Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Weak order for the discretization of the stochastic heat equation

Authors: Arnaud Debussche and Jacques Printems
Journal: Math. Comp. 78 (2009), 845-863
MSC (2000): Primary 60H15, 60H35, 65C30, 65M60
Published electronically: October 7, 2008
MathSciNet review: 2476562
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the approximation of the distribution of $ X_t$ Hilbert-valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as

$\displaystyle \mathrm{d} X_t+AX_t \, \mathrm{d} t = Q^{1/2} \mathrm{d} W(t), \quad X_0=x \in H, \quad t\in[0,T], $

driven by a Gaussian space time noise whose covariance operator $ Q$ is given. We assume that $ A^{-\alpha}$ is a finite trace operator for some $ \alpha>0$ and that $ Q$ is bounded from $ H$ into $ D(A^\beta)$ for some $ \beta\geq 0$. It is not required to be nuclear or to commute with $ A$.

The discretization is achieved thanks to finite element methods in space (parameter $ h>0$) and a $ \theta$-method in time (parameter $ \Delta t=T/N$). We define a discrete solution $ X^n_h$ and for suitable functions $ \varphi$ defined on $ H$, we show that

$\displaystyle \vert\mathbb{E} \, \varphi(X^N_h) - \mathbb{E} \, \varphi(X_T) \vert = O(h^{2\gamma} + \Delta t^\gamma) $

where $ \gamma<1- \alpha + \beta$. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations.

References [Enhancements On Off] (What's this?)

  • 1. E.J. ALLEN, S.J. NOVOSEL, Z. ZHANG, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics Stochastics Rep., 64 (1998), 117-142. MR 1637047 (99d:60067)
  • 2. J. H. BRAMBLE, A. H. SCHATZ, V. THOMÉE, L. B. WAHLBIN, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal. 14 (1977), 218-241. MR 0448926 (56:7231)
  • 3. E. BUCKWAR, T. SHARDLOW, Weak approximation of stochastic differential delay equations, IMA J. Numer. Anal. 25 (2005), 57-86. MR 2110235 (2006a:65012)
  • 4. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam-London-New-York, 1978. MR 0520174 (58:25001)
  • 5. G. DA PRATO and J. ZABCZYK, Stochastic Equations in Infinite Dimensions, Encyclopedia of mathematics and its applications 44, Cambridge University Press, 1992. MR 1207136 (95g:60073)
  • 6. G. DA PRATO and J. ZABCZYK, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society, Lecture Note Series 293, Cambridge University Press, 2002. MR 1985790 (2004e:47058)
  • 7. A.M. DAVIE, J.G. GAINES, Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Math. Comp. 70 (2001), 121-134 MR 1803132 (2001h:65012)
  • 8. A. DE BOUARD, A. DEBUSSCHE, Weak and strong order of convergence of a semi discrete scheme for the stochastic Nonlinear Schrodinger equation, Appl. Math. and Optim., 54 (2006), 369-399. MR 2268663 (2008g:60208)
  • 9. M. GEISSERT, M. KOVACS, S. LARSSON, Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise, Preprint.
  • 10. I. C. GOKHBERG, M. G. KREĬN, Introduction to the theory of linear nonselfadjoint operators in Hilbert space, Amer. Math. Soc., Providence, RI, 1970.
  • 11. W. GRECKSCH, P.E. KLOEDEN, Time-discretised Galerkin approximations of parabolic stochastic PDEs, Bull. Austral. Math. Soc. 54 (1996), 79-85. MR 1402994 (97g:60080)
  • 12. I. GYÖNGY, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I, Potential Anal. 9 (1998), 1-25. MR 1644183 (99j:60091)
  • 13. I. GYÖNGY, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II, Potential Anal. 11 (1999), 1-37 MR 1699161 (2000g:60106)
  • 14. I. GYÖNGY, A. MILLET, On discretization schemes for stochastic evolution equations, Potential Analysis 23 (2005), 99-134. MR 2139212 (2006a:60115)
  • 15. I. GYÖNGY, A. MILLET, Rate of Convergence of Implicit Approximations for stochastic evolution equations, Stochastic Differential Equations: Theory and Applications. A volume in Honor of Professor Boris L. Rosovskii, Interdisciplinary Mathematical Sciences, Vol. 2, World Scientific (2007), 281-310.
  • 16. I. GYÖNGY, A. MILLET, Rate of convergence of space time approximations for stochastic evolution equations, Preprint (2007).
  • 17. I. GYÖNGY, D. NUALART, Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise, Potential Anal. 7 (1997), 725-757. MR 1480861 (98m:60097)
  • 18. E. HAUSENBLAS, Approximation for semilinear stochastic evolution equations in Banach spaces, Journal in Comp. and Appl. Math., 147 (2002), 485-516. MR 1933610 (2003j:35338)
  • 19. E. HAUSENBLAS, Approximation for semilinear stochastic evolution equations, Potential Analysis, 18 (2003), 141-186. MR 1953619 (2003m:60167)
  • 20. E. HAUSENBLAS, Weak approximation of stochastic partial differential equations. in Stochastic analysis and related topics VIII. Silivri workshop, Progress in Probability. U. Capar and A. Üstünel editors. Basel: Birkhäuser, 2003. MR 2189620 (2006k:60114)
  • 21. C. JOHNSON, S. LARSSON, V. THOMÉE, L. B. WALHBIN, Error estimates for spatially discrete approximations of semilinear parabolic equations with non smooth initial data, Math. Comput., 49, (1987), 331-357. MR 906175 (88k:65100)
  • 22. P.E. KLOEDEN, E. PLATEN, Numerical solution of stochastic differential equations, Applications of Mathematics, 23, Springer-Verlag, New York, 1992. MR 1214374 (94b:60069)
  • 23. M.-N. LE ROUX, Semidiscretization in Time for Parabolic Problems, Math. Comput., 33 (1979), 919-931. MR 528047 (80f:65101)
  • 24. G. LORD, J. ROUGEMONT, A Numerical Scheme for stochastic PDEs with Gevrey Regularity, IMA J. Num. Anal., 24 (2004), 587-604. MR 2094572 (2005d:60102)
  • 25. A. MILLET, P.L. MORIEN, On implicit and explicit discretization schemes for parabolic SPDEs in any dimension, Stoch. Proc. and Appl. 115 (2005), n$ ^{}o$ 7, 1073-1106. MR 2147242 (2006b:60141)
  • 26. G. N. MILSTEIN, Numerical integration of stochastic differential equations, Mathematics and its Applications, 313, Kluwer Academic Publishers, Dordrrecht, 1995. MR 1335454 (96e:65003)
  • 27. G. N. MILSTEIN, M. V. TRETYAKOV, Stochastic numerics for mathematical physics, Scientific Computation series, Springer-Verlag, 2004. MR 2069903 (2005f:60004)
  • 28. J. PRINTEMS, On the discretization in time of parabolic stochastic partial differential equations, Math. Model. and Numer. Anal., 35 (2001), 1055-1078. MR 1873517 (2002j:60116)
  • 29. T. SHARDLOW, Numerical methods for stochastic parabolic PDEs, Numer. Funct. Anal. Optim., 20 (1999), 121-145. MR 1683281 (2000g:65004)
  • 30. G. STRANG, G.J. FIX, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, 1973. MR 0443377 (56:1747)
  • 31. D. TALAY, Probabilistic numerical methods for partial differential equations: elements of analysis, Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), 148-196, Lecture Notes in Math., 1627, Springer, Berlin, 1996. MR 1431302 (98j:60092)
  • 32. V. THOMÉE, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, 1997. MR 1479170 (98m:65007)
  • 33. Y. YAN, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 1363-1384. MR 2182132 (2007a:65013)
  • 34. Y. YAN, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise, BIT 44 (2004), 829-847. MR 2211047 (2007c:60065)
  • 35. J.B. WALSH Finite element methods for parabolic stochastic PDE's, Potential Anal. 23 (2005), 1-43. MR 2136207 (2006b:60155)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 60H15, 60H35, 65C30, 65M60

Retrieve articles in all journals with MSC (2000): 60H15, 60H35, 65C30, 65M60

Additional Information

Arnaud Debussche
Affiliation: IRMAR et ENS de Cachan, antenne de Bretagne, Campus de Ker Lann, avenue Robert Schumann, 35170 Bruz, France

Jacques Printems
Affiliation: Laboratoire d’Analyse et de Mathématiques Appliquées, CNRS UMR 8050, Université de Paris XII, 61, avenue du Général de Gaulle, 94010 Créteil, France

Keywords: Weak order, stochastic heat equation, finite element, Euler scheme
Received by editor(s): October 30, 2007
Received by editor(s) in revised form: May 7, 2008
Published electronically: October 7, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society