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

Authors:
T. Butler;C. Dawson;T. Wildey
Affiliations:
tbutler@ices.utexas.edu and clint@ices.utexas.edu;-;tmwilde@sandia.gov
Venue:
SIAM Journal on Scientific Computing
Year:
2011

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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.