Goal-Oriented Error Estimation and Adaptivity for Stochastic Collocation FEM

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.