Sparse polynomial chaos expansions of vector-valued response quantities

Abstract

Sparse polynomial chaos expansions have recently emerged in uncertainty quantification analysis as a tool to solve high dimensional problems, e.g. stochastic problems involving a few dozens to a few hundred random variables. Based on penalized regression analysis and the so-called least angle regression algorithm the method has proven efficiency in a number of applications. The approach was so far rather limited to scalar output quantities, e.g. quantities of interest that are post-processed from the solution of a stochastic partial differential equation (SPDE). In this paper we extend this approach to vector output quantities in order to obtain the complete solution field. This is carried out by using principal component analysis before computing the PC expansions of the various components. As a whole a complete non intrusive framework is obtained that is only based on a set of deterministic solutions of the underlying deterministic problem. The approach is illustrated by the computation of the displacement field of a tension rod with lognormal, spatially variable Young’s modulus. The problem exhibits 62 stochastic dimensions.