Efficient Solvers for Stochastic Finite Element Saddle Point Problems

Transcription

1 Efficient Solvers for Stochastic Finite Element Saddle Point Problems Catherine E. Powell School of Mathematics University of Manchester, UK Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 1/3

2 Collaborators: Elisabeth Ullmann, University of Freiberg, Germany. Andrew Gordon, University of Manchester, UK. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 2/3

3 Deterministic PDEs Galerkin mixed finite element approximation for two field PDE problems on D R 2,3, leads to variational problems of the form find u h (x) V h and p h (x) W h such that: a (u h, v) + b(p h, v) = g (v), v V h b(w, u h ) = f (w), w W h and saddle point systems A BT B 0 u p = g f for which the study of efficient solvers is mature. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 3/3

4 PDEs with Random Data Stochastic Galerkin mixed finite element approximation for two field PDE problems with random data, on D Γ, with Γ R M, leads to find u hd (x, ξ) V h S d and p hd (x, ξ) W h S d such that: â (u hd, v) + ˆb(p hd, v) = ĝ (v), v V h S d ˆb(w, uhd ) = ˆf (w), w W h S d and saddle point systems  ˆB T ˆB 0 u p = g f which are much larger and more complicated to solve. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 4/3

5 Model Problem: Darcy Flow T p = u, in D u = 0, in D p = g, on D D T p n = 0, on D N = D\ D D Solution variables: p = p(x) hydraulic head (pressure) u = u(x) velocity field Data: T = T(x) transmissivity coefficients g = g(x) boundary data D R d spatial domain Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 5/3

6 Outline Stochastic Galerkin methods for: T (x, ω) 1 u(x, ω) p (x, ω) = 0, u(x, ω) = 0 Solving single very large saddle point system Stochastic collocation methods for: T (x, ω) 1 u(x, ω) p (x, ω) = 0, u(x, ω) = 0 Solving many small deterministic saddle point systems (see A. Gordon s poster) Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 6/3

7 Stochastic PDE Problem Let T(x, ω) : D Ω R be a correlated random field. Approximate T 1 (x, ω) by a function T 1 M (x, ξ) involving a finite number of random variables ξ = (ξ 1,..., ξ M ) taking values in Γ = M i=1 Γ i R M. Find p(x, ξ), u(x, ξ) such that P -a.s. T 1 M (x, ξ)u(x, ξ) p (x, ξ) = 0, u(x, ξ) = 0 in D Γ, p(x, ξ) = g(x) on D D Γ, u(x, ξ) n = 0 on D N Γ. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 7/3

8 Finite-dimensional Noise Assumption Popular choices are: Truncated Karhunen-Loève expansion (linear) T 1 M (x, ξ) = µ(x) + M k=1 t k (x) ξ k, t k (x) = λ k c k (x) {λ k, c k (x)}, k = 1, 2,... are the eigenvalues and eigenfunctions of covariance {ξ 1, ξ 2,...} are uncorrelated, with mean zero and unit variance Truncated Polynomial Chaos expansion (nonlinear) T 1 M (x, ξ(ω)) = µ(x) + N k=1 t k (x) ψ k (ξ), t k (x) = T 1 ψ k (ξ) Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 8/3

9 Example - KL expansion Consider the covariance function C T 1(x, y) = exp ( x y 1 ) D = [0, 1] [0, 1] and Gaussian random variables. Realisations of T 1 M with M = 5, 20, 50, with µ(x) = Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 9/3

10 Polynomial Chaos If the random variables are independent then the joint density function has the form: ρ(ξ) = i ρ i (ξ i ) and E [g(ξ)] = g(ξ) = Γ ρ(y)g(y) dy. Orthonormal polynomials in ξ are constructed via ψ i (ξ) = M s=1 ψ is (ξ s ) where ψ is (ξ s ) is a univariate polynomial of degree i s, orthonormal w.r.t to, = E [ ]. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 10/3

11 Weak Formulation V = L 2 ρ(γ, H(div; D)) W = L 2 ρ(γ, L 2 (D)) We seek u(x, ξ) V and p(x, ξ) W such that: ( T 1 M u, v ) + (p, v) = (g, v n) ΓD, (w, u) = 0, v(x, ξ) V and w(x, ξ) W. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 11/3

12 Stochastic Galerkin Equations Find u hd (x, ξ) V h S d and p hd (x, ξ) W h S d satisfying: ( ) T 1 M u hd, v + (p hd, v) = (g, v n) ΓD, (w, u hd ) = 0, v V h S d and w W h S d. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 12/3

13 Stochastic Galerkin Equations Find u hd (x, ξ) V h S d and p hd (x, ξ) W h S d satisfying: ( ) T 1 M u hd, v + (p hd, v) = (g, v n) ΓD, (w, u hd ) = 0, v V h S d and w W h S d. V h H(div; D), W h L 2 (D) are a deterministic inf-sup stable pairing S d L 2 (Γ) Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 12/3

14 Total degree polynomials For S d we choose M-variate polynomials of total degree d. Use orthonormal polynomial chaos basis S d = span { } ψ 1 (ξ),..., ψ Nξ (ξ), N ξ = (M + d)! M!d! M = 1 M = 5 M = 10 d = 1 d = 2 d = 3 d = 1 d = 2 d = 3 d = 1 d = 2 d = Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 13/3

15 Saddle point systems V h = span {ϕ i (x)} N u i=1, W h = span {φ j (x)} N p j=1, S d = span {ψ k (ξ)} N ξ k=1 Â ˆB T u = g ˆB 0 p f (never assembled) of dimension N x N ξ N x N ξ where N x = N u + N p and Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 14/3

16 Saddle point systems V h = span {ϕ i (x)} N u i=1, W h = span {φ j (x)} N p j=1, S d = span {ψ k (ξ)} N ξ k=1  ˆB T u = g ˆB 0 p f (never assembled) of dimension N x N ξ N x N ξ where N x = N u + N p and  ˆB T ˆB 0 = I A 0 + N k=1 G k A k I B T I B 0 [A 0 ] ij = D µ(x) ϕ i ϕ j dx, [A k ] ij = D t k (x) ϕ i ϕ j dx. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 14/3

17 Linear (KL) Case T 1 M (x, ξ) = µ(x) + M k=1 t k (x)ξ k N = M (the number of random variables) G k matrices have at most two non-zeros entries per row. [G k ] rs = ξ k ψ r (ξ)ψ s (ξ), k = 1 : M N k=1 G k is sparse so  is block sparse. Matrix-vector products are cheap  O(M(N u N ξ )) ˆB, ˆB T O(N p N ξ ) Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 15/3

18 Matrix structure M = 2, d = 2 (left) and M = 4, d = 2 (right) Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 16/3

19 Nonlinear (PC) Case T 1 M (x, ξ) = µ(x) + N k=1 t k (x)ψ k N = (M+2d)! M!(2d)! >> N ξ G k matrices have non-trivial sparsity pattern [G k ] rs = ψ k (ξ)ψ r (ξ)ψ s (ξ), k = 1 : N M k=1 G k is dense so  is block dense. Matrix-vector products are expensive  O(N(N u N ξ )) O(N(N u N 2 ξ )) ˆB, ˆB T O(N p N ξ ) Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 17/3

20 Matrix structure Want to perform as few matrix vector products with this as possible! Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 18/3

21 (Minres) preconditioning strategies There are two well-known classes of block-diagonal preconditioners. Schur-Complement Preconditioning  0 0 ˆB 1 ˆBT Augmented Preconditioning  + γ 1 ˆBT Ŵ 1 ˆB 0 0 γŵ Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 19/3

22 (Minres) preconditioning strategies There are two well-known classes of block-diagonal preconditioners. Schur-Complement Preconditioning  0 0 ˆB 1 ˆBT Augmented Preconditioning  + γ 1 ˆBT Ŵ 1 ˆB 0 0 γŵ Both require cheap, robust approximations to  Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 20/3

23 Schur-Complement Preconditioning Approximate  L diag(a 0) where L is s.p.d. A practical preconditioner is: P = L diag(a 0) 0 0 ˆB (L diag(a0 )) 1 ˆBT = L diag(a 0) 0 0 L ( Bdiag(A 0 ) 1 B T ). Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 21/3

24 Schur-Complement Preconditioning Approximate  L diag(a 0) where L is s.p.d. A practical preconditioner is: P = L diag(a 0) 0 0 ˆB (L diag(a0 )) 1 ˆBT = L diag(a 0) 0 0 L ( Bdiag(A 0 ) 1 B T ). For the model problem, Bdiag(A 0 ) 1 B T µ and optimal deterministic elliptic PDE solvers (eg. AMG) can be applied. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 21/3

25 Schur-Complement Preconditioning Approximate  L diag(a 0) where L is s.p.d. A practical preconditioner is: P = L diag(a 0) 0 0 ˆB (L diag(a0 )) 1 ˆBT = L diag(a 0) 0 0 L ( Bdiag(A 0 ) 1 B T ). For the model problem, Bdiag(A 0 ) 1 B T µ and optimal deterministic elliptic PDE solvers (eg. AMG) can be applied. Simplest choice: L = I. Approximating P 1 r requires: One solve with a diagonal matrix N ξ V-cycles of multigrid (e.g. AMG) on a deterministic matrix of dimension N p. Cost is O(N ξ (N u + N p )). Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 21/3

26 Convergence analysis P = I diag(a 0) 0 0 I amg(bdiag(a 0 ) 1 B T ) Theorem Let 0 < ˆν 1... ˆν N be the eigenvalues of diag(â) 1 Â. The eigenvalues of the preconditioned saddle-point matrix lie in the union of the bounded intervals, [ ( ) 1 ˆν 1 ˆν Θ2, 1 ( )] ˆν N ˆν 2N 2 + 4θ2 [ ˆν 1, 1 ( )] ˆν N + ˆν 2N 2 + 4Θ2. When L = I, this preconditioner is mean-based. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 22/3

27 Linear (KL) case Theorem E.g. if lowest-order Raviart-Thomas elements are used, the eigenvalues of diag(â) 1 Â lie in the bounded interval [ 1 2 c dτ, c dτ ] where τ = 3σ 2µ M k=1 λk t k (x), σ and µ are the standard deviation and mean of T 1 (x, ω), and c d is a constant (possibly) depending on d. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 23/3

28 Linear (KL) case Theorem E.g. if lowest-order Raviart-Thomas elements are used, the eigenvalues of diag(â) 1 Â lie in the bounded interval [ 1 2 c dτ, c dτ ] where τ = 3σ 2µ M k=1 λk t k (x), σ and µ are the standard deviation and mean of T 1 (x, ω), and c d is a constant (possibly) depending on d. When T 1 M σ µ is a KL expansion this is OK because is small for well-posed problem. c d = O(1) (bounded rvs), c d = O( d) (Gaussian rvs) Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 23/3

29 Example - KL case Let D = [0, 1] [0, 1], f = 0 with mixed bcs. We choose a Bessel covariance function for the random input with µ(x) = 1 and λ = 1 M = 6 random variables capture 98% of the variance. u n =0 p=1 p=0 u n =0 Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 24/3

30 Mean of numerical solution Pressure (left), Flux (right) Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 25/3

31 Variance of numerical solution Pressure (left), y component (middle) and x component (right) of the Flux x 10 3 x Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 26/3

32 Preconditioned minres iterations d N ξ (N u + N p ) 344,064 1,032,192 2,580,480 5,677,056 σ µ σ µ σ µ = 0.1 total iters N V 1,260 3,864 10,080 22,176 total solve time 14.0s 45.35s s s = 0.2 total iters N V 1,540 4,956 13,020 29,106 total solve time 17.18s 58.51s s = 0.3 total iters N V 1,848 6,216 16,800 39,732 total solve time 20.66s 72.97s s Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 27/3

33 Nonlinear (PC case) Theorem E.g. if lowest-order Raviart-Thomas elements are used, the eigenvalues of diag(â) 1 Â lie in the bounded interval [ 1 2 c dτ, c dτ ] where, for lognormal diffusion coefficients in particular, τ depends nonlinearly on σ G c d = O (exp(dm)) The Schur-complement preconditioner is too weak for large d and σ G requiring an excessive number of matrix-vector products. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 28/3

34 Preconditioned minres iterations Previous test problem. T M (x, ξ) = exp(g M (x, ξ)) where underlying Gaussian field G(x, ω) has Bessel covariance function. M = 5. d N ξ (N u + N p ) 31, , , ,184 N σ G = 0.2 total iters total solve time 2s 22s 245s 2979s σ G = 0.4 total iters total solve time 3s 35s 534s 6321s σ G = 0.6 total iters total solve time 3s 53s 1369s 13794s σ G = 0.8 total iters total solve time 4s 84s 2604s 29808s Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 29/3

35 Augmented Preconditioning Consider the ideal preconditioner P = Â + γ 1 ˆBT Ŵ 1 ˆB 0 0 γŵ Theorem Let Ŵ = I M, where w T (I M) w = w 2 L 2 (Γ,L 2. The eigenvalues of the (D)) preconditioned saddle-point matrix are bounded and lie in ( 1, ˆβ 2 t min γ + ˆβ 2 t min ] {1} where ˆβ is the inf-sup constant and T M t min a.e. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 30/3

36 Augmented Preconditioning Suppose we approximate  by L A 0 where L is any s.p.d N ξ N ξ matrix and replace Ŵ with L 1 M then, P = L ( A 0 + γ 1 B T M 1 B ) 0 0 L 1 γm Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 31/3

37 Augmented Preconditioning Suppose we approximate  by L A 0 where L is any s.p.d N ξ N ξ matrix and replace Ŵ with L 1 M then, P = L ( A 0 + γ 1 B T M 1 B ) 0 0 L 1 γm For this model problem, v T ( A 0 + γ 1 B T M 1 B ) v = ( T 1 M v, v ) + γ 1 ( v, v). The matrix in red is a discretisation of a weighted H(div) operator on the velocity space. Optimal deterministic solvers (geometric multigrid) can be exploited. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 31/3

38 Augmented Preconditioning Simplest choice: L = I ( mean-based preconditioner) P = L ( A 0 + γ 1 B T M 1 B ) 0 0 L γm Approximating P 1 r requires: N ξ solves with M and N p multiplications with L. N ξ V-cycles of multigrid (e.g. Arnold-Falk-Winther) on a deterministic matrix of dimension N u and N u solves with L. Even if L is dense, still cheaper than a matrix-vector product with saddle point matrix! Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 32/3

39 Convergence analysis P = L ( A 0 + γ 1 B T M 1 B ) 0 0 L γm Spectral bounds for preconditioned system matrix depend on choice of L and the efficiency of the approximation  L A 0 and γ. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 33/3

40 Convergence analysis P = L ( A 0 + γ 1 B T M 1 B ) 0 0 L γm Spectral bounds for preconditioned system matrix depend on choice of L and the efficiency of the approximation  L A 0 and γ. However, asymptotically, as γ 0, we can show all N ξ N p negative eigenvalues cluster at 1. N ξ N p positive eigenvalues cluster at +1 N ξ (N u N p ) positive eigenvalues lie in [ˆν 1, ˆν n ] where ˆν 1 v T Âv v T (L A 0 ) v ˆν n Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 33/3

41 Preconditioned minres iterations Previous test problem. Now T M (x, ξ) = exp(g M (x, ξ)) where underlying Gaussian field G(x, ω) has Bessel covariance function. M = 5. Choose L = I and γ = d N ξ (N u + N p ) 31, , , ,184 N σ G = 0.2 total iters total solve time 3s 9s 65s 399s σ G = 0.4 total iters total solve time 3s 12s 97s 668s σ G = 0.6 total iters total solve time 4s 17s 140s 1063s σ G = 0.8 total iters total solve time 5s 19s 203s 1658s Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 34/3

42 Improved choice of L Could use Pitsianis-Van Loan so-called best Kronecker product approximation : L = argmin{h R N ξ N ξ : Â H A 0 F } L is the symmetric and positive definite matrix L = I + N k=1 tr(a T k A 0) tr(a T 0 A 0) G k Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 35/3

43 Preconditioned minres iterations Previous test problem. Now T M (x, ξ) = exp(g M (x, ξ)) where underlying Gaussian field G(x, ω) has Bessel covariance function. M = 5. d N ξ (N u + N p ) 31, , , ,184 N σ G = 0.2 total iters total solve time 3s 8s 54s 328s σ G = 0.4 total iters total solve time 3s 11s 73s 429s σ G = 0.6 total iters total solve time 4s 12s 86s 576s σ G = 0.8 total iters total solve time 4s 14s 106s 723s Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 36/3

44 Average time in seconds per minres iteration and (av. mat-vec time + av. preconditioner time) d=2 d=3 d=4 L = I 0.58 ( ) 4.00 ( ) ( ) Improved L 0.59 ( ) 4.08 ( ) ( ) Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 37/3

45 Summary If random inputs are KL expansions: Mean-based preconditioners are cheap and robust within the range of statistical parameters where the problem is well-posed. Schur-complement preconditioners are OK as matrix-vector products are cheap. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 38/3

46 Summary If random inputs are KL expansions: Mean-based preconditioners are cheap and robust within the range of statistical parameters where the problem is well-posed. Schur-complement preconditioners are OK as matrix-vector products are cheap. If the random inputs are PC expansions Mean-based preconditioners are still cheap but are inadequate as σ and d increase. Augmented preconditioners are promising because fewer matrix-vector products are required Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 38/3

47 Summary If random inputs are KL expansions: Mean-based preconditioners are cheap and robust within the range of statistical parameters where the problem is well-posed. Schur-complement preconditioners are OK as matrix-vector products are cheap. If the random inputs are PC expansions Mean-based preconditioners are still cheap but are inadequate as σ and d increase. Augmented preconditioners are promising because fewer matrix-vector products are required But: Also need to look at stopping criteria for iteration. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 38/3

48 References C.E.Powell, E. Ullmann, Preconditioning stochastic Galerkin saddle point systems, Submitted. H. Elman, D. Furnival, C.E. Powell, H(div) preconditioning for a mixed finite element formulation of the stochastic diffusion equation. Math. Comp O.G. Ernst, C.E. Powell, D. Silvester and E. Ullmann, Efficient solvers for a Linear Stochastic Galerkin Mixed Formulation of the Steady-State Diffusion Equation, SIAM J. Sci. Comput C.E. Powell, H. Elman, Block-diagonal preconditioning for spectral stochastic finite element systems, IMA J. Numer Anal. 2008/9. Efficient Solvers for Stochastic Finite Element Saddle Point Problems p. 39/3