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 weighted least-squares 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, M = O(N log(N )) 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.

Approximating smooth, multivariate functions on irregular domains

“…One solution to this problem is to employ discrete measures, supported over a fine grid that suitably fills Ω. This strategy, which uses ideas of [25], has been recently developed in [5,36]. Yet this procedure requires the domain Ω to be known in advance, and requires a fine grid to first be generated.…”