On the convergence of generalized polynomial chaos expansions

• Published in: ESAIM. Mathematical modelling and numerical analysis Vol. 46; no. 2; pp. 317 - 339
• Main Authors: Ernst, Oliver G., Mugler, Antje, Starkloff, Hans-Jörg, Ullmann, Elisabeth
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.03.2012
• Subjects: 33C45; 35R60; 40A30; 41A10; 60H35; 65N30; Approximation; Chaos theory; Complement; Convergence; determinate measure; Equations with random data; Exact sciences and technology; generalized polynomial chaos; Integrals; Mathematical analysis; Mathematical models; Mathematics; moment problem; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; polynomial chaos; Polynomials; Random variables; Sciences and techniques of general use; Special functions; spectral elements; stochastic Galerkin method; Stochastic models; Studies; Wiener integral; Wiener–Hermite expansion; Chaos; Numerical analysis; Boundary value problem; Initial value problem; Probability distribution; Mathematical model; Random variable; Partial differential equation; Orthogonal polynomial; Convergence
• ISSN: 0764-583X; 1290-3841
• DOI: 10.1051/m2an/2011045

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.