Adaptive Smolyak Pseudospectral Approximations

Abstract

Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for nonintrusive pseudospectral approximation, based on Smolyak's algorithm with generalized sparse grids. We rigorously analyze and extend the nonadaptive method proposed in [P. G. Constantine, M. S. Eldred, and E. T. Phipps, Comput. Methods Appl. Mech. Engrg., 229--232 (2012), pp. 1--12], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that $\mathcal{O}(1)$ errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal aliasing and makes far more effective use of sparse function evaluations. These results are applicable to broad choices of quadrature rule and generalized sparse grids. Exploiting this flexibility, we introduce a greedy heuristic for adaptive refinement of the pseudospectral approximation. We numerically demonstrate convergence of the algorithm on the Genz test functions, and illustrate the accuracy and efficiency of the adaptive approach on a realistic chemical kinetics problem.