Goal-oriented adaptive multilevel stochastic Galerkin FEM

This paper is concerned with the numerical approximation of quantities of interest associated with solutions to parametric elliptic partial differential equations (PDEs). 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 the functionals of solutions to this class of parametric PDEs. The algorithm applies to bounded linear goal functionals as well as to continuously G\^ateaux differentiable nonlinear functionals. 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 error estimates in the goal functional converge to zero. Furthermore, in the case of bounded linear goal functionals, we show that, under an appropriate saturation assumption, our goal-oriented adaptive strategy yields optimal convergence rates with respect to the overall dimension of the underlying multilevel approximations spaces.