NDLI: Least Squares Approximation of Polynomial Chaos Expansions With Optimized Grid Points

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Ghili, Saman Iaccarino, Gianluca
Copyright Year	2017
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.
Sponsorship	Lawrence Livermore National Laboratory
Starting Page	A1991
Ending Page	A2019
Page Count	29
File Format	PDF
ISSN	10648275
DOI	10.1137/15M1028303
e-ISSN	10957197
Issue Number	5
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-09-14
Access Restriction	Subscribed
Subject Keyword	Algorithms for functional approximation pseudospectral approximation uncertainty quantification Approximation by polynomials polynomial chaos expansion least squares approximation cubature rules Quadrature and cubature formulas Multidimensional problems
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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