Coherence Motivated Sampling and Convergence Analysis of Least-Squares   Polynomial Chaos Regression
  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. 
Jerrad Hampton, Alireza Doostan
math.ST stat.TH