Least Squares Approximation of Polynomial Chaos Expansions With Optimized Grid Points
Pages A1991 - A2019
Abstract
In approximating a function by its truncated polynomial chaos expansion (PCE), the target function is projected onto a space spanned by a finite set of orthonormal basis polynomials. Nonintrusive or pseudospectral methods approximate this projection using point evaluations of the target function. One nonintrusive strategy uses the weighted least squares approximation. For this method, like many other nonintrusive methods, the quality of the results depends crucially on the points and weights. The aim of this paper is to find points and weights that lead to excellent accuracy and stability of the weighted least squares approximation. We start by studying the connection between some of the most efficient cubature rules and weighted least squares approximations, and we use this connection to formulate an optimization problem for finding the desired points and weights. We also give a practical algorithm for solving this optimization problem. We then use these points and weights in our numerical experiments to approximate the PCE coefficients for various target functions and polynomial approximation spaces. The results suggest that by using these points and weights, an optimal convergence rate can be achieved even when the number of points scales linearly with (and is only slightly lager than) the dimension of the polynomial space.
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Published: 01 January 2017
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