Commun. Comput. Phys., 5 (2009), pp. 242-272.

Fast Numerical Methods for Stochastic Computations: A Review

Dongbin Xiu 1*

1 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA.

Received 18 January 2008; Accepted (in revised version) 20 May 2008
Available online 1 August 2008


This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations. The focus is on efficient high-order methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework of gPC is reviewed, along with its Galerkin and collocation approaches for solving stochastic equations. Properties of these methods are summarized by using results from literature. This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multi-dimensional random spaces.

AMS subject classifications: 41A10, 60H35, 65C30, 65C50

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Key words: Stochastic differential equations, generalized polynomial chaos, uncertainty quantification, spectral methods.

*Corresponding author.
Email: (D. Xiu)

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