Volume 36, Issue 2
Weak Error Estimates for Trajectories of SPDEs Under Spectral Galerkin Discretization

Charles-Edouard Bréhier, Martin Hairer & Andrew M. Stuart

J. Comp. Math., 36 (2018), pp. 159-182.

Published online: 2018-04

[An open-access article; the PDF is free to any online user.]

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  • Abstract

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories: test functions are regular, but depend on the values of the process on the interval [0, T].

We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Itô map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Itô map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest.

We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

  • Keywords

Stochastic Partial Differential Equations, Weak approximation, Spectral Galerkin discretization.

  • AMS Subject Headings

60H15, 65C30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

brehier@math.univ-lyon1.fr (Charles-Edouard Bréhier)

m.hairer@imperial.ac.uk (Martin Hairer)

astuart@caltech.edu (Andrew M. Stuart)

  • BibTex
  • RIS
  • TXT
@Article{JCM-36-159, author = {Bréhier , Charles-Edouard and Hairer , Martin and Stuart , Andrew M. }, title = {Weak Error Estimates for Trajectories of SPDEs Under Spectral Galerkin Discretization}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {2}, pages = {159--182}, abstract = {

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories: test functions are regular, but depend on the values of the process on the interval [0, T].

We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Itô map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Itô map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest.

We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1607-m2016-0539}, url = {http://global-sci.org/intro/article_detail/jcm/12254.html} }
TY - JOUR T1 - Weak Error Estimates for Trajectories of SPDEs Under Spectral Galerkin Discretization AU - Bréhier , Charles-Edouard AU - Hairer , Martin AU - Stuart , Andrew M. JO - Journal of Computational Mathematics VL - 2 SP - 159 EP - 182 PY - 2018 DA - 2018/04 SN - 36 DO - http://doi.org/10.4208/jcm.1607-m2016-0539 UR - https://global-sci.org/intro/article_detail/jcm/12254.html KW - Stochastic Partial Differential Equations, Weak approximation, Spectral Galerkin discretization. AB -

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories: test functions are regular, but depend on the values of the process on the interval [0, T].

We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Itô map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Itô map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest.

We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

Charles-Edouard Bréhier, Martin Hairer & Andrew M. Stuart. (2020). Weak Error Estimates for Trajectories of SPDEs Under Spectral Galerkin Discretization. Journal of Computational Mathematics. 36 (2). 159-182. doi:10.4208/jcm.1607-m2016-0539
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