    January  2016, 36(1): 481-497. doi: 10.3934/dcds.2016.36.481

## Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus

 1 School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

Received  August 2014 Revised  March 2015 Published  June 2015

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.
Citation: Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481
##### References:
  A. Andersson, R. Kruse and S. Larsson, Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE,, preprint , ().   Google Scholar  A. Andersson and S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation,, preprint , ().   Google Scholar  R. Anton, D. Cohen, S. Larsson and X. Wang, Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise,, preprint , ().   Google Scholar  X. Bardina, M. Jolis and L. Quer-Sardanyons, Weak convergence for the stochastic heat equation driven by gaussian white noise, Electron. J. Probab, 15 (2010), 1267-1295. doi: 10.1214/EJP.v15-792.  Google Scholar  C. E. Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Anal., 40 (2014), 1-40. doi: 10.1007/s11118-013-9338-9.  Google Scholar  D. Cohen, S. Larsson and M. Sigg, A trigonometric method for the linear stochastic wave equation, SIAM J. Numer. Anal., 51 (2013), 204-222. doi: 10.1137/12087030X.  Google Scholar  D. Cohen and M. Sigg, Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations, Numer. Math., 121 (2012), 1-29. doi: 10.1007/s00211-011-0426-8.  Google Scholar  D. Conus, A. Jentzen and R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients,, preprint , ().   Google Scholar  G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.  Google Scholar  G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar  A. de Bouard and A. Debussche, Weak and strong order of convergence of a semi discrete scheme for the stochastic nonlinear Schrödinger equation, Appl. Math. Opt., 54 (2006), 369-399. doi: 10.1007/s00245-006-0875-0.  Google Scholar  A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp., 80 (2011), 89-117. doi: 10.1090/S0025-5718-2010-02395-6.  Google Scholar  A. Debussche and J. Printems, Weak order for the discretization of the stochastic heat equation, Math. Comp., 78 (2009), 845-863. doi: 10.1090/S0025-5718-08-02184-4.  Google Scholar  M. Geissert, M. Kovács and S. Larsson, Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise, BIT, 49 (2009), 343-356. doi: 10.1007/s10543-009-0227-y.  Google Scholar  M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numerica, 19 (2010), 209-286. doi: 10.1017/S0962492910000048.  Google Scholar  A. Jentzen, P. Kloeden and G. Winkel, Efficient simulation of nonlinear parabolic SPDEs with additive noise, Ann. Appl. Probab., 21 (2011), 908-950. doi: 10.1214/10-AAP711.  Google Scholar  A. Jentzen and P. E. Kloeden, 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 (2009), 649-667. doi: 10.1098/rspa.2008.0325.  Google Scholar  A. Jentzen and R. Kurniawan, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients,, preprint, ().   Google Scholar  P. E. Kloeden, G. J. Lord, A. Neuenkirch and T. Shardlow, The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds, J. Comput. Appl. Math., 235 (2011), 1245-1260. doi: 10.1016/j.cam.2010.08.011.  Google Scholar  M. Kovács, S. Larsson and F. Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise, BIT, 52 (2012), 85-108. doi: 10.1007/s10543-011-0344-2.  Google Scholar  M. Kovács, S. Larsson and F. Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes, BIT, 53 (2013), 497-525. doi: 10.1007/s10543-012-0405-1.  Google Scholar  R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-02231-4.  Google Scholar  F. Lindner and R. L. Schilling, Weak order for the discretization of the stochastic heat equation driven by impulsive noise, Potential Anal., 38 (2013), 345-379. doi: 10.1007/s11118-012-9276-y.  Google Scholar  G. J. Lord and A. Tambue, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise, IMA J. Numer. Anal., 33 (2013), 515-543. doi: 10.1093/imanum/drr059.  Google Scholar  T. Shardlow, Weak convergence of a numerical method for a stochastic heat equation, BIT, 43 (2003), 179-193. doi: 10.1023/A:1023661308243.  Google Scholar  V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, 2006. Google Scholar  X. Wang, An exponential integrator scheme for time discretization of nonlinear stochastic wave equation, J. Sci. Comput., 64 (2015), 234-263. doi: 10.1007/s10915-014-9931-0.  Google Scholar  X. Wang and S. Gan, A Runge-Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise, Numer. Algorithms, 62 (2013), 193-223. doi: 10.1007/s11075-012-9568-8.  Google Scholar  X. Wang and S. Gan, Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise, J. Math. Anal. Appl., 398 (2013), 151-169. doi: 10.1016/j.jmaa.2012.08.038.  Google Scholar

show all references

##### References:
  A. Andersson, R. Kruse and S. Larsson, Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE,, preprint , ().   Google Scholar  A. Andersson and S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation,, preprint , ().   Google Scholar  R. Anton, D. Cohen, S. Larsson and X. Wang, Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise,, preprint , ().   Google Scholar  X. Bardina, M. Jolis and L. Quer-Sardanyons, Weak convergence for the stochastic heat equation driven by gaussian white noise, Electron. J. Probab, 15 (2010), 1267-1295. doi: 10.1214/EJP.v15-792.  Google Scholar  C. E. Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Anal., 40 (2014), 1-40. doi: 10.1007/s11118-013-9338-9.  Google Scholar  D. Cohen, S. Larsson and M. Sigg, A trigonometric method for the linear stochastic wave equation, SIAM J. Numer. Anal., 51 (2013), 204-222. doi: 10.1137/12087030X.  Google Scholar  D. Cohen and M. Sigg, Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations, Numer. Math., 121 (2012), 1-29. doi: 10.1007/s00211-011-0426-8.  Google Scholar  D. Conus, A. Jentzen and R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients,, preprint , ().   Google Scholar  G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.  Google Scholar  G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar  A. de Bouard and A. Debussche, Weak and strong order of convergence of a semi discrete scheme for the stochastic nonlinear Schrödinger equation, Appl. Math. Opt., 54 (2006), 369-399. doi: 10.1007/s00245-006-0875-0.  Google Scholar  A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp., 80 (2011), 89-117. doi: 10.1090/S0025-5718-2010-02395-6.  Google Scholar  A. Debussche and J. Printems, Weak order for the discretization of the stochastic heat equation, Math. Comp., 78 (2009), 845-863. doi: 10.1090/S0025-5718-08-02184-4.  Google Scholar  M. Geissert, M. Kovács and S. Larsson, Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise, BIT, 49 (2009), 343-356. doi: 10.1007/s10543-009-0227-y.  Google Scholar  M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numerica, 19 (2010), 209-286. doi: 10.1017/S0962492910000048.  Google Scholar  A. Jentzen, P. Kloeden and G. Winkel, Efficient simulation of nonlinear parabolic SPDEs with additive noise, Ann. Appl. Probab., 21 (2011), 908-950. doi: 10.1214/10-AAP711.  Google Scholar  A. Jentzen and P. E. Kloeden, 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 (2009), 649-667. doi: 10.1098/rspa.2008.0325.  Google Scholar  A. Jentzen and R. Kurniawan, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients,, preprint, ().   Google Scholar  P. E. Kloeden, G. J. Lord, A. Neuenkirch and T. Shardlow, The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds, J. Comput. Appl. Math., 235 (2011), 1245-1260. doi: 10.1016/j.cam.2010.08.011.  Google Scholar  M. Kovács, S. Larsson and F. Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise, BIT, 52 (2012), 85-108. doi: 10.1007/s10543-011-0344-2.  Google Scholar  M. Kovács, S. Larsson and F. Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes, BIT, 53 (2013), 497-525. doi: 10.1007/s10543-012-0405-1.  Google Scholar  R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-02231-4.  Google Scholar  F. Lindner and R. L. Schilling, Weak order for the discretization of the stochastic heat equation driven by impulsive noise, Potential Anal., 38 (2013), 345-379. doi: 10.1007/s11118-012-9276-y.  Google Scholar  G. J. Lord and A. Tambue, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise, IMA J. Numer. Anal., 33 (2013), 515-543. doi: 10.1093/imanum/drr059.  Google Scholar  T. Shardlow, Weak convergence of a numerical method for a stochastic heat equation, BIT, 43 (2003), 179-193. doi: 10.1023/A:1023661308243.  Google Scholar  V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, 2006. Google Scholar  X. Wang, An exponential integrator scheme for time discretization of nonlinear stochastic wave equation, J. Sci. Comput., 64 (2015), 234-263. doi: 10.1007/s10915-014-9931-0.  Google Scholar  X. Wang and S. Gan, A Runge-Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise, Numer. Algorithms, 62 (2013), 193-223. doi: 10.1007/s11075-012-9568-8.  Google Scholar  X. Wang and S. Gan, Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise, J. Math. Anal. Appl., 398 (2013), 151-169. doi: 10.1016/j.jmaa.2012.08.038.  Google Scholar
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