Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression

By Jerrad Hampton, Alireza Doostan. 2014

158 citations recorded by the source corpus.

Independent sampling of orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models using Polynomial Chaos (PC) expansions. It is known that bounding the spectral radius of a random matrix consisting of PC samples yields a bound on the number of samples necessary to identify coefficients in the PC expansion via solution to a least squares regression problem. We present a related analysis which guarantees a mean square convergence using a coherence parameter of the sampled PC basis that may be both analytically bounded and computationally estimated. Utilizing asymptotic results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under each 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 PC basis. 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 such sampling schemes with identical support, and which guarantees recovery with a number of samples that is, up to a logarithmic factor, linear in the number of basis functions considered. 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 all observed cases the coherence-optimal sampling leads to similar or considerably improved accuracy over the other considered sampling distributions.

Subfields: Civil and Structural Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8362053acd16de660f29f66d1c477559cd8eb7af

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Citations in the corpus

Randomized numerical linear algebra: Foundations and algorithms (2020, background): “For some examples, see (Rauhut and Ward 2012, Cohen, Davenport and Leviatan 2013, Hampton and Doostan 2015, Rauhut and Ward 2016, Cohen and Migliorati 2017, Arras, Bachmayr and Cohen 2019, Avron, Kapralov, Musco, Musco, Velingker and Zandieh 2019, Chen and Price 2019).”
Least squares polynomial chaos expansion: A review of sampling strategies (2017, methodology): “In [44], the coherence parameter µ given by µ ( Y ) := sup is used to bound (probabilistically) the deviation of M from the identity for stability purposes as discussed in Section 3.1.”