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Zbl 1014.65004
Xiu, Dongbin; Karniadakis, George Em
The Wiener--Askey polynomial chaos for stochastic differential equations. (English)
[J] SIAM J. Sci. Comput. 24, No.2, 619-644 (2002). ISSN 1064-8275; ISSN 1095-7197/e

Summary: We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos. Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error. Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte Carlo simulations for low dimensional stochastic inputs.

MSC 2000:: *65C30 Stochastic differential and integral equations
65C05 Monte Carlo methods
60H10 Stochastic ordinary differential equations
34F05 ODE with randomness
60H35 Computational methods for stochastic equations

Keywords: polynomial chaos; Askey scheme; orthogonal polynomials; stochastic differential equations; spectral methods; Galerkin projection; exponential convergence; numerical examples; Monte Carlo simulations

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