Weak approximation of stochastic partial differential equations: the nonlinear case

Debussche, Arnaud

doi:10.1090/S0025-5718-2010-02395-6

Weak approximation of stochastic partial differential equations: the nonlinear case
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Math. Comp. 80 (2011), 89-117 Request permission

Abstract:

We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that, as is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case of a semilinear stochastic heat equation driven by a space-time white noise.

References

E. J. Allen, S. J. Novosel, and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics Stochastics Rep. 64 (1998), no. 1-2, 117–142. MR 1637047, DOI 10.1080/17442509808834159
V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Related Fields 104 (1996), no. 1, 43–60. MR 1367666, DOI 10.1007/BF01303802
Evelyn Buckwar and Tony Shardlow, Weak approximation of stochastic differential delay equations, IMA J. Numer. Anal. 25 (2005), no. 1, 57–86. MR 2110235, DOI 10.1093/imanum/drh012
A. De Bouard and A. Debussche, A semi-discrete scheme for the stochastic nonlinear Schrödinger equation, Numer. Math. 96 (2004), no. 4, 733–770. MR 2036364, DOI 10.1007/s00211-003-0494-5
Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
A. M. Davie and J. G. Gaines, Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Math. Comp. 70 (2001), no. 233, 121–134. MR 1803132, DOI 10.1090/S0025-5718-00-01224-2
Anne de Bouard and Arnaud Debussche, Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation, Appl. Math. Optim. 54 (2006), no. 3, 369–399. MR 2268663, DOI 10.1007/s00245-006-0875-0
Arnaud Debussche and Jacques Printems, Weak order for the discretization of the stochastic heat equation, Math. Comp. 78 (2009), no. 266, 845–863. MR 2476562, DOI 10.1090/S0025-5718-08-02184-4
Matthias Geissert, Mihály Kovács, and Stig Larsson, Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise, BIT 49 (2009), no. 2, 343–356. MR 2507605, DOI 10.1007/s10543-009-0227-y
I.C. Gohberg, M.G. Krejn Introduction to the theory of linear nonselfadjoint operators in Hilbert space , Amer. Math. Soc., Providence, RI, 1970.
W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs, Bull. Austral. Math. Soc. 54 (1996), no. 1, 79–85. MR 1402994, DOI 10.1017/S0004972700015094
István Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I, Potential Anal. 9 (1998), no. 1, 1–25. MR 1644183, DOI 10.1023/A:1008615012377
István Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II, Potential Anal. 11 (1999), no. 1, 1–37. MR 1699161, DOI 10.1023/A:1008699504438
István Gyöngy and Annie Millet, On discretization schemes for stochastic evolution equations, Potential Anal. 23 (2005), no. 2, 99–134. MR 2139212, DOI 10.1007/s11118-004-5393-6
István Gyöngy and Annie Millet, Rate of convergence of implicit approximations for stochastic evolution equations, Stochastic differential equations: theory and applications, Interdiscip. Math. Sci., vol. 2, World Sci. Publ., Hackensack, NJ, 2007, pp. 281–310. MR 2393581, DOI 10.1142/9789812770639_{0}011
István Gyöngy and Annie Millet, Rate of convergence of space time approximations for stochastic evolution equations, Potential Anal. 30 (2009), no. 1, 29–64. MR 2465711, DOI 10.1007/s11118-008-9105-5
István Gyöngy and David Nualart, Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise, Potential Anal. 7 (1997), no. 4, 725–757. MR 1480861, DOI 10.1023/A:1017998901460
Erika Hausenblas, Numerical analysis of semilinear stochastic evolution equations in Banach spaces, J. Comput. Appl. Math. 147 (2002), no. 2, 485–516. MR 1933610, DOI 10.1016/S0377-0427(02)00483-1
Erika Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal. 18 (2003), no. 2, 141–186. MR 1953619, DOI 10.1023/A:1020552804087
Erika Hausenblas, Weak approximation for semilinear stochastic evolution equations, Stochastic analysis and related topics VIII, Progr. Probab., vol. 53, Birkhäuser, Basel, 2003, pp. 111–128. MR 2189620
Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374, DOI 10.1007/978-3-662-12616-5
Arturo Kohatsu-Higa, Weak approximations. A Malliavin calculus approach, Math. Comp. 70 (2001), no. 233, 135–172. MR 1680895, DOI 10.1090/S0025-5718-00-01201-1
Gabriel J. Lord and Jacques Rougemont, A numerical scheme for stochastic PDEs with Gevrey regularity, IMA J. Numer. Anal. 24 (2004), no. 4, 587–604. MR 2094572, DOI 10.1093/imanum/24.4.587
Annie Millet and Pierre-Luc Morien, On implicit and explicit discretization schemes for parabolic SPDEs in any dimension, Stochastic Process. Appl. 115 (2005), no. 7, 1073–1106. MR 2147242, DOI 10.1016/j.spa.2005.02.004
G. N. Mil′šteĭn, A method with second order accuracy for the integration of stochastic differential equations, Teor. Verojatnost. i Primenen. 23 (1978), no. 2, 414–419 (Russian, with English summary). MR 0517998
G. N. Mil′shteĭn, Weak approximation of solutions of systems of stochastic differential equations, Teor. Veroyatnost. i Primenen. 30 (1985), no. 4, 706–721 (Russian). MR 816284
G. N. Milstein, Numerical integration of stochastic differential equations, Mathematics and its Applications, vol. 313, Kluwer Academic Publishers Group, Dordrecht, 1995. Translated and revised from the 1988 Russian original. MR 1335454, DOI 10.1007/978-94-015-8455-5
G. N. Milstein and M. V. Tretyakov, Stochastic numerics for mathematical physics, Scientific Computation, Springer-Verlag, Berlin, 2004. MR 2069903, DOI 10.1007/978-3-662-10063-9
David Nualart, The Malliavin calculus and related topics, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1344217, DOI 10.1007/978-1-4757-2437-0
Jacques Printems, On the discretization in time of parabolic stochastic partial differential equations, M2AN Math. Model. Numer. Anal. 35 (2001), no. 6, 1055–1078. MR 1873517, DOI 10.1051/m2an:2001148
Tony Shardlow, Numerical methods for stochastic parabolic PDEs, Numer. Funct. Anal. Optim. 20 (1999), no. 1-2, 121–145. MR 1683281, DOI 10.1080/01630569908816884
Anders Szepessy, Raúl Tempone, and Georgios E. Zouraris, Adaptive weak approximation of stochastic differential equations, Comm. Pure Appl. Math. 54 (2001), no. 10, 1169–1214. MR 1843985, DOI 10.1002/cpa.10000
Denis Talay, Probabilistic numerical methods for partial differential equations: elements of analysis, Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995) Lecture Notes in Math., vol. 1627, Springer, Berlin, 1996, pp. 148–196. MR 1431302, DOI 10.1007/BFb0093180
Denis Talay, Discrétisation d’une équation différentielle stochastique et calcul approché d’espérances de fonctionnelles de la solution, RAIRO Modél. Math. Anal. Numér. 20 (1986), no. 1, 141–179 (French, with English summary). MR 844521, DOI 10.1051/m2an/1986200101411
Yubin Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal. 43 (2005), no. 4, 1363–1384. MR 2182132, DOI 10.1137/040605278
Yubin Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise, BIT 44 (2004), no. 4, 829–847. MR 2211047, DOI 10.1007/s10543-004-3755-5
J. B. Walsh, Finite element methods for parabolic stochastic PDE’s, Potential Anal. 23 (2005), no. 1, 1–43. MR 2136207, DOI 10.1007/s11118-004-2950-y