Finite element approximation of the stochastic heat equation with additive noise

Transcription

1 p. 1/32 Finite element approximation of the stochastic heat equation with additive noise Stig Larsson

2 p. 2/32 Outline Stochastic heat equation with additive noise du u dt = dw, x D, t > u =, x D, t > u() = u Abstract framework Mild solution Finite element approximation Strong convergence Weak convergence

3 p. 3/32 Co-workers Yubin Yan Mihály Kovács Fredrik Lindgren Matthias Geissert (Darmstadt)

4 p. 4/32 Stochastic heat equation The stochastic heat equation is u(x,t) t u(x, t) = f(u(x, t)) + g(u(x, t))ẇ(x, t), x Ω Rd, t > u(x, t) =, x Ω, t > u(x, ) = u (x), x Ω I know four approaches to SPDE: Martingale measure approach: Walsh 1986 Variational approach: Pardoux 1972, Krylov and Rozowski 1979 Semigroup approach: Da Prato and Zabczyk 1992 Wick product approach: Øksendal 1996 We use the semigroup approach. It is one way of giving a rigorous meaning to the equation.

5 p. 5/32 Semigroup approach First we change to stochastic notation: Ω D R d, x ξ D R d, u(x, t) X t (ξ), so that the stochastic heat equation can be written in the form Define dx t X t dt = f(x t ) dt + g(x t ) dw t H = L 2 (D), A =, D(A) = H 2 (D) H 1 (D) so that we can write { dx t + AX t dt = f(x t ) dt + g(x t ) dw t X = x

6 p. 6/32 Semigroup approach Let {e ta } t be the analytic semigroup generated by A, that is, u(t) = e ta u = e tλ j (u, φ j )φ j j=1 is the solution of the homogeneous heat equation { u + Au =, t > u() = u The solution of the non-homogeneous equation { u + Au = f, t > u() = u is then given by u(t) = e ta u + e (t s)a f(s) dt

7 p. 7/32 Analytic semigroup e ta v = e tλ j (v, φ j )φ j j=1 can be extended as a holomorphic function of t smoothing property: A β e ta v Ct β v, β A 1/2 e sa v 2 ds C v 2 Theory of semigroups of bounded linear operators

8 p. 8/32 Semigroup approach The stochastic heat equation: { dx t + AX t dt = f(x t ) dt + g(x t ) dw t X = x It is given a rigorous meaning as the integral equation: X t = e ta x + e (t s)a f(x s ) ds + e (t s)a g(x s ) dw s Such a solution is called a mild solution. Other approaches: different formulations and solution concepts. We need to define the stochastic integral.

9 p. 9/32 Semigroup approach Linear equation with additive noise : { dx + AX dt = dw, t > X() = X H = L 2 (D),, (, ), D R d, bounded domain A =, D(A) = H 2 (D) H(D) 1 X(t), H-valued stochastic process on probability space (Ω, F, P) W(t), H-valued Wiener process with filtration F t E(t) = e ta, analytic semigroup generated by A Mild solution (stochastic convolution): X(t) = E(t)X + E(t s) dw(s), t

10 p. 1/32 Q-Wiener process covariance operator Q : H H, self-adjoint, positive definite, bounded, linear operator Qe j = γ j e j, γ j >, {e j } j=1 ON basis β j (t), independent identically distributed, real-valued, Brownian motions W(t) = γ 1/2 j β j (t)e j j=1 Two important cases: Tr(Q) <. W(t) converges in L 2 (Ω, H): E γ 1/2 2 j β j (t)e j = γ j E ( β j (t) 2) = t j=1 j=1 γ j = t Tr(Q) < Q = I, white noise. W(t) is not H-valued, since Tr(I) =, but converges in a weaker sense j=1

11 p. 11/32 Q-Wiener process If Tr(Q) < : W() = continuous paths independent increments Gaussian law P (W(t) W(s)) 1 = N(, (t s)q), s t W(t) generates a filtration F t so that it becomes a square integrable H-valued martingale. We can integrate with respect to W.

12 p. 12/32 Stochastic integral X(t) = E(t)X + E(t s) dw(s), t We can define the stochastic integral E B(s)Q 1/2 2 HS ds < Isometry property E B(s) dw(s) 2 = E B(s)Q 1/2 2 HS ds B(s) dw(s) if Hilbert-Schmidt operator B: B 2 HS = Bϕ l 2 <, {ϕ l } arbitrary orthonormal basis in H l=1 Da Prato and Zabczyk, Stochastic Equations in Infinite Dimensions

13 p. 13/32 Regularity v β = A β/2 v, Ḣ β = D(A β/2 ), β R for example: Ḣ 2 = H 2 (D) H(D), 1 Ḣ 1 = H(D) 1 v 2 L 2 (Ω,Ḣβ ) = E( v 2 β) = A β/2 v 2 dξ dp(ω), β R Ω Theorem. If A (β 1)/2 Q 1/2 HS < for some β, then ( ) X(t) L2 (Ω,Ḣ β ) C X L2 (Ω,Ḣ β ) + A(β 1)/2 Q 1/2 HS D Two cases: If Q 1/2 2 HS = Tr(Q) <, then β = 1 If Q = I, d = 1, A = 2 ξ 2, then A (β 1)/2 HS < for β < 1/2 A (β 1)/2 2 HS = j λ (1 β) j j j (1 β)2/d < iff d = 1, β < 1/2

14 p. 14/32 Proof with X = X(t) 2 L 2 (Ω,Ḣ β ) = E ( = = = C A β/2 E(t s)dw(s) 2 ) A β/2 E(s)Q 1/2 2 HS ds A 1/2 E(s)A (β 1)/2 Q 1/2 2 HS ds k=1 A 1/2 E(s)A (β 1)/2 Q 1/2 φ k 2 ds A (β 1)/2 Q 1/2 φ k 2 k=1 = C A (β 1)/2 Q 1/2 2 HS A 1/2 E(s)v 2 ds C v 2

15 p. 15/32 Regularity of W Theorem. If A (β 1)/2 Q 1/2 HS < for some β, then W(t) L 2 (Ω, Ḣ β 1) ) L 2 (Ω, Ḣ 1 ) Proof. W(t) = dw(s) ( W(t) 2 L 2 (Ω,Ḣ β 1 ) = E A (β 1)/2 dw(s) 2 ) = A (β 1)/2 Q 1/2 2 HS ds = t A (β 1)/2 Q 1/2 2 HS < If Q 1/2 2 HS = Tr(Q) <, then β = 1 and W(t) L 2(Ω, H) If Q = I, d = 1, A = 2 x 2, then A (β 1)/2 HS < for β < 1/2

16 p. 16/32 The finite element method Deterministic heat equation, with u t = u t : { u t u = f, x D, t > u =, x D, t > (u, v) = D uv dξ (u t, v) ( u, v) }{{} =( u, v) = (f, v), v H 1 (D) = Ḣ 1 weak form: { u(t) H 1 (D), u() = u (u t, v) + ( u, v) = (f, v), v H 1 (D)

17 p. 17/32 The finite element method weak form: { u(t) H 1 (D), u() = u (u t, v) + ( u, v) = (f, v), v H 1 (D) triangulations {T h } <h<1, mesh size h function spaces {S h } <h<1, S h H 1 (D) = Ḣ1 { u h (t) S h, (u h (), v) = (u, v), v S h (u h,t, v) + ( u h, v) = (f, v), v S h (u h,t, v) + ( u h, v) }{{} =(A h u h,v) { u h,t + A h u h = P h f, t > u h () = P h u = (f, v), v S h }{{} =(P h f,v) continuous piecewise linear functions the same abstract framework

18 p. 18/32 The finite element method triangulations {T h } <h<1, mesh size h finite element spaces {S h } <h<1 S h H(D) 1 = Ḣ1 S h continuous piecewise linear functions A h : S h S h, discrete Laplacian, (A h ψ, χ) = ( ψ, χ), χ S h P h : L 2 S h, orthogonal projection, (P h f, χ) = (f, χ), χ S h { Xh (t) S h, X h () = P h X dx h + A h X h dt = P h dw P h W(t) is Q h -Wiener process with Q h = P h QP h. Mild solution, with E h (t) = e ta h, X h (t) = E h (t)p h X + E h (t s)p h dw(s)

19 p. 19/32 Error estimates for the deterministic problem { ut + Au =, t > u() = v u(t) = E(t)v { uh,t + A h u h =, t > u h () = P h v u h (t) = E h (t)p h v Denote F h (t)v = E h (t)p h v E(t)v, v β = A β/2 v We have, for β 2, F h (t)v Ch β v β, t ( 1/2 F h (s)v ds) 2 Ch β v β 1, t V. Thomée, Galerkin Finite Element Methods for Parabolic Problems

20 p. 2/32 Strong convergence in L 2 norm Theorem. If A (β 1)/2 Q 1/2 HS < for some β [, 2], then X h (t) X(t) L2 (Ω,H) Ch β( X L2 (Ω,Ḣ β ) + A(β 1)/2 Q 1/2 HS ) Recall: X h (t) X(t) L2 (Ω,H) = ( ) 1/2 E( X h (t) X(t) 2 ) Two cases: If Q 1/2 2 HS = Tr(Q) <, then the convergence rate is O(h). If Q = I, d = 1, A = 2 ξ 2, then the rate is almost O(h 1/2 ). No result for Q = I, d 2.

21 p. 21/32 Strong convergence: proof X(t) = E(t)X + X h (t) = E h (t)p h X + F h (t) = E h (t)p h E(t) E(t s) dw(s) E h (t s) P h dw(s) X h (t) X(t) = F h (t)x + F h (t s) dw(s) = e 1 (t) + e 2 (t) F h (t)x Ch β X β (deterministic error estimate) = e 1 (t) L2 (Ω,H) Ch β X L2 (Ω,Ḣ β )

22 p. 22/32 Strong convergence: proof E B(s) dw(s) 2 t = E B(s)Q 1/2 2 HS ds (isometry) ( 1/2 F h (s)v ds) 2 Ch β v β 1, with v = Q 1/2 ϕ l (deterministic) = e 2 (t) 2 L 2 (Ω,H) = E F h (t s) dw(s) 2 = = l=1 F h (t s)q 1/2 ϕ l 2 ds C F h (t s)q 1/2 2 HS ds h 2β Q 1/2 ϕ l 2 β 1 l=1 = Ch 2β A (β 1)/2 Q 1/2 ϕ l 2 = Ch 2β A (β 1)/2 Q 1/2 2 HS l=1 If Tr(Q) <, we may choose β = 1, otherwise β < 1.

23 p. 23/32 Time discretization { dx + AX dt = dw, t > X() = X The implicit Euler method: k = t, t n = nk, W n = W(t n ) W(t n 1 ) { X n h S h, X h = P h X X n h X n 1 h + ka h X n h = P h W n, X n h = E kh X n 1 h + E kh P h W n, E kh = (I + ka h ) 1 X n h = E n khp h X + n j=1 E n j+1 kh P h W j X(t n ) = E(t n )X + n E(t n s) dw(s)

24 p. 24/32 Error estimates for deterministic parabolic problem Denote F n = E n kh P h E(t n ) We have the following estimates for β 2: F n v C(k β/2 + h β ) v β ( n k F j v 2) 1/2 C(k β/2 + h β ) v β 1 j=1

25 p. 25/32 Strong convergence Theorem. If A (β 1)/2 Q 1/2 HS < for some β [, 2], then, with e n = Xh n X(t n), ) e n L2 (Ω,H) C(k β/2 + h β ) ( X L2(Ω,Ḣ β) + A(β 1)/2 Q 1/2 HS J. Printems (21) (only time-discretization) Y. Yan, BIT (24), SIAM J. Numer. Anal (25)

26 p. 26/32 Weak convergence If A (β 1)/2 Q 1/2 HS < for some β [, 2] and g C 2 b (H,R), then ( ) E g(xh) n g(x(t n )) C(k β + h 2β ) twice the rate of strong convergence Debussche and Printems 27 Geissert, Kovács, and Larsson 27 (with somewhat stronger assumption on g and only spatial discretization) the proof is based on Itô s formula and the Kolmogorov equation in infinite dimension

27 p. 27/32 Weak convergence { } u(t, x) = E g(z(t, t, x)) { ( = E g E(T t)x + T t )} E(T s) dw(s) Kolmogorov equation: u t = u/ t, u x = u/ x, u xx = 2 u/ x ) 2 u t (t, x) (Ax, u x (t, x)) (u Tr xx (t, x)q =, x D(A), t < T, u(t, x) = g(x) ( ) E u(t, X h (T)) = Eg(X h (T)), assume X() = X h () ( ) ( ) ( ) ( ) E u(, X h ()) = E u(, X()) = E g(z(t,, X )) = E g(x(t)) ( Eg(X h (T)) Eg(X(T)) = E u(t, X h (T)) {Itô formula} = E {Kolmogorov eq} = E T T ( ) u(, X h ()) ( ) u t (A h X h, u x ) Tr(u xxq h ) dt ( ) ((A A h )X h, u x ) Tr(u xx(q h Q)) dt

28 p. 28/32 Weak convergence { } u(t, x) = E g(z(t, t, x)) { ( = E g E(T t)x + ( ) u x (t, x) = E E(T t)g (Z(T, t, x)), ( ) u xx (t, x) = E E(T t)g (Z(T, t, x))e(t t) T t )} E(T s) dw(s) Eg(X h (T)) Eg(X(T)) = E T ( ((A Ah )X h (t), u x (t, X h (t)) ) Tr( u xx (t, X h (t))(q h Q) )) dt finite element and semigroup techniques Malliavin calculus??

29 p. 29/32 Ongoing and future work wavelet expansion of noise nonlinear equations dx + AX dt = f(x) dt + g(x) dw adaptive methods based on a posteriori error estimates Cahn-Hilliard-Cook equation dx + A 2 X dt = Af(X) dt + dw Malliavin calculus stochastic wave equation (the semigroup is not analytic) more general noise: H-valued Lévy process

30 p. 3/32 Non-future work We do not study PDE with (smooth) random coefficients u t (a(x, ω) u) = f(x, t, ω)

31 p. 31/32 Implementation Euler s method { U n S h, U = P h u U n U n 1 + ta h U n = P h W n (U n U n 1, χ) + t( U n, χ) = ( W n, χ), χ S h U n (x) = N h k=1 N h k=1 U n k (ϕ k, ϕ j ) + t U n k ϕ k (x), χ = ϕ j, finite element basis functions N h k=1 MU n + tku n = MU n 1 + b n U n k ( ϕ k, ϕ j ) = N h k=1 U n 1 k (ϕ k, ϕ j ) + ( W n, ϕ j )

32 p. 32/32 Implementation How to simulate b n j = ( W n, ϕ j ) = (W(t n ) W(t n 1 ), ϕ j )? Covariance of b n : E(b n i b n j ) = E ( ( W n, ϕ i )( W n, ϕ j ) ) = t(qϕ i, ϕ j ) In other words: E(b n b n ) = tq, Q ij = (Qϕ i, ϕ j ) Cholesky factorization: Q = LL T Take b n = tlβ n, where β n, n = 1, 2,..., are N(, 1) Then E(b n b n ) = E(b n (b n ) T ) = te(lβ n (Lβ n ) T ) = tle(β n (β n ) T )L T = tll T = tq