Martin, Andreas; Prigarin, Sergej M.; Winkler, Gerhard. Working Paper Exact and fast numerical algorithms for the stochastic wave equation

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1 econstor Der Open-Access-Publiationsserver der ZBW Leibniz-Informationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics Martin, Andreas; Prigarin, Sergej M.; Winler, Gerhard Woring Paper Exact and fast numerical algorithms for the stochastic wave equation Discussion paper // Sonderforschungsbereich 386 der Ludwig-Maximilians-Universität München, No. 349 Provided in Cooperation with: Collaborative Research Center SFB 386: Statistical Analysis of discrete structures - Applications in Biometrics and Econometrics, LMU Munich Suggested Citation: Martin, Andreas; Prigarin, Sergej M.; Winler, Gerhard 003 : Exact and fast numerical algorithms for the stochastic wave equation, Discussion paper // Sonderforschungsbereich 386 der Ludwig-Maximilians-Universität München, No. 349, nbn-resolving.de/urn:nbn:de:bvb:19-epub-175- This Version is available at: Nutzungsbedingungen: Die ZBW räumt Ihnen als Nutzerin/Nutzer das unentgeltliche, räumlich unbeschränte und zeitlich auf die Dauer des Schutzrechts beschränte einfache Recht ein, das ausgewählte Wer im Rahmen der unter nachzulesenden vollständigen Nutzungsbedingungen zu vervielfältigen, mit denen die Nutzerin/der Nutzer sich durch die erste Nutzung einverstanden erlärt. Terms of use: The ZBW grants you, the user, the non-exclusive right to use the selected wor free of charge, territorially unrestricted and within the time limit of the term of the property rights according to the terms specified at By the first use of the selected wor the user agrees and declares to comply with these terms of use. zbw Leibniz-Informationszentrum Wirtschaft Leibniz Information Centre for Economics

2 Exact and Fast Numerical Algorithms for the Stochastic Wave Equation Andreas Martin, Sergei M. Prigarin and Gerhard Winler April 9, 003 Abstract On the basis of integral representations we propose fast numerical methods to solve the Cauchy problem for the stochastic wave equation without boundaries and with the Dirichlet boundary conditions. The algorithms are exact in a probabilistic sense. 1 Statement of the problem and integral representations Consider the stochastic wave equation t t, x a t, x gt, x ft, x dw t, x 1 x with random initial conditions, where W is a Gaussian white noise on the plane. We give shortly the precise statement of the problem. The aim of AMS Subject Classification. Primary, 60h15, 68U0. Key words and phrases. stochastic wave equation, numerical simulation, isonormal process. This wor was partly supported by a DAAD Dotorandenstipendium im Rahmen des gemeinsamen Hochschulsonderprogramms III von Bund und Ländern. Institute of Biomathematics and Biometry, GSF-National Research Centre for Environment and Health, Ingolstädter Landstraße 1, Neuherberg/München, Germany. martin@gsf.de Institute of Computational Mathematics and Mathematical Geophysics, pr. Lavrentieva 6, Novosibirs, , Russia. smp@osmf.sscc.ru Institute of Biomathematics and Biometry, GSF-National Research Centre for Environment and Health, Ingolstädter Landstraße 1, Neuherberg/München, Germany. gwinler@gsf.de 1

3 1 STATEMENT OF THE PROBLEM the paper is to construct numerical algorithms in order to solve the Cauchy problem for equation 1 without boundaries and with the Dirichlet boundary conditions, i.e., to simulate realizations of the random field. First, we present the necessary formalism for the stochastic wave equation and integral representations for the solutions that underlay our numerical methods. The details can be found in [5], [1] and []. Let Ω, F, P be a complete probability space, W be an isonormal process indexed by L R R, BR R, λ and {F t ; t R } be a nullset augmented filtration which satisfies W 1 A is F t -measurable whenever A B[0, t] R with λa < ; W 1 A is independent of F t whenever A Bt, R with λa <. Moreover, let F {F x; x R} and H {Hx; x R} be almost surely continuous and F-measurable processes such that E{ F R x }ϕx dx and R E{ Hx }ϕx dx are finite for any smooth ϕ with compact support. Finally, assume f and g to be real valued functions on R R such that fϕ is square integrable and gϕ is integrable for any smooth ϕ with compact support, respectively. Similar to the deterministic case, the solution of equation 1 with initial conditions F, H on R R can be represented in terms of Green s function G a hereafter a > 0, { 1 if 0 s t and x y at s G a t, x, s, y 0 otherwise. The solution is given by t, x u 0 t, x G a t, x, s, ygs, y dy ds R R G a t, x, s, yfs, y dw s, y, R R where u 0 t, x F x at F x at/ xat Hy/ dy. x at Remar 1.1 We require the integrability conditions on F and H to have first and second moments for the random field. An example for a homogeneous wave equation with random initial conditions is studied in [4].

4 1 STATEMENT OF THE PROBLEM 3 Remar 1. The stochastic integral in is defined by hs, y dw s, y : W h R R for any h L R R, BR R, λ and we use the integral notation for convenience. Remar 1.3 It holds 0, F almost surely, and for f 0 we have / t0, H. However, for f 0 the last equation has only formal meaning, because the random field is not differentiable. For the Cauchy problem on R [0, 1] with Dirichlet boundary conditions Green s function Σ Ga has the following form Σ Ga t, x, s, y [ G a t, x ; s, y1 [0,1] y ] G a t, x ; s, y1 [0,1] y, and instead of u 0 we consider u 0,D t, x 1 [ F x at1 0,1 x at F x at1 0,1 x at 1 F x at1 0,1 x at ] F x at1 0,1 x at [ xat 1 Hy dy x at 0 xat 1 x at 0 In this case the solution of 1 with initial conditions F, H is t, x u 0,D t, x 1 R 0 1 R 0 ] Hy dy. Σ Ga t, x, s, ygs, y dy ds 3 Σ Ga t, x, s, yfs, y dw s, y. In both cases the solutions are Gaussian random fields and the integral representations give the basis to construct exact numerical solutions.

5 NUMERICAL ALGORITHMS 4 Numerical algorithms A realization of the solution will be simulated on a grid G {t j, x jh, ah} with j {0, 1/, 1, 3/,...}, {..., 1, 1/, 0, 1/, 1,...}, where h is a time step, ah is a spatial step and for a grid point the indices j, are simultaneously either integers or fractional numbers. In addition to the values j jh, ah on the grid G the algorithm operates with the values µ j on an adjacent grid G {t j, x jh, ah} with the same sets of indices, but one of the indices is always integer while the other is fractional. Moreover, µ j is given by µj x ah/ x ah/ H jy dy, where H j y is the second part of the initial conditions for the Cauchy problem at time t j see Remar Cauchy problem without boundaries The algorithm uses the Marov property of with respect to time, i.e. the values for the next time level depend only on the values obtained at the previous time level. Therefore the complexity of the algorithm is linear. Initialisation. Set j 0. The values 0 are taen from the initial condition, i.e. 0 F ah 0, ah t 0, x. Moreover, set µ 0 Hy dy. x ah/ x ah/ Step 1. Equation 3 gives an explicit rule to calculate j1/ : j1/ 1 1/ j 1 j 1 µj 1/ where ξ j1/ 1/ tj1/ x1 t j x tj1/ x1 t j is a Gaussian random variable with mean tj1/ x1 t j and variance tj1/ x ξ j1/ 1/, 4 G a t j1/, x 1/, s, ygs, y dy ds x G a t j1/, x 1/, s, yfs, y dw s, y G a t j1/, x 1/, s, ygs, y dy ds t j x1 x G at j1/, x 1/, s, yf s, y dy ds.

6 NUMERICAL ALGORITHMS 5 Step. The next step is to calculate µ j1/ 1 can be expressed in two different ways, that j1 1 and where j1 1 j1 1 1 j1/ 1/ j1/ 3/. To this end, one observes 1 µj1/ 1 ξ j j 1 j µ j 1/ 3/ µj 6 ξ j1/ 1/ ξj1/ 3/ δj 1 ξj1 1, δ j 1 1 tj1/ 1 x1 as t j t j x 1 as t j tj1/ x1 as t j t j x 1 as t j gs, y dy ds 7 fs, y dw s, y is a Gaussian random variable with mean given by the first double integral in 7 and variance 1 tj1/ x1 as t j f s, y dy ds. t j x 1 as t j A combination of 5 and 6 yields µ j1/ 1 [ 1 j 1 j ξ j1/ 1/ ξj1/ 3/ δ j 1 1 µ j 1/ µj 3/ j1/ 1/ 3/ ] j1/. Thus, at step two the values µ j1/ 1 are simulated according to expression 8. Note, that all the components of the expression, except δ j 1, are defined at the previous steps of the algorithm, and the random variable ξ j1 1 appears in 5 and 6 but cancels out in 8. Cycling. Set j j 1/ and go bac to step 1. Proposition.1 The sequence j, on the grid G fulfils j, t j, x with probability one. Proof. On the grid G Green s function G α can be decomposed into triangles with corners given by the grid points. Thus induction on the time level t j gives the assertion. 8

7 NUMERICAL ALGORITHMS 6. Cauchy problem with boundary conditions This algorithm is a modification of the previous one. As ah is a step size in space, here we assume that N 1/ah is an integer. Initialisation. Set j 0 and determine 0 for {0,..., N} and for {0,..., N 1} by the initial conditions F and H, respectively. µ 0 1/ Step 1. The values j1/ 1/ for {0,..., N 1} are calculated according to equation 4. Step. The values µ j1/ 1 for {0,..., N } are calculated according to equation 8. Step 3. The values j1 to equation 4 while j1 0 and j1 N i.e., j1 0 j1 N 0. for {1,..., N 1} are calculated according are defined by the boundary conditions, Step 4. Now, the values µ j1 1 have to be calculated. For { 1, 3,..., N 5 equation 8 is used. To find µ j1 1/ and µj1 N 1/ the boundary conditions should be applied. The value j3/ 1/ can be expressed in two different ways note that j1 0 0, j3/ 1/ j3/ 1/ 1 j1 0 j j1 1 1 µj1 1/ 1 j1/ 1/ j1/ 3/ ξ j1 1 δ j1/ 1/ ξ j3/ 1/. µj1 1/ ξ j3/ 1/, ξ j3/ 1/ 1 µj1/ 1 From these two representations we obtain the formula for simulation, [ 1 µ j1 1/ j1/ 1/ j1/ 3/ 1 µ j1/ 1 ξ j1 j1/ 1 δ 1/ 1 ] j1 1. Similarly, for µ j1 N 1/ we have µ j1 N 1/ [ 1 j1/ N 3/ j1/ N 1/ 1 j1/ δ ξ j1 N 1 N 1/ 1 j1 N 1 ]. µ j1/ N 1 Cycling. Set j j 1 and go bac to Step 1. The sequence j, fulfils j, t j, x with probability one as before. }

8 3 CONCLUSION 7.3 Results of simulation Results of numerical simulations are presented in Figs. 1 and. In addition, in Fig. 3 we present an example of the sample size dependence of the empirical correlation between two points of the solution for the Cauchy problem without boundaries. Figure 1: Realization of the solution of the Cauchy problem without boundaries; zero initial conditions and harmonic initial conditions. Figure : Realization of the solution of the Cauchy problem with Dirichlet boundary conditions; zero initial conditions and harmonic initial conditions. 3 Conclusion The two proposed methods are exact in the sense that the finite-dimensional distributions of the field on the grid G coincide with the finite-dimensional distributions of the numerical solution. Moreover, the complexity of the algorithms is linear and they are therefore attractive for numerical simulations. Remar 3.1 Exact algorithms to solve the stochastic Klein-Gordon equation / t t, x / x t, x αt, x dw t, x

9 REFERENCES 8 Figure 3: Empirical correlations with confidence intervals of level 0.95 via the sample size; exact correlation is equal to 0.5. were discussed in [3]. But the algorithms proposed in that paper don t have the Marov property and, practically, result in a general simulation of Gaussian random vectors with correlated components. Because of this fact, it can turn out that the algorithms are not feasible for large grids. References [1] R. Carmona and D. Nualart. Random nonlinear wave equations: Smoothness of the solutions. Probab. Theory Relat. Fields, 794: , [] A. Martin. Hyperbolic stochastic partial differential equations: Small balls and simulation; propagation of singularities. PhD thesis, Aachen: Shaer Verlag. Neuherberg: Institut f. Biomathemati u. Biometrie, GSF - Forschungszentrum f. Umwelt u. Gesundheit, 00. [3] A. Martin, S.M. Prigarin, and G. Winler. Exact numerical algorithms for linear stochastic wave equation and stochastic Klein-Gordon equation. In G.A. Mihailov ed. et al., Proceedings to the International Conference on Computational Mathematics. ICM&MG Publisher, Prospect Lavrentieva 6, Novosibirs, Russia. 00. [4] E. Orsingher. Vibrations with random initial conditions. Boll. Unione Mat. Ital., VI. Ser., B, 4: , [5] J.B. Walsh. An introduction to stochastic partial differential equations. In École d été de probabilités de Saint-Flour IV , Lect. Notes Math. 1180,