  We use the ideas of goal-oriented error estimation and adaptivity to design
and implement an efficient adaptive algorithm for approximating linear
quantities of interest derived from solutions to elliptic partial differential
equations (PDEs) with parametric or uncertain inputs. In the algorithm, the
stochastic Galerkin finite element method (sGFEM) is used to approximate the
solutions to primal and dual problems that depend on a countably infinite
number of uncertain parameters. Adaptive refinement is guided by an innovative
strategy that combines the error reduction indicators computed for spatial and
parametric components of the primal and dual solutions. The key theoretical
ingredient is a novel two-level a posteriori estimate of the energy error in
sGFEM approximations. We prove that this error estimate is reliable and
efficient. The effectiveness of the goal-oriented error estimation strategy and
the performance of the goal-oriented adaptive algorithm are tested numerically
for three representative model problems with parametric coefficients and for
three quantities of interest (including the approximation of pointwise values).
