Weighted discrete least-squares polynomial approximation using randomized quadratures

JOURNAL OF COMPUTATIONAL PHYSICS

卷:298

Article

Zhou, Tao¹  Narayan, Akil^2,3  Xiu, Dongbin^2,3 

[1] Chinese Acad Sci, AMSS, Inst Computat Math & Sci Engn Comp, Beijing, Peoples R China

[2] Univ Utah, Dept Math, Salt Lake City, UT 84112 USA

[3] Univ Utah, Sci Comp & Imaging Inst, Salt Lake City, UT 84112 USA

关键词: Least squares method;    Orthogonal polynomials;    Generalized polynomial chaos;    Uncertainty quantification;   

DOI  :  10.1016/j.jcp.2015.06.042

来源: Elsevier

【 摘 要 】

We discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation. The problem stems from uncertainty quantification (UQ), where the independent variables of the functions are random variables with specified probability measure. We propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product Gaussian quadrature points. We analyze the stability properties of this method and prove that the method is asymptotically stable, provided that the number of points scales linearly (up to a logarithmic factor) with the cardinality of the polynomial space. Specific results in both bounded and unbounded domains are obtained, along with a convergence result for Chebyshev measure. Numerical examples are provided to verify the theoretical results. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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