Foundations of the stochastic Galerkin method

Transcription

1 Foundations of the stochastic Galerkin method Claude Jeffrey Gittelson ETH Zurich, Seminar for Applied Mathematics Pro*oc Workshop 2009 in isentis

2 Stochastic diffusion equation R d Lipschitz, for ω Ω, (a(ω,x) u(ω,x)) = f(x) x, u(ω,x) = 0 x. Probability space (Ω,A,P). Parametric weak formulation: For ω Ω, find u(ω) H0 1 () s.t. a(ω,x) u(ω,x) v(x)dx = f(x)v(x)dx for all v H 1 0 (). C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 1

3 Outline stochastic problem on transform parameter domain weak formulation stochastic basis finite element approximation Ω H0 1() R N H0 1() L 2 ρ (RN ;H0 1()) l 2 (Λ;H0 1()) V j C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 2

4 Transformation to a deterministic problem Random variable Y = (Y m ) m N : Ω R N, a(ω,x) = a Y (Y (ω),x) ω Ω. Assume (Y m ) m N independent. ρ := Y (P) = ρ m, m=1 ρ m := Y m (P) Then u(ω) = u Y (Y (ω)) for all ω Ω, where u Y (y) H0 1() solves a Y (y,x) u Y (y,x) v(x)dx = f(x)v(x)dx for all v H 1 0 () and all y RN. C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 3

5 Example: Karhunen-Loève expansion efine a(ω,x) = a 0 (x) + λm ϕ m (x)y m (ω). m=1 (ϕ m ) m N are eigenfunctions of the covariance operator of a(, ) with eigenvalues λ m. (Y m ) m N are uncorrelated random variables. a Y (y,x) := a 0 (x) + Then a(ω,x) = a Y (Y (ω),x). λm ϕ m (x)y m. m=1 C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 4

6 Example: Karhunen-Loève expansion Alternative: Karhunen-Loève expansion of log(a(ω, x)), ( ) a(ω,x) = a 0 (x)exp λm ϕ m (x)y m (ω) m=1 m=1 ) = a 0 (x) exp( λm ϕ m (x)y m (ω). efine ( ) a Y (y,x) := a 0 (x)exp λm ϕ m (x)y m m=1. Then a(ω,x) = a Y (Y (ω),x). C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 5

7 Weak formulation For all y R N,Find u Y L 2 ρ(r N ;H0 1 ()) such that a Y (y) u Y (y) v(y)dxdρ(y) = f v(y)dxdρ(y) R N for v H 1 0 ().for all v L2 ρ (RN ;H 1 0 ()). Product structure: L 2 ρ(r N ;H 1 0 ()) = L 2 ρ(r N ) H 1 0 (). deterministic space H 1 0 (), stochastic space L 2 ρ(r N ). R N C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 6

8 Weak formulation Assume f H 1 () and a Y (, ) is uniformly bounded, 0 < a a Y (y,x) a < x, y R N. Theorem u Y is the unique element of L 2 ρ (RN ;H0 1 ()) such that a Y u Y v dxdρ = fv dxdρ R N for all v L 2 ρ(r N ;H0 1 ()). Furthermore, R N u Y L ρ (R N ;H 1 0 ()) 1 a f H 1 (). C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 7

9 Stochastic semi-discretization Let (ϕ µ ) µ Λ be an orthonormal basis of L 2 ρ(r N ) and u Y (y,x) = u µ (x)ϕ µ (y), (u µ ) µ Λ l 2( Λ;H0() 1 ). µ Λ efine a νµ (x) := a Y (y,x)ϕ µ (y)ϕ ν (y)dρ(y), µ,ν Λ, R N and f ν (x) := f(x) ϕ ν (y)dρ(y), ν Λ. R N Then using the test function v(x)ϕ ν (y) for v H0 1 (), ν Λ, a νµ (x) u µ (x) v(x)dx = f ν (x)v(x)dx. µ Λ System of deterministic equations for the coefficients of u Y. C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 8

10 Stochastic semi-discretization By Parseval s identity, u Y L 2 ρ (R N ;H 1 0 ()) = (u µ) µ Λ l 2 (Λ;H 1 0 ()). Theorem u Y L 2 ρ (RN ;H0 1 ()) satisfies a Y u Y vdxdρ = R N R N fvdxdρ v L 2 ρ (R N ;H 1 0 () ) if and only if (u µ ) µ Λ l 2 (Λ;H0 1 ()) satisfies a νµ u µ v dx = f ν v dx v H0 1 (), ν Λ. µ Λ C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 9

11 Tensor-product stochastic basis Construct orthonormal basis (ϕ µ ) µ Λ of L 2 ρ(r N ) with elements of the form ϕ µ (y) = ϕ m 1 ν 1 (y m1 )... ϕ m k ν k (y mk ). (ϕ m ν ) ν Λ m is an orthonormal basis of L 2 ρ m (R), such as polynomials piecewise polynomials wavelets C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 10

12 Example: orthonormal polynomials (P m n ) n N0 orthonormal polynomial basis of L 2 ρ m (R), P m 0 = 1, deg P m n = n Λ := l 1 (N; N 0 ) P µ := Pµ m m, m=1 ( P µ (y) = m supp m=1 µ µ Λ ) Pµ m m (y m ) 3 C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 11

13 Example: Haar wavelets C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 12

14 General tensor-product construction Let (ϕ m ν ) ν Λm be an orthonormal basis of L 2 ρ m (R) with ϕ m 0 = 1. Then (ϕ m ν ) ν m, m := Λ m \{0} is an orthonormal basis of L 2 ρ m (R)/R. For I F(N) a finite subset of N, ϕ µ := m I ϕ m µ m, µ I := m I m. efine W I := span{ϕ µ ; µ I } L 2 ρ (RN ). Then m I L 2 ρ m (R) = J I W J = #I k=0 J I #J=k W J. C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 13

15 General tensor-product construction Theorem (ϕ µ ) µ Λ, Λ := I F(N) I, is an orthonormal basis of L 2 ρ (RN ). Proof. (ϕ µ ) µ I is an orthonormal basis of W I for all I F(N) and L 2 ρ (R N) = I F(N) W I = k=0 I N #I=k W I. C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 14

16 Finite element approximation efine nested finite element spaces {0} = V 0 V 1... V j V j+1... H 1 0 (). For a sequence j = (j(µ)) µ Λ l 1 (Λ; N 0 ), define V j := v = ( v µ ϕ µ v = (v µ ) µ Λ ; v µ V j(µ) µ Λ L2 ρ R N ;H µ Λ The Galerkin projection ũ V j is determined by a νµ (x) ũ µ (x) ṽ(x)dx = f ν (x)ṽ(x)dx µ Λ for all ṽ V j(ν), ν Λ. C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 15

17 Finite element approximation Theorem There is a unique ũ V j such that a νµ ũ µ ṽ dx = f ν ṽ dx ṽ V j(ν), ν Λ. µ Λ It satisfies the bound and is quasi-optimal, ũ L 2 ρ (R N ;H 1 0 ()) 1 a f H 1 () a ũ u Y L 2 ρ (R N ;H0 1()) a inf w u Y L 2 ew V ρ (R N ;H0 1()). j C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 16

18 Summary We constructed tensor-product orthonormal bases of L 2 ρ(r N ), and used these to recast a stochastic boundary value problem as an infinite system of deterministic equations, which can be solved by Galerkin approximation using standard finite element spaces. Thank you for your attention! C.J. Gittelson Pro*oc Workshop 2009 in isentis p. 17