On the convergence of generalized polynomial chaos expansions

Oliver G. Ernst; Antje Mugler; Hans Jörg Starkloff; Elisabeth Ullmann

doi:10.1051/m2an/2011045

On the convergence of generalized polynomial chaos expansions

Oliver G. Ernst, Antje Mugler, Hans Jörg Starkloff, Elisabeth Ullmann

Research output: Contribution to journal › Article › peer-review

316 Scopus citations

Abstract

A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.

Original language	English
Pages (from-to)	317-339
Number of pages	23
Journal	Mathematical Modelling and Numerical Analysis
Volume	46
Issue number	2
DOIs	https://doi.org/10.1051/m2an/2011045
State	Published - Mar 2012
Externally published	Yes

Keywords

Determinate measure
Equations with random data
Generalized polynomial chaos
Moment problem
Polynomial chaos
Spectral elements
Stochastic Galerkin method
Wiener integral
Wiener-Hermite expansion

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abstract = "A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.",

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AB - A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.

KW - Determinate measure

KW - Equations with random data

KW - Generalized polynomial chaos

KW - Moment problem

KW - Polynomial chaos

KW - Spectral elements

KW - Stochastic Galerkin method

KW - Wiener integral

KW - Wiener-Hermite expansion

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JF - Mathematical Modelling and Numerical Analysis

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ER -

On the convergence of generalized polynomial chaos expansions

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Keywords

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