Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs

Transcription

1 Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs Roman Andreev ETH ZÜRICH / 29 JAN 29

2 TOC of the Talk Motivation & Set-Up Model Problem Stochastic Galerkin FEM Conclusions & Outlook

3 Motivation & Set-Up Goal Goal: model and solve a diffusion-reaction system with coefficients a and c a source f an unknown solution u (a u) + cu = f being uncertain, that is probabilistic quantities, as in, e.g., groundwater flow computations

4 Motivation & Set-Up Notation D open bounded domain in R d (Ω, F, P) probability space a : Ω D R random diffusion coefficient c : Ω D R random reaction coefficient f : D R source

5 Motivation & Set-Up Assumptions Assume D has Lipschitz boundary a and c are measurable positivity boundedness ess inf a > and ess inf c Ω D Ω D ess sup Ω D a < and ess sup c < Ω D deterministic source of finite second moment f L 2 (D)

6 Model Problem Strong Formulation Parametric problems Dirichlet: find u : Ω D R s.t. (a u) + cu = f in Ω D and u = on Ω D parabolic: find u : Ω D [, T] R s.t. u t (a u) + cu = f in Ω D (, T) with initial and boundary conditions hyperbolic: find u : Ω D [, T] R s.t. u tt (a u) + cu = in Ω D (, T) with initial and boundary conditions

7 Model Problem Hyperbolic Example: Vibrating String I Elasticity tensor E(u(t)), t = (E ± 1.96 Var)(u(t)), t = x

8 Model Problem Hyperbolic Example: Vibrating String II Elasticity tensor E(u(t)), t = 1.5 (E ± 1.96 Var)(u(t)), t = x

9 Model Problem Focus: Weak Formulation for the Stochastic Dirichlet Problem Variational problem Define the Hilbert space H 1 := {v : Ω H1 (D) s.t. v H 1 < } where v 2 H 1 ] := E [ v 2H 1(D) = v(ω, x) 2 dxdp(ω) Ω D Well-posed weak formulation: find u H 1 s.t. v H1 [ ] [ ] [ ] E a u, v L 2 (D) + E cu, v L 2 (D) = E f, v L 2 (D)

10 Model Problem Solution Strategies Note that H 1 = L2 (Ω, dp; H 1 (D)) = L 2 (Ω, dp) H 1 (D) L 2 (Ω, dp; H 1 (D)) suggests non-intrusive solvers such as Monte Carlo, collocation L 2 (Ω, dp) H 1 (D) suggests intrusive solvers such as stochastic Galerkin FEM

11 Stochastic Galerkin FEM Meta [1] M. Bieri, Ch. Schwab, Sparse high order FEM for elliptic spdes, CMAME, in press [2] R. Andreev, Sparse Wavelet-Galerkin Methods for Stochastic Diffusion Problems, BSc thesis [3] M. Bieri, R. Andreev, Ch. Schwab, Sparse Tensor Discretization of Elliptic spdes, subm. [4] This work: spatial dimension one, only reaction term piecewise quadratic spatial FEM (simple) parallelization strategy non-stationary problems

12 Stochastic Galerkin FEM Overview Discretization of L 2 (Ω, dp) and H 1 (D) for H 1 (D) we use a wavelet basis {ψ i i = (j,l) I} for L 2 (Ω, dp) we construct an orthonormal polynomial basis {P α α Λ}

13 Stochastic Galerkin FEM Prewavelet Basis on Interval l = l = l =

14 Stochastic Galerkin FEM Biorthogonal pw Quadratic Wavelet Basis on Interval l = l = l =

15 Stochastic Galerkin FEM Karhunen-Loève Expansion I We assume that c, Ω = R R... and a(ω, x) = a(ξ, x) = ā(x) + m Nξ m λm ϕ m (x) where ξ is a random vector ( noise ) with distribution ρ : Ω R {ϕ m m N} is an orthonormal basis for L 2 (D) λ 1 λ 2... with λ m m Such expansions (e.g., the Karhunen-Loève expansion) exist for a large class of industrially relevant random processes

16 Stochastic Galerkin FEM Karhunen-Loève Expansion II Assume that {ξ m m N} are independent random variables, i.e., ρ = ρ 1 ρ 2... Assume that supp ρ is compact, for simplicity ρ m 1 2 χ [ 1,1] As basis for L 2 (Ω, dp) = Lρ 2 (Ω) we take ρ-orthonormal polynomials P α := L α1 L α2..., α = (α 1,α 2,...) Λ := N N where L k is the ρ k -orthonormal (Legendre) polynomial of degree k

17 Stochastic Galerkin FEM Overview II Discretization of L 2 (Ω, dp) and H 1 (D) H 1 (D) = clos span{ψ i i I} L 2 (Ω, dp) = L 2 ρ (Ω) = clos span{p α α Λ}

18 Stochastic Galerkin FEM Hierarchy Hierarchy of degrees of freedom For I l := {i I i 2 = l}, l N we have #I l 2 dl Partition Λ = Λ Λ 1... such that #Λ l 2 γl with a steering parameter γ > 1

19 Stochastic Galerkin FEM Full Tensor Product Subspace: Definition Finite dimensional variational problem Define full tensor product subspaces of H 1 = L 2 (Ω, dp) H 1 (D) as V L := span{p α ψ i (α, i) Λ lω I ld } l Ω, l D L Finite dimensional problem: find u L V L s.t. v V L [ ] [ ] E a u L, v L 2 (D) = E f, v L 2 (D)

20 Stochastic Galerkin FEM Sparse Tensor Product Subspace: Definition Finite dimensional variational problem Define sparse tensor product subspaces of H 1 = L 2 (Ω, dp) H 1 (D) as V L := span{p α ψ i (α, i) Λ lω I ld } l Ω +l D L Finite dimensional problem: find û L V L s.t. v V L [ ] [ ] E a û L, v L 2 (D) = E f, v L 2 (D)

21

22 Stochastic Galerkin FEM Convergence Rates Proposition (see [3]): assuming finite elements of order at least p in physical space < r < s 3 2 λ m ϕ m L (D) m s that u A r ((H p H 1 )(D)), i.e., uniformly in N N inf Λ Λ # Λ=N α/ Λ E [P α u] 2 (H p H 1 )(D) N 2r γ = p/r and optimal (best-n-term) choice of Λ l we obtain u u L H 1 C(u,β)(dimV L ) β, u û L H 1 C(u, ˆβ)L 1+ˆβ(dim V L ) ˆβ, β = (1/r + d/p) 1 ˆβ = min{r, p/d}

23 Stochastic Galerkin FEM Hierarchy: Heuristics I Recall (a u) = f with a = ā + m N ξ m λm ϕ m Therefore, in 1d, u = F ā + m N ξ m λm ϕ m and (see [1]) u L (Ω D) α Λ c α m N µ αm m }{{} µ α with µ m = λm ϕ m L (D) ess inf D ā

24 Stochastic Galerkin FEM Hierarchy: Heuristics II Lemma (see [3]): for < t < s 1 α Λ there exists C(t, d) such that for δ 1,δ 2 > and η m := R m Rm, 2 < R m < ess inf D ā ess inf Ω D a m (1+δ 1 ) λm ϕ m L (D) ζ(1+δ 1 ) µ αm m := η (1 δ 2)α m m with a constant independent of α we have E [P α u] H 1 (D) C(t, d,#suppα) m suppα µ αm m } {{ } µ α

25 Stochastic Galerkin FEM Hierarchy Hierarchy of degrees of freedom For I l := {i I i 2 = l}, l N we have #I l 2 dl Choose a decreasing threshold sequence ǫ l(n ) s.t. Λ l := {α Λ ǫ l > µ α ǫ l+1 }, l N satisfies #Λ l 2 γl with the steering parameter γ > 1

26 Numerical Examples Example I: Convergence Example I set-up diffusion coefficient on D = ( 1, 1) given by a (s) alg (ξ, x) = 1 + ξ m m N λ (s) m sin(mπx) with algebraic Karhunen-Loève eigenvalue decay λ (s) m = 1 1 ζ(s) (m ) s for s > 1 right hand side f(x) = 1 + x, x D heuristics I for computing {Λ l } l N pw quadratic spatial FEM

27 Numerical Examples Example I: Convergence w.r.t. #I l H 1 error First order FEM N 1 Second order FEM N N = #(I... I L )

28 Numerical Examples Example I: Convergence w.r.t. #Λ l H 1 error Algebraic decay, s = 2 Algebraic decay, s = 3 Algebraic decay, s = 4 Exponential decay #(A... A L )

29 Numerical Examples Example I: Convergence w.r.t. #Λ l - EOC Interpolated EOC 1.5 Exponential decay Algebraic decay, s = 4 Algebraic decay, s = 3 Algebraic decay, s = #(A... A L )

30 Numerical Examples Example I: Convergence of Tensor Product Approximations (s = 4) 1 2 N = dimv L N = dim V L H 1 error N

31 Numerical Examples Example II: Reaction Term Example II set-up diffusion coefficient a(ξ, x) = 1 + m N ξ a m λm sin(mπx), λm = 2 m reaction coefficient c(ξ, x) = 1 + m N ξ c m λm sin(mπx) ξ = (ξ a m,ξ c m) m N vector of i.i.d. r.v. s with p.d.f. 1 2 χ [ 1,1] rest as in Example I

32 Numerical Examples Example II: Degrees of Freedom l Total #Λ a l #Λ c l #I l Table: Number of degrees of freedom per level for the reference solution of Example II. FTP: 2 1 9, STP: d.o.f. s.

33 Numerical Examples Example II: Convergence H 1 error dim V L

34 Conclusions & Outlook Conclusions data-adaptive hierarchy of stochastic d.o.f. s possible sparse tensorization improves the convergence rate Outlook L-shaped domain efficient parallelization adaptive methods for stochastic PDEs

35 Conclusions & Outlook Conclusions data-adaptive hierarchy of stochastic d.o.f. s possible sparse tensorization improves the convergence rate Outlook L-shaped domain efficient parallelization adaptive methods for stochastic PDEs

36 Thank You!