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Numerical solutions of stochastic PDEs driven by arbitrary type of noise

Article

Abstract

So far the theory and numerical practice of stochastic partial differential equations (SPDEs) have dealt almost exclusively with Gaussian noise or Lévy noise. Recently, Mikulevicius and Rozovskii (Stoch Partial Differ Equ Anal Comput 4:319–360, 2016) proposed a distribution-free Skorokhod–Malliavin calculus framework that is based on generalized stochastic polynomial chaos expansion, and is compatible with arbitrary driving noise. In this paper, we conduct systematic investigation on numerical results of these newly developed distribution-free SPDEs, exhibiting the efficiency of truncated polynomial chaos solutions in approximating moments and distributions. We obtain an estimate for the mean square truncation error in the linear case. The theoretical convergence rate, also verified by numerical experiments, is exponential with respect to polynomial order and cubic with respect to number of random variables included.

Keywords

Distribution-free Stochastic PDE Stochastic polynomial chaos Wick product Skorokhod integral 

Notes

Acknowledgements

The authors would like to thank Michael Tretyakov and Zhongqiang Zhang for helpful discussions on the relation between commutativity and K-version convergence.

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Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

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