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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 |
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. |
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