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In this paper we present an algorithm for adaptive sparse grid approximationsof quantities of interest computed from discretized partial differentialequations. We use adjoint-based a posteriori error estimates of the physicaldiscretization error and the interpolation error in the sparse grid to enhancethe sparse grid approximation and to drive adaptivity of the sparse grid.Utilizing these error estimates provides significantly more accurate functionalvalues for random samples of the sparse grid approximation. We also demonstratethat alternative refinement strategies based upon a posteriori error estimatescan lead to further increases in accuracy in the approximation over traditionalhierarchical surplus based strategies. Throughout this paper we also provideand test a framework for balancing the physical discretization error with thestochastic interpolation error of the enhanced sparse grid approximation.

Enhancing adaptive sparse grid approximations and improving refinement strategies using adjoint-based a posteriori error estimates