  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.
