Articlebronze2011

On the convergence of generalized polynomial chaos expansions

Oliver G. Ernst · Antje Mugler · Hans-Jörg Starkloff · Elisabeth Ullmann

414 citations0/0 files storedESAIM Mathematical Modelling and Numerical Analysis

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Probabilistic and Robust Engineering Design Mathematical functions and polynomials Mathematical Approximation and Integration

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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References

1991Stochastic Finite Elements: A Spectral Approach 2002The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations 2005High-Order Collocation Methods for Differential Equations with Random Inputs 2003Modeling uncertainty in flow simulations via generalized polynomial chaos 2004Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations 1951Multiple Wiener Integral 2004Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure 2002Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos

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