  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.
