  In this paper we present an algorithm for adaptive sparse grid approximations
of quantities of interest computed from discretized partial differential
equations. We use adjoint-based a posteriori error estimates of the physical
discretization error and the interpolation error in the sparse grid to enhance
the sparse grid approximation and to drive adaptivity of the sparse grid.
Utilizing these error estimates provides significantly more accurate functional
values for random samples of the sparse grid approximation. We also demonstrate
that alternative refinement strategies based upon a posteriori error estimates
can lead to further increases in accuracy in the approximation over traditional
hierarchical surplus based strategies. Throughout this paper we also provide
and test a framework for balancing the physical discretization error with the
stochastic interpolation error of the enhanced sparse grid approximation.
