Near-Optimal Sampling Strategies for Multivariate Function Approximation on General Domains

Abstract

In this paper, we address the problem of approximating a multivariate function defined on a general domain in d dimensions from sample points. We consider weighted least-squares approximation in an arbitrary finite-dimensional space P from independent random samples taken according to a suitable measure. In general, least-squares approximations can be inaccurate and ill-conditioned when the number of sample points M is close to N = dim(P). To counteract this, we introduce a novel method for sampling in general domains which leads to provably accurate and well-conditioned approximations. The resulting sampling measure is discrete and therefore straightforward to sample from. Our main result shows near-optimal sample complexity for this procedure; specifically,(Formula presented) samples suffice for a well-conditioned and accurate approximation. Numerical experiments on polynomial approximation in general domains confirm the benefits of this method over standard sampling.