Article2011

A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions

Troy D. Butler · Clint N. Dawson · Timothy Michael Wildey

34 citations0/0 files storedSIAM Journal on Scientific Computing

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Abstract

We develop computable a posteriori error estimates for linear functionals of a solution to a general nonlinear stochastic differential equation with random model/source parameters. These error estimates are based on a variational analysis applied to stochastic Galerkin methods for forward and adjoint problems. The result is a representation for the error estimate as a polynomial in the random model/source parameter. The advantage of this method is that we use polynomial chaos representations for the forward and adjoint systems to cheaply produce error estimates by simple evaluation of a polynomial. By comparison, the typical method of producing such estimates requires repeated forward/adjoint solves for each new choice of random parameter. We present numerical examples showing that there is excellent agreement between these methods.

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References

2002The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations 2005An adaptive multi-element generalized polynomial chaos method for stochastic differential equations 2006Stochastic spectral methods for efficient Bayesian solution of inverse problems 2004Uncertainty propagation using Wiener–Haar expansions 2001Solution of stochastic partial differential equations using Galerkin finite element techniques 2004Multi-resolution analysis of Wiener-type uncertainty propagation schemes 1999Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach 2005Beyond Wiener–Askey Expansions: Handling Arbitrary PDFs

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2018Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification 2016Efficient Adaptive Stochastic Galerkin Methods for Parametric Operator Equations 2013Propagation of Uncertainties Using Improved Surrogate Models 2015A posteriori error estimation for elliptic partial differential equations with small uncertainties 2015

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