Compressive Sampling of Polynomial Chaos Expansions: Convergence Analysis and Sampling Strategies

TL;DR

This paper analyzes convergence and sampling strategies for Polynomial Chaos expansions in uncertainty quantification, proposing coherence-based bounds and a coherence-optimal MCMC sampling method that improves accuracy in high-dimensional and high-order problems.

Contribution

It introduces coherence bounds for polynomial bases, importance sampling distributions, and a coherence-optimal MCMC method for improved sampling efficiency.

Findings

Coherence bounds are derived for Hermite and Legendre polynomials.

Importance sampling reduces dependence on polynomial order.

Coherence-optimal sampling improves accuracy in high-dimensional problems.

Abstract

Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with high-dimensional random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as {\it coherence}, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an $\ell_1$-minimization problem. Utilizing asymptotic results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under the 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 {\it 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.