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

Butler, T.; Dawson, C.; Wildey, T.

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

Type

Article

Authors

Butler, T.
Dawson, C.
Wildey, T.

Date

2011-01

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. © 2011 Society for Industrial and Applied Mathematics.

Citation

Butler T, Dawson C, Wildey T (2011) A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions. SIAM Journal on Scientific Computing 33: 1267–1291. Available: http://dx.doi.org/10.1137/100795760.

Acknowledgements

Submitted to the journal's Methods and Algorithms for Scientific Computing section May 18, 2010; accepted for publication (in revised form) March 2, 2011; published electronically June 7, 2011. This work was made possible with funding from the King Abdullah University of Science and Technology (KAUST).Sandia National Labs, Albuquerque, NM 87185 (tmwilde@sandia.gov). Sandia is a multiprogram laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the United States Department of Energy's National Nuclear Security Administration under contract DE-AC04-94-AL85000. This author's work was partially supported by NSF grant DMS 0618679.