NDLI: Local Polynomial Chaos Expansion for Linear Differential Equations with High Dimensional Random Inputs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Chen, Yi Gittelson, Claude Jakeman, John Xiu, Dongbin
Copyright Year	2015
Abstract	In this paper we present a localized polynomial chaos expansion for partial differential equations (PDE) with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. The local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and, more importantly, in a much lower dimensional random space. In a postprocesing stage, accurate samples of the original stochastic problems are obtained from the samples of the local solutions by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient. In this paper we present the general mathematical framework of the methodology and use numerical examples to demonstrate the properties of the method.
Starting Page	A79
Ending Page	A102
Page Count	24
File Format	PDF
ISSN	10648275
DOI	10.1137/140970100
e-ISSN	10957197
Issue Number	1
Volume Number	37
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-01-08
Access Restriction	Subscribed
Subject Keyword	Interpolation uncertainty quantification Approximation by polynomials domain decomposition stochastic differential equation Orthogonal functions and polynomials, general theory generalized polynomial chaos
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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