Goal-oriented adaptivity for multilevel stochastic Galerkin FEM with nonlinear goal functionals

Abstract: This paper is concerned with the numerical approximation of quantities of interest associated with solutions to parametric elliptic partial differential equations (PDEs). The key novelty of this work is in its focus on the quantities of interest represented by continuously G^ateaux differentiable nonlinear functionals. We consider a class of parametric elliptic PDEs where the underlying differential operator has affine dependence on a countably infinite number of uncertain parameters. We design a goal-oriented adaptive algorithm for approximating nonlinear functionals of solutions to this class of parametric PDEs. In the algorithm, the approximations of parametric solutions to the primal and dual problems are computed using the multilevel stochastic Galerkin finite element method (SGFEM) and the adaptive refinement process is guided by reliable spatial and parametric error indicators that identify the dominant sources of error. We prove that the proposed algorithm generates multilevel SGFEM approximations for which the estimates of the error in the goal functional converge to zero. Numerical experiments for a selection of test problems and nonlinear quantities of interest demonstrate that the proposed goal-oriented adaptive strategy yields optimal convergence rates (for both the error estimates and the reference errors in the quantities of interest) with respect to the overall dimension of the underlying multilevel approximations spaces.