Nonparametric density estimation for randomly perturbed elliptic problems II: Applications and adaptive modeling

Abstract

In this paper, we develop and apply an efficient adaptive algorithm for computing the propagation of uncertainty into a quantity of interest computed from numerical solutions of an elliptic partial differential equation with a randomly perturbed diffusion coefficient. The algorithm is well-suited for problems for which limited information about the random perturbations is available and when an approximation of the probability distribution of the output is desired. We employ a nonparametric density estimation approach based on a very efficient method for computing random samples of elliptic problems described and analyzed in (SIAM J. Sci. Comput. 2008. DOI: JCOMP-D-08-00261). We fully develop the adaptive algorithm suggested by the analysis in that paper, discuss details of its implementation, and illustrate its behavior using a realistic data set. Finally, we extend the analysis to include a 'modeling error' term that accounts for the effects of the resolution of the statistical description of the random variation. We modify the adaptive algorithm to adapt the resolution of the statistical description and illustrate the behavior of the adaptive algorithm in several examples. © 2009 John Wiley & Sons, Ltd.