Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

JOURNAL OF COMPUTATIONAL PHYSICS

卷:280

Article

Hampton, Jerrad¹  Doostan, Alireza¹ 

[1] Univ Colorado, Dept Aerosp Engn Sci, Boulder, CO 80309 USA

关键词: Compressive sampling;    Polynomial chaos;    Sparse approximation;    l(1)-minimization;    Markov Chain Monte Carlo;    Hermite polynomials;    Legendre polynomials;    Stochastic PDEs;    Uncertainty quantification;   

DOI  :  10.1016/j.jcp.2014.09.019

来源: Elsevier

【 摘 要 】

Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an l(1)-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free