Goal-Oriented Error Estimation and Adaptivity for Stochastic Collocation FEM

TL;DR

This paper develops a goal-oriented adaptive strategy with novel error estimates for stochastic collocation FEM, improving the approximation of quantities of interest in elliptic PDEs with random inputs.

Contribution

It introduces a new adaptive method with reliable a posteriori error estimates and correction terms for stochastic collocation FEM targeting QoIs in elliptic PDEs.

Findings

Adaptive algorithm effectively approximates QoIs.

Error estimates improve accuracy in stochastic collocation.

Method handles affine and nonaffine parametric coefficients.

Abstract

We propose and analyze a general goal-oriented adaptive strategy for approximating quantities of interest (QoIs) associated with solutions to linear elliptic partial differential equations with random inputs. The QoIs are represented by bounded linear or continuously G\^ateaux differentiable nonlinear goal functionals, and the approximations are computed using the sparse grid stochastic collocation finite element method (SC-FEM). The proposed adaptive strategy relies on novel reliable a posteriori estimates of the errors in approximating QoIs. One of the key features of our error estimation approach is the introduction of a correction term into the approximation of QoIs in order to compensate for the lack of (global) Galerkin orthogonality in the SC-FEM setting. Computational results generated using the proposed adaptive algorithm are presented in the paper for representative elliptic problems with affine and nonaffine parametric coefficient dependence and for a range of linear and nonlinear goal functionals.