Solving the Stochastic Steady-State Diffusion Problem Using Multigrid

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1 Solving the Stochastic Steady-State Diffusion Problem Using Multigrid Tengfei Su Applied Mathematics and Scientific Computing Program Advisor: Howard Elman Department of Computer Science May 5, 2016 Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 1 / 31

2 Outline 1 Project Review 2 Multigrid Two-Grid Method Algorithm Validation 3 Low-Rank Approximation Motivation Algorithm Numerical Results 4 Conclusions Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 2 / 31

3 Project Goal The stochastic steady-state diffusion equation { (c(x, ω) u(x, ω)) = f (x) in D Ω u(x, ω) = 0 with stochastic coefficient c(x, ω) : D Ω R. Approaches Monte Carlo method (MCM) Stochastic finite element method (SFEM) [Ghanem & Spanos, 2003] Solver: Multigrid [Elman & Furnival, 2007] on D Ω Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 3 / 31

4 Stochastic FEM Karhunen-Loève expansion [Loève, 1960] m c(x, ω) c 0 (x) + λi c i (x)ξ i (ω). i=1 Weak form Γ ρ(ξ) D c(x, ξ) u(x, ξ) v(x, ξ)dxdξ = ρ(ξ) f (x)v(x, ξ)dxdξ, Γ D for v(x, ω) V = H 1 0 (D) L2 (Γ). Here ρ(ξ) is the joint density function, and Γ is the joint image of {ξ i } m i=1. Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 4 / 31

5 Stochastic FEM Finite-dimensional subspace V hp = S T = span{φ(x)ψ(ξ), φ S, ψ T } with basis functions φ(x): piecewise linear/bilinear basis functions ψ(ξ): m-dimensional orthonormal polynomials, total order not exceeding p [Xiu & Karniadakis, 2003]. SFEM solution Γ ψ r (ξ)ψ s (ξ)ρ(ξ)dξ = δ rs, M = u hp (x, ξ) = N j=1 s=1 M u js φ j (x)ψ s (ξ) (m + p)! m!p! Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 5 / 31

6 Galerkin System Find u R MN, such that [Powell & Elman, 2009] Au = f, where u = [u 11, u 21,..., u N1,..., u 1M, u 2M,..., u NM ] T, m A = G 0 K 0 + G i K i, f = g 0 f 0. i=1 Block structure of A (m = 4, p = 1, 2, 3) Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 6 / 31

7 Outline 1 Project Review 2 Multigrid Two-Grid Method Algorithm Validation 3 Low-Rank Approximation Motivation Algorithm Numerical Results 4 Conclusions Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 7 / 31

8 Two-Grid Method Two-gird correction scheme [Elman & Furnival, 2007] Choose initial guess u (0) for i = 0 until convergence for k steps u (i) u (i) + Q 1 (f Au (i) ) end r = R(f Au (i) ) solve Āē = r u (i+1) = u (i) + Pē for k steps u (i+1) u (i+1) + Q 1 (f Au (i+1) ) end end Multigrid: Apply the two-grid method recursively Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 8 / 31

9 Two-Grid Method Two grid spaces V hp = T p S h, V 2h,p = T p S 2h Prolongation and restriction operators Construction of Ā P = I P, R = I P T Ā = G 0 K 2h 0 + Damped Jacobi smoother m i=1 G i K 2h i Q = 1 ω D, D = diag(a) = I diag(k 0) Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 9 / 31

10 Algorithm Algorithm 1: Multigrid for stochastic Galerkin systems initialization: i = 0, r (0) = f, r 0 = f while r > tol r 0 & i maxit do /* solve residual equation */ e (i) = MgIter(A, 0, r (i), level) u (i+1) = u (i) + e (i) r (i+1) = f Au (i+1) r = r (i+1), i = i + 1 function u (1) = MgIter(A, u (0), f, level) if level == 2 then /* coarsest grid level */ u (1) = A\f else /* apply one MG iteration */ for k steps do u (0) u (0) + Q 1 (f Au (0) ) r = R(f Au (0) ) ē = MgIter(Ā, 0, r, level 1) u (1) = u (0) + Pē for k steps do u (1) u (1) + Q 1 (f Au (1) ) Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 10 / 31

11 Model Problem D = ( 1, 1) 2, f = 1. Covariance function KL expansion r(x, y) = σ 2 exp( 1 b x 1 y 1 1 b x 2 y 2 ) c(x, ω) = c 0 (x) + 3 m λi c i (x)ξ i (ω) i=1 Uniform distribution (assuming independence) ξ i U( 1, 1), ρ(ξ) = 1 2 m Take c 0 (x) = 1, σ = 0.3, b = 2, multigrid tol = 10 6 Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 11 / 31

12 Validation: Convergence Performance Independent of h m = 5, p = 3 m = 3, p = 5 h n h n Independent of m, p (h = 2 3 ) (m = 3) p n (p = 3) m n Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 12 / 31

13 Validation: Monte Carlo Monte Carlo method ξ i U( 1, 1), (K 0 + Compute mean and variance E[u MC ] = 1 q q r=1 Var[u MC ] = 1 q 1 u r MC m ξ i K i )u = f 0 i=1 q (umc r E[u MC ]) 2 r=1 Monte Carlo: h = 2 4, m = 3, q = 1, 000, 000 Multigrid: h = 2 4, m = 3, p = 9, tol = Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 13 / 31

14 Validation: Monte Carlo E[u FE ] E[u MC ] 2 E[u MC ] 2 = , Var[u FE ] Var[u MC ] 2 Var[u MC ] 2 = Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 14 / 31

15 Outline 1 Project Review 2 Multigrid Two-Grid Method Algorithm Validation 3 Low-Rank Approximation Motivation Algorithm Numerical Results 4 Conclusions Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 15 / 31

16 Motivation Galerkin system SFEM solution Au = (G 0 K 0 + m G i K i )u = f. i=1 u = [u 11,..., u N1,..., u 1M,..., u NM ] T Write into matrix form u 11 u 12 u 1M u 21 u 22 u 2M U = mat(u) = u N1 u N2 u NM A(U) = K 0 UG T 0 + m i=1 K i UG T i = F. Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 16 / 31

17 Motivation Decay of singular values for the solution matrix U (h = 2 6, m = 5, p = 3) U can be well approximated by a low-rank matrix! Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 17 / 31

18 Motivation Low-rank approximation U U k = V k Wk T, V k R N k, W k R M k, k N, M m (K 0 V k )(G 0 W k ) T + (K i V k )(G i W k ) T = F l Fr T. i=1 Memory and computational cost NM v.s. k(n + M) Low-rank matrix iterates are convenient to implement m A(VW T ) = (K 0 V )(G 0 W ) T + (K i V )(G i W ) T i=1 = [K 0 V, K 1 V,..., K m V ][G 0 W, G 1 W,..., G m W ] T Matrix rank grows rapidly: k (m + 1)k. The iterates should be truncated in every iteration. Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 18 / 31

19 Algorithm Algorithm 2: Multigrid with low-rank truncations initialization: i = 0, R (0) = F in low-rank format, r 0 = F while r > tol r 0 & i maxit do /* solve residual equation */ E (i) = MgIter(A, 0, R (i), level) U (i+1) = U (i) + E (i), U (i+1) = T abs (U (i+1) ) R (i+1) = F A(U (i+1) ), R (i+1) = T abs (R (i+1) ) r = R (i+1), i = i + 1 function U (1) = MgIter(A, U (0), F, level) if level == 2 then /* coarsest grid level */ solve A(U (1) ) = F directly else /* apply one MG iteration */ for k steps do U (0) U (0) + S (F A(U (0) )), U (0) = T rel (U (0) ) R = F A(U (0) ), R = Trel ( R) R = R R, Ē = MgIter(Ā, 0, R, level 1) U (1) = U (0) + PĒ for k steps do U (1) U (1) + S (F A(U (1) )), U (1) = T rel (U (1) ) Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 19 / 31

20 Low-rank truncation The truncation operator Ũ = T (U) [Kressner & Tobler, 2011] U = VW T, V R N k, W R M k Ũ = Ṽ W T, Ṽ R N k, W R M k Computation QR factorization: V = Q V R V, W = Q W R W where R V, R W R k k SVD: R V R T W = ˆV diag(σ 1,..., σ k )Ŵ T Truncation σ 2 k σk 2 ɛ rel σ σ2 k k = max{k σ k ɛ abs } Ṽ = Q V ˆV (:, 1 : k), W = Q W ˆV (:, 1 : k)diag(σ 1,..., σ k ) Cost: O((M + N + k)k 2 ) Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 20 / 31

21 Model Problem D = ( 1, 1) 2, f = 1. Covariance function KL expansion r(x, y) = σ 2 exp( 1 b x 1 y 1 1 b x 2 y 2 ) c(x, ω) = c 0 (x) + 3 m λi c k (x)ξ i (ω) i=1 Uniform distribution (assuming independence) ξ i U( 1, 1), ρ(ξ) = 1 2 m Take c 0 (x) = 1. m and b are chosen such that m λ i / λ i 95% i=1 i=1 Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 21 / 31

22 Implementation IFISS (Incompressible Flow and Iterative Solver Software, Silvester, et al) and SIFISS: Generate the Galerkin systems MATLAB R2015a 1.6 GHz Intel Core i5, 4 GB memory Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 22 / 31

23 Numerical Results Case 1: Standard deviation σ Decay of singular values of solution matrix U Relative residuals for low-rank approximations Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 23 / 31

24 Numerical Results Case 1: Standard deviation σ (N = 16129, M = 56) Truncation No truncation Rank 6 2 σ = Iterations Elapsed time Rel residual 3.0e-6 6.2e-4 1.2e-6 2.2e-4 Rank 13 6 σ = 0.01 Iterations Elapsed time Rel residual 1.1e-5 6.0e-4 1.6e-5 2.2e-4 Rank σ = 0.1 Iterations Elapsed time Rel residual 1.7e-5 1.1e-3 1.9e-5 2.4e-4 Rank σ = 0.3 Iterations Elapsed time Rel residual 1.8e-5 1.7e-3 1.4e-5 1.7e-3 Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 24 / 31

25 Numerical Results Case 2: Spatial dimension N (σ = 0.01, M = 56) Truncation No truncation Rank 13 6 nc = 7 Iterations N = Elapsed time Rel residual 1.1e-5 6.0e-4 1.6e-5 2.2e-4 Rank 16 6 nc = 8 Iterations N = Elapsed time Rel residual 1.3e-5 5.8e-4 1.8e-5 2.3e-4 Rank 11 5 nc = 9 Iterations N = Elapsed time Rel residual 4.6e-5 1.7e-3 1.9e-5 3.3e-3 Rank 11 2 nc = 10 Iterations N = Elapsed time Rel residual 4.5e-5 6.7e-3 1.9e-5 3.3e-3 Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 25 / 31

26 Numerical Results Case 3: Stochastic dimension M (σ = 0.01, N = 16129, p = 3) Truncation No truncation Rank 13 6 m = 5 Iterations M = 56 Elapsed time Rel residual 1.1e-5 6.0e-4 1.6e-5 2.2e-4 Rank 24 8 m = 7 Iterations M = 84 Elapsed time Rel residual 5.5e-6 4.6e-4 1.2e-6 2.2e-4 Rank m = 9 Iterations M = 220 Elapsed time Rel residual 4.0e-6 1.7e-4 1.2e-6 2.2e-4 Rank m = 11 Iterations M = 364 Elapsed time Rel residual 2.4e-6 7.5e-5 1.2e-6 1.6e-5 Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 26 / 31

27 Outline 1 Project Review 2 Multigrid Two-Grid Method Algorithm Validation 3 Low-Rank Approximation Motivation Algorithm Numerical Results 4 Conclusions Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 27 / 31

28 Conclusions Summary The multigrid solver works well for the stochastic Galerkin systems and can be more efficient than the Monte Carlo mehtod. The solution matrix can be better approximated by a low-rank matrix when the standard deviation in the random coefficient is small. Low-rank truncations reduce computing time for multigrid, especially when the spatial degree of freedom is large. Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 28 / 31

29 Future Work Low-rank truncation based on singular values Cost: O((N + M + k)k 2 ) QR factorization can be expensive (nc = 7, m = 5, p = 3) σ QR 51.2% 67.7% 73.2% SVD 3.4% 2.2% 2.2% Better truncation strategy: take advantage of grid hierarchy? Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 29 / 31

30 References 1 Ghanem, R. G. & Spanos, P. D. (2003). Stochastic Finite Elements: A Spectral Approach. New York: Dover Publications. 2 Elman, H. & Furnival D. (2007). Solving the stochastic steady-state diffusion problem using multigrid. IMA Journal of Numerical Analysis 27, Loève, M. (1960). Probability Theory. New York: Van Nostrand. 4 Xiu, D. & Karniadakis G. M. (2003). Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics, 187, Powell, C. & Elman H. (2009). Block-diagonal preconditioning for spectral stochastic finite-element systems. IMA Journal of Numerical Analysis 29, Kressner D. & Tobler C. (2011). Low-rank tensor Krylov subspace methods for parametrized linear systems. SIAM Journal of Matrix Analysis and Applications 32.4, Silvester, D., Elman, H. & Ramage, A. Incompressible Flow and Iterative Solver Software, Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 30 / 31

31 The End Thank you! Questions? Tengfei Su Solving the Stochastic Steady-State Diffusion Problem Using Multigrid 31 / 31