Numerical discretisations of stochastic wave equations

Transcription

1 Numerical discretisations of stochastic wave equations David Cohen Matematik och matematisk statistik \ UMIT Research Lab, Umeå universitet Institut für Mathematik, Universität Innsbruck Joint works with R. Anton, S. Larsson, L. Quer-Sardanyons, M. Sigg, X. Wang Swedish Research Council (VR) project nr

2 Outline I. Crash course on SPDEs II. The stochastic wave equation III. Numerical discretisations IV. Ongoing and future works

3 Thanks to I. Crash course on SPDEs

4 Motivation We can see a stochastic wave equation ((x, t) [, 1] [, T]) u tt (x, t) u xx (x, t) = RANDOM PERTURBATION as an infinite system of SDE (pseudo-spectral method) dẋ j (t) + ωj 2 X j (t) dt = dβ j (t), j Z, where β j (t) are standard Brownian motions for t [, T]: β j () = a.s. For s < t T we have β j (t) β j (s) N(, t s) = t sn(, 1). For s t v w T the increments β j (t) β j (s) and β j (w) β j (v) are independent.

5 SPDEs: Notations and definitions Mainly TWO approaches to define SPDEs: Functional setting (SDE in Hilbert space) and random-field approach. Let D R d, d = 1, 2, 3, be a nice domain. Let us first consider the linear stochastic wave equation with additive noise in the Hilbert space U := L 2 (D): d u u dt = dw in D (, T), u = in D (, T), u(, ) = u, u(, ) = v in D. Here u = u(x, t) is a U-valued stochastic process, that is u : [, T] Ω U = L 2 (D), u(t) := u(t, ω) : D R, where (Ω, F, P) is our probability space. We will now define a Fourier series for the infinite dimensional Wiener process W(t).

6 Definition of the noise Let Q L(U) be a bounded, linear, symmetric, non-negative operator The operator Q has eigenpairs {(γ j, e j )} j=1 with orthonormal basis {e j } j=1 of U. Theorem. The Wiener process with covariance operator Q is given by W(t) = j=1 γ 1/2 j e j β j (t), where β j (t) are i.i.d. standard Brownian motion. The eig. values γ j > of the operator Q determine the spatial correlation of the noise.

7 Covariance operator Recall definition of noise: W(t) = j=1 Consider two types of covariance operator: Cylindrical Wiener process, e.g. Q = I: γ 1/2 j e j β j (t). W(t) = e j β j (t). j=1 Noise is white in space and time. Operator Q is trace-class if Tr(Q) = γ j <. j=1 This gives noise with some spatial correlation.

8 Stochastic integrals With this definition of the noise, we (Da Prato, Zabczyk, f.ex.) can define the stochastic Itô integral together with Itô s isometry [ t E t Φ(s) dw(s) Φ(s) dw(s) 2 U ] = t Φ(s)Q 1/2 2 HS ds, where we recall that U = L 2 (D) and the Hilbert-Schmidt norm on ( 1/2 compact operators T HS := Tr(TT ) = Tφ j U) 2 with {φj } j=1 an ON basis in U. j=1

9 Last slide of the crash course... Since we will deal with mean-square error bounds, the following norm will be useful v L2 (Ω,U) := E[ v 2 U] 1/2, where we recall that U = L 2 (D) and that E is the mathematical expectation on our probability space (Ω, F, P).

10 No... this was not the last slide :-) The second approach is based on another definition of the noise. Problem (1d for simplicity): 2 u t (t, x) = 2 u (t, x) + Ẇ(t, x). 2 x2 Here, W(t, x) is a Brownian sheet (multi-parameter version of Brownian motion). That is, the noise term Ẇ(t, x) is a mean zero Gaussian noise with spatial correlation k(, ), i.e. E[Ẇ(t, x)ẇ(s, y)] = δ(t s)k(x, y), where δ is a Dirac delta function at the origin.

11 Thanks to A. Grandchamp II. The stochastic wave equation

12 Motivation: motion of DNA molecule in a liquid Motion of a strand of DNA floating in a liquid (Gonzalez, Maddocks 21, Dalang 29): DNA molecule string system of 3 wave equations in R 3. Liquid particles hit DNA stochastic motion.

13 The stochastic wave equation Consider the stochastic wave equation d u u dt = f (u) dt + g(u) dw in D (, T), u = in D (, T), u(, ) = u, u(, ) = v in D, where u = u(x, t), D R d, d = 1, 2, 3, is a bounded convex domain with polygonal boundary D. The stochastic process {W(t)} t is an L 2 (D)-valued (possibly cylindrical) Q-Wiener process. We set Λ = with D( ) = H 2 (D) H 1 (D).

14 Abstract formulation of the problem Recall the problem: d u u dt = f (u) dt + g(u) dw and Λ =. Define the velocity of the solution u 2 := u 1 := u and rewrite the above problem as where X := and X := dx(t) = AX(t) dt + F(X(t)) dt + G(X(t)) dw(t), t >, X() = X, [ u1 [ u 2 u v ]. ], A := [ ] [ ] [ ] I, F(X) :=, G(X) :=, Λ f (u 1 ) g(u 1 ) The operator A is the generator of a strongly continuous semigroup E(t) = e ta.

15 Exact solution of the stochastic wave equation The stochastic wave equation dx(t) = AX(t) dt + F(X(t)) dt + G(X(t)) dw(t), t >, X() = X, has a unique mild solution (recall E(t) = e ta ) X(t) = E(t)X + t E(t s)f(x(s)) ds + This is the variation-of-constants formula :-) t E(t s)g(x(s)) dw(s). Obs. Here, one needs some regularity assumptions on the noise, f and g.

16 1d stochastic sine-gordon Problem: For (x, t) [, 1] [, 1], consider the SPDE with space-time white-noise d u u dt = sin(u) dt sin(u) dw, u(, t) = u(1, t) =, u(x, ) = cos(π(x 1/2)), u(x, ) = x sine-gordon x

17 Thanks to III. Numerical discretisations

18 Results (I)... spoiler alert... Problem: d u u dt = dw. Spatial discretisation by standard linear FEM with mesh h. Time discretisation by stochastic trigonometric method with step size k. Theorems (C., Larsson, Sigg 213) Exact linear growth of the energy for the num. solution. Mean-square order of convergence at most one (depends on the regularity of the noise) U n 1 u h,1(t n ) L2 (Ω,U) Ck min{β,1} Λ (β 1)/2 Q 1/2 HS, where u h,1 is the FE solution of our problem. Mean-square errors for the full discretisation. FEM done in Kovács, Larsson, Saedpanah 21

19 Results (II) Problem: d u u dt = f (u) dt + g(u) dw. Spatial discretisation by standard linear FEM with mesh h. Time discretisation by stochastic trigonometric method with step size k. Theorems (Anton, C., Larsson, Wang 216) Mean-square errors for the full discretisation U n 1 u 1(t n ) L2 (Ω,U) C (h 2β 3 + k min(β,1)) for β [, 3], U n 2 u 2(t n ) L2 (Ω,U) C (h 2(β 1) 3 + k min(β 1,1)) for β [1, 4]. Almost linear growth of the energy for the num. solution of the nonlinear problem with additive noise.

20 Results (III) for random-field Problem 2 u(t, x) = 2 u(t, x) + f (u(t, x)) + σ(u(t, x)) 2 W (t, x). t 2 x 2 x t Spatial discretisation by centered FD with mesh x. Time discretisation by stochastic trigonometric method with step size t. Theorem (C., Quer-Sardanyons 215). Consider, for simplicity, the initial values u = v =. Assume f and σ satisfy a global Lipschitz and linear growth condition. Let p 1. Then, the following estimate of the error for the full discretisation holds: sup (t,x) [,T] [,1] ( E [ u M,N (t, x) u(t, x) 2p]) 1 2p C 1 ( x) 1 3 ε + C 2 ( t) 1 2, for all small enough ε >. The constants C 1 and C 2 are positive and do not depend neither on M nor on N.

21 Stochastic trigonometric method (Hilb. sp.) (I) Recall: The FE problem for the linear wave equation with additive noise reads (P h orthogonal projection, B = [, I]) dx h (t) = A h X h (t) dt + P h B dw(t), X h () = X h,, t >. Use variation-of-constants formula for the exact solution: where X h (t) = E h (t)x h, + E h (t) = e ta h = [ t E h (t s)p h B dw(s), ] C h (t) Λ 1/2 h S h (t) h S h (t) C h (t) Λ 1/2 with C h (t) = cos(tλ 1/2 h ) and S h (t) = sin(tλ 1/2 h ).

22 Stochastic trigonometric method (II) Discretise the stochastic integral in the var.-of-const. formula X h (k) = E h (k)x h, + k E h (k s)p h B dw(s) to obtain U n+1 = E h (k)u n + E h (k)p h B W n, that is, [ ] [ ] U n+1 [U 1 C U n+1 = h (k) Λ 1/2 ] [ ] h S h (k) n 1 Λ 1/2 2 Λ 1/2 h S h (k) C h (k) U2 n + h S h (k) P h W n, C h (k) where W n = W(t n+1 ) W(t n ) denotes the Wiener increments. Get an approximation Uj n u h,j (t n ) of the exact solution of our FE problem at the discrete times t n = nk. Similar ideas used for the semi-linear case with multiplicative noise. Chap. XIII of the yellow bible by Hairer, Lubich, Wanner 26

23 A trace formula Theorem (C., Larsson, Sigg 213) Expected value of the energy of the FE solution X h (t) = [u h,1 (t), u h,2 (t)] grows linearly with time: [ E [ = E ( Λ 1/2 h u h,1 (t) 2 L 2 (D) + u h,2(t) 2 L 2 (D) ( 1/2 Λ h u h, 2 L 2 (D) + v h, 2 L 2 (D)) ] + 1tTr(P 2 hqp h ). We have the same result for the num. sol. given by the stochastic trigonometric method: [ ( 1 1/2 E 2 Λ h U1 n 2 L 2 (D) + ) ] Un 2 2 L 2 (D) [ ( 1/2 = E Λ h u h, 2 L 2 (D) + v h, 2 L 2 (D)) ] + 1t 2 ntr(p h QP h ). 1 2 Obs. These are long-time results for the numerical solutions. ) ]

24 A trace formula: Numerical experiments (I) We consider the linear stochastic wave equation d u u dt = dw, (x, t) (, 1) (, 1), u(, t) = u(1, t) =, t (, 1), u(x, ) = cos(π(x 1/2)), u(x, ) =, x (, 1), with a Q-Wiener process W(t) with covariance operator Q = Λ 1/2. Where we recall Λ =.

25 A trace formula: Numerical experiments (II) 3 Exact Stochastic Trigonometric method h=.1, k=.1, M=15 samples 25 2 Energy t

26 A trace formula: Numerical experiments (III) BEM with k=.1 SV with k=.5 CNM with k=.1 Exact h=.1, M=1 samples 35 Energy t

27 Mean-square errors: Numerical experiments (I) We consider the 1d sine-gordon equation driven by a multiplicative space-time white noise ((x, t) (, 1) (,.5)) d u(x, t) u(x, t) dt = sin(u(x, t)) dt + u(x, t) dw(x, t), u(, t) = u(1, t) =, t (,.5), u(x, ) = cos(π(x 1/2)), u(x, ) =, x (, 1), with a space-time white noise with Q = I (β < 1/2).

28 Mean-square errors: Numerical experiments (II) Error 1-2 Slope 1/2 Error STM Slope 1/4 Error SEM Slope 1/3 Error CNM k Parameters: True sol. with STM with k exact = 2 11 and h exact = 2 9. M s = 25 for the expectations.

29 @ Universal Pictures IV. Ongoing and future works

30 Ongoing and future works (I) With R. Anton (Umeå University). Exponential methods for the time discretisation of stochastic Schrödinger equations idu ( u + F(u))dt = G(u)dW in R d [, T] u() = u, where u = u(x, t), with t and x R d, is a complex valued random process. With R. Anton and L. Quer-Sardanyons (UA Barcelona). Exponential methods for parabolic problems (random-field approach).

31 Ongoing and future works (II) With G. Dujardin (Inria Lille Nord-Europe). Exponential integrators for nonlinear Schrödinger equations with white noise dispersion idu + c u dβ + u 2σ u dt = u() = u, where u = u(x, t), with t and x R d, is a complex valued random process; c and σ are positive real numbers; and β = β(t) is a real valued Brownian motion. Look for mean-square error estimates and mass-preserving exponential integrators.

32 Thanks to

33 Main ingredients of the proofs Recall: u M,N (t, x) = t 1 + t 1 { G M (t κ T N(s), x, y)f ( u M,N (κ T N(s), κ M (y)) )} dy ds { G M (t κ T N(s), x, y)σ ( u M,N (κ T N(s), κ M (y)) )} W(ds, dy). 1 Write u M,N (t, x) u(t, x) = u M,N (t, x) u M (t, x) + u M (t, x) u(t, x) and use the spatial error estimate from Quer-Sardanyons, Sanz-Solé 26 to get the order C 1 ( x) 1 3 ε for all small enough ε >. 2 Next, consider u M,N (t, x) u M (t, x) and use a Gronwall argument. 3 Use properties of G M (t, x, y), of u M (t, x), of u M,N (t, x), assumptions on f and σ, Hölder s inequality, Burkholder-Davis-Gundy s inequality and finally Gronwall s inequality to bound the error of the fully-discrete solution.

34 Main steps for the proofs (I) First, by definition of u M and u M,N, consider the difference u M,N (t, x) u M (t, x) = t 1 {G M (t κ TN(s), x, y)f ( u M,N (κ TN(s), κ M (y)) ) G M (t s, x, y)f ( u M (s, κ M (y)) )} dy ds t 1 + {G M (t κ TN(s), x, y)σ ( u M,N (κ TN(s), κ M (y)) ) G M (t s, x, y)σ ( u M (s, κ M (y)) )} W(ds, dy). Next, add and subtract some terms in order to be able to use the properties of f and σ.

35 Main steps for the proofs (II) The first expression u M,N (t, x) u M (t, x) = t 1 {G M (t κ T N(s), x, y)f ( u M,N (κ T N(s), κ M (y)) ) G M (t s, x, y)f ( u M (s, κ M (y)) )} dy ds + noisy blabla can be decomposed as the sum of 3 terms: D 1 := D 2 := D 3 := t 1 t 1 t 1 G M (t κ T N(s), x, y) { f ( u M,N (κ T N(s), κ M (y)) ) f ( u M (κ T N(s), κ M (y)) )} dy ds, { G M (t κ T N(s), x, y) G M (t s, x, y) } f ( u M (κ T N(s), κ M (y)) ) dy ds, G M (t s, x, y) { f ( u M (κ T N(s), κ M (y)) ) f ( u M (s, κ M (y)) )} dy ds.

36 Main steps for the proofs (III) The second expression t 1 { G M (t κ T N (s), x, y)σ( u M,N (κ T N (s), κ M(y)) ) G M (t s, x, y)σ ( u M (s, κ M (y)) )} W(ds, dy). can be decomposed as the sum of 3 terms: D 4 := D 5 := D 6 := t 1 t 1 t 1 G M (t κ T N(s), x, y) { σ ( u M,N (κ T N(s), κ M (y)) ) σ ( u M (κ T N(s), κ M (y)) )} W(ds, dy), { G M (t κ T N(s), x, y) G M (t s, x, y) } σ ( u M (κ T N(s), κ M (y)) ) W(ds, dy), G M (t s, x, y) { σ ( u M (κ T N(s), κ M (y)) ) σ ( u M (s, κ M (y)) )} W(ds, dy).

37 Main steps for the proofs (IV) We now have to estimate each of the terms D 1,..., D 6. Estimates for D 6 : D 6 := t 1 G M (t s, x, y) { σ ( u M (κ T N(s), κ M (y)) ) σ ( u M (s, κ M (y)) )} W(ds, dy). By Burkholder-Davis-Gundy s and Hölder s inequalities, and Lipschitz condition on σ, one obtains E [ D6 2p] C t 1 G M (t s, x, y) 2 dy sup E[ u M (κ T N(s), x) u M (s, x) 2p ] ds. x [,1] The Hölder continuity property of the process u M (t, x) gives E [ t D6 2p] C (κ T N(s) s) p ds C ( t) p.

38 Main steps for the proofs (V) All these estimates together give sup E [ u M,N (t, x) u M (t, x) 2p] C 1 ( t) p (t,x) [,T] [,1] + C 2 t sup E [ u M,N (r, x) u M (r, x) 2p] ds, (r,x) [,s] [,1] for some positive constants C 1 and C 2 independent of N and M. An application of Gronwall s lemma finally lead to ( [ sup E u M,N (t, x) u M (t, x) 2p]) 1 2p C ( t) 1 2. (t,x) [,T] [,1] This concludes the proof of the theorem.

39 Thanks for your attention!!