 cos(2φ) + cosh(2ωt) | cos(φ + iωt)| .(37)
At late times, we correctly recover the classical inflationary expansion q ∝ exp(ωt) of Equation (25). In this limit, the phase of ψ approximates the classical action (27) (up to a global phase, which fixes the initial value) and loses its explicit dependence on time, thus reducing to the classical action (18). On the other hand, when we approach the time t = 0, where the state is maximally contracted for the given value of φ, the expectation value of the scale variable deviates from the classical one, which is directed at asymptotic convergence to zero: we observe instead the state of the de Sitter universe bouncing back from an earlier phase of contraction (see Figure 1). Figure 1: Time evolution of the expectation value of the scale variable q for φ = π/4. We set ω = 1.
Conclusions
In this short note, we have proposed to extend to the quantum regime the use of the multi-fingered time originating from the semi-classical WKB approximation of geometrodynamics. We have done so by rewriting the classical HJE equation for vacuum space to include an explicit time dependency on Hamilton's principal function, and requiring this dependency to vanish globally. The quantization provides a WDW equation that describes the evolution of the state with respect to classical multi-fingered time. Quantum matter fields can be included by appropriately augmenting the Hamiltonian operator. The classical limit will still be granted.
The main purpose of this work was to provide a formal WDW equation, (19), with a clear and working notion of time. This result, however, does not help with the original mathematical difficulties of the WDW equation, such as the indefiniteness of the superspace metric, or the divergence of functional derivatives in the full theory. However, we showed by example of a minisuperspace model of flat de Sitter universe that exact normalizable solutions can be easily found and interpreted. In particular, for the de Sitter universe, we found a wellbehaved Gaussian solution, which shows a state that bounces back at the time of maximal contraction, thus avoiding the classical asymptotic regression to nihil.
The next step along this line of research could be the formalization of our heuristic approach both in the Hamiltonian and the Lagrangian formalism, where the wave functional could be constructed in terms of a path integral over foliations of constant multi-fingered time. On the side of application, it would be fundamental to work out exact solutions that include quantum matter degrees of freedom. Expanding on the de Sitter model introduced here by adding perturbations could be relevant for early inflationary cosmology. Furthermore, beyond cosmology, an application to time evolution during the last stage of gravitational collapse could be of interest.
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