American Institute of Mathematical Sciences

Advanced
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

Xiaojie Wang 1,

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.
Keywords: exponential Euler scheme, weak convergence., additive noise, Semi-linear stochastic partial differential equations.
Mathematics Subject Classification: Primary: 60H35, 60H15; Secondary: 65C3.
Citation: Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481
References:
[1]

A. Andersson, R. Kruse and S. Larsson, Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE,, preprint , (). 

[2]

A. Andersson and S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation,, preprint , (). 

[3]

R. Anton, D. Cohen, S. Larsson and X. Wang, Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise,, preprint , (). 

[4]

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.

[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 (2014), 1-40. doi: 10.1007/s11118-013-9338-9.

[6]

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.

[7]

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.

[8]

D. Conus, A. Jentzen and R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients,, preprint , (). 

[9]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.

[10]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numerica, 19 (2010), 209-286. doi: 10.1017/S0962492910000048.

[16]

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.

[17]

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.

[18]

A. Jentzen and R. Kurniawan, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients,, preprint, (). 

[19]

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.

[20]

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.

[21]

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.

[22]

R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-02231-4.

[23]

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.

[24]

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.

[25]

T. Shardlow, Weak convergence of a numerical method for a stochastic heat equation, BIT, 43 (2003), 179-193. doi: 10.1023/A:1023661308243.

[26]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, 2006.

[27]

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.

[28]

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.

[29]

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.

show all references

References:
[1]

A. Andersson, R. Kruse and S. Larsson, Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE,, preprint , (). 

[2]

A. Andersson and S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation,, preprint , (). 

[3]

R. Anton, D. Cohen, S. Larsson and X. Wang, Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise,, preprint , (). 

[4]

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.

[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 (2014), 1-40. doi: 10.1007/s11118-013-9338-9.

[6]

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.

[7]

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.

[8]

D. Conus, A. Jentzen and R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients,, preprint , (). 

[9]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.

[10]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numerica, 19 (2010), 209-286. doi: 10.1017/S0962492910000048.

[16]

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.

[17]

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.

[18]

A. Jentzen and R. Kurniawan, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients,, preprint, (). 

[19]

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.

[20]

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.

[21]

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.

[22]

R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-02231-4.

[23]

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.

[24]

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.

[25]

T. Shardlow, Weak convergence of a numerical method for a stochastic heat equation, BIT, 43 (2003), 179-193. doi: 10.1023/A:1023661308243.

[26]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, 2006.

[27]

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.

[28]

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.

[29]

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.

[1]

Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353
[2]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[3]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23
[4]

Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control and Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013
[5]

Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure and Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631
[6]

Wenqiang Zhao. Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2499-2526. doi: 10.3934/dcdsb.2018065
[7]

Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations and Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017
[8]

Jesus Idelfonso Díaz, Jean Michel Rakotoson. On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037
[9]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209
[10]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[11]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[12]

Boling Guo, Guoli Zhou. On the backward uniqueness of the stochastic primitive equations with additive noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3157-3174. doi: 10.3934/dcdsb.2018305
[13]

Xiang Lv. Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2313-2323. doi: 10.3934/dcdsb.2021133
[14]

Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255
[15]

Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029
[16]

Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems and Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51
[17]

Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100
[18]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[19]

Xiaojun Li, Xiliang Li, Kening Lu. Random attractors for stochastic parabolic equations with additive noise in weighted spaces. Communications on Pure and Applied Analysis, 2018, 17 (3) : 729-749. doi: 10.3934/cpaa.2018038
[20]

Xiaobin Yao. Asymptotic behavior for stochastic plate equations with memory and additive noise on unbounded domains. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 443-468. doi: 10.3934/dcdsb.2021050

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (213)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy