Learning with Discrete Least Squares on Multivariate Polynomial Spaces Using Evaluations at Random or Low-Discrepancy Point Sets

Abstract

We review the results achieved in previous works [1, 2, 6, 8, 10---12] concerning the analysis of stability and accuracy of discrete least-squares approximation on multivariate polynomial spaces with noiseless evaluations at random points, and the results from [9] concerning the case of noiseless evaluations at low-discrepancy point sets. Afterwards, we present some numerical examples that confirm our theoretical findings and give some insights on their potential applications. The purpose of the numerical section is twofold: on the one hand we compare the performance of discrete least squares using random points versus low-discrepancy points; on the other hand we point out further directions of research, by showing what happens when we choose fewer evaluation points than those prescribed by our theoretical analysis.