×

Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. (English) Zbl 1322.65021

Summary: This paper deals with the weak error estimates of the exponential Euler method for semi-linear stochastic partial differential equations (SPDEs). A weak error representation formula is first derived for the exponential integrator scheme in the context of truncated SPDEs. The obtained formula, that enjoys the absence of the irregular term involved with the unbounded operator, is then applied to a parabolic SPDE. Under certain mild assumptions on the nonlinearity, we treat a full discretization based on the spectral Galerkin spatial approximation and provide an easy weak error analysis, which does not rely on Malliavin calculus.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35K59 Quasilinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A. Andersson, Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE,, preprint <a href= · Zbl 1357.60063
[2] A. Andersson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation,, preprint <a href=
[3] R. Anton, Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise,, preprint <a href=
[4] X. Bardina, Weak convergence for the stochastic heat equation driven by gaussian white noise,, Electron. J. Probab, 15, 1267 (2010) · Zbl 1225.60100 · doi:10.1214/EJP.v15-792
[5] C. E. Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise,, Potential Anal., 40, 1 (2014) · Zbl 1286.35010 · doi:10.1007/s11118-013-9338-9
[6] D. Cohen, A trigonometric method for the linear stochastic wave equation,, SIAM J. Numer. Anal., 51, 204 (2013) · Zbl 1273.65010 · doi:10.1137/12087030X
[7] D. Cohen, Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations,, Numer. Math., 121, 1 (2012) · Zbl 1247.65004 · doi:10.1007/s00211-011-0426-8
[8] D. Conus, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients,, preprint <a href= · Zbl 1477.65020
[9] G. Da Prato, <em>Ergodicity for Infinite Dimensional Systems</em>,, Cambridge University Press (1996) · Zbl 0849.60052 · doi:10.1017/CBO9780511662829
[10] G. Da Prato, <em>Stochastic Equations in Infinite Dimensions</em>,, Cambridge University Press (1992) · Zbl 1140.60034 · doi:10.1017/CBO9780511666223
[11] A. de Bouard, Weak and strong order of convergence of a semi discrete scheme for the stochastic nonlinear Schrödinger equation,, Appl. Math. Opt., 54, 369 (2006) · Zbl 1109.60051 · doi:10.1007/s00245-006-0875-0
[12] A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case,, Math. Comp., 80, 89 (2011) · Zbl 1217.65011 · doi:10.1090/S0025-5718-2010-02395-6
[13] A. Debussche, Weak order for the discretization of the stochastic heat equation,, Math. Comp., 78, 845 (2009) · Zbl 1215.60043 · doi:10.1090/S0025-5718-08-02184-4
[14] M. Geissert, Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise,, BIT, 49, 343 (2009) · Zbl 1171.65005 · doi:10.1007/s10543-009-0227-y
[15] M. Hochbruck, Exponential integrators,, Acta Numerica, 19, 209 (2010) · Zbl 1242.65109 · doi:10.1017/S0962492910000048
[16] A. Jentzen, Efficient simulation of nonlinear parabolic SPDEs with additive noise,, Ann. Appl. Probab., 21, 908 (2011) · Zbl 1223.60050 · doi:10.1214/10-AAP711
[17] A. Jentzen, Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465, 649 (2009) · Zbl 1186.65011 · doi:10.1098/rspa.2008.0325
[18] A. Jentzen, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients,, preprint · Zbl 1482.65143
[19] P. E. Kloeden, The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds,, J. Comput. Appl. Math., 235, 1245 (2011) · Zbl 1208.65017 · doi:10.1016/j.cam.2010.08.011
[20] M. Kovács, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise,, BIT, 52, 85 (2012) · Zbl 1242.65010 · doi:10.1007/s10543-011-0344-2
[21] M. Kovács, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes,, BIT, 53, 497 (2013) · Zbl 1280.65009 · doi:10.1007/s10543-012-0405-1
[22] R. Kruse, <em>Strong and Weak Approximation of Semilinear Stochastic Evolution Equations</em>,, Springer (2014) · Zbl 1285.60002 · doi:10.1007/978-3-319-02231-4
[23] F. Lindner, Weak order for the discretization of the stochastic heat equation driven by impulsive noise,, Potential Anal., 38, 345 (2013) · Zbl 1263.60058 · doi:10.1007/s11118-012-9276-y
[24] G. J. Lord, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise,, IMA J. Numer. Anal., 33, 515 (2013) · Zbl 1272.65010 · doi:10.1093/imanum/drr059
[25] T. Shardlow, Weak convergence of a numerical method for a stochastic heat equation,, BIT, 43, 179 (2003) · Zbl 1031.65015 · doi:10.1023/A:1023661308243
[26] V. Thomée, <em>Galerkin Finite Element Methods for Parabolic Problems</em>,, Springer-Verlag (2006) · Zbl 1105.65102
[27] X. Wang, An exponential integrator scheme for time discretization of nonlinear stochastic wave equation,, J. Sci. Comput., 64, 234 (2015) · Zbl 1322.65022 · doi:10.1007/s10915-014-9931-0
[28] X. Wang, A Runge-Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise,, Numer. Algorithms, 62, 193 (2013) · Zbl 1267.65007 · doi:10.1007/s11075-012-9568-8
[29] X. Wang, Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise,, J. Math. Anal. Appl., 398, 151 (2013) · Zbl 1260.65006 · doi:10.1016/j.jmaa.2012.08.038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.