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

Abstract

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 logarithmic factors, 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.

Time‐dependent global sensitivity analysis with active subspaces for a lithium ion battery model

“…Therefore, we advise oversampling—that is, choosing N > m +1. Recent work by Hampton and Doostan in deterministic least‐squares approximation with random evaluations shows that choosing produces a linear model that behaves like the best linear approximation in the continuous, mean‐squared sense.…”

An overview of uncertainty quantification techniques with application to oceanic and oil‐spill simulations

“…The choice of sampling points is a delicate matter since an ill‐conditioned matrix may lead to an unstable calculation. A number of recent articles [ Hampton and Doostan , ; Zhou et al ., ; Shin and Xiu , ; P. Seshadri et al, Optimal quadrature subsampling for least squares polynomial approximations, arXiv:1601.05470, 2016] have proposed optimal and stable sampling strategies for least squares polynomial approximations in high‐dimensional spaces.…”

Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits

“…Various coherence-based random sampling schemes has also been derived from the properties of Hermite and Legendre polynomials as proposed in [73].…”