Weighted approximate Fekete points: Sampling for least-squares polynomial approximation

TL;DR

This paper introduces a weighted greedy algorithm for selecting sample points in multidimensional spaces to improve least-squares polynomial approximation, demonstrating theoretical and practical advantages over existing methods.

Contribution

It presents a novel weighted approximate Fekete points method using Christoffel function weights, with proven optimality properties and broad applicability.

Findings

Outperforms existing sampling designs in various dimensions

Almost always generates optimally-conditioned linear systems in 1D

Effective for both low and high-dimensional domains

Abstract

We propose and analyze a weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least-squares polynomial approximations on $L^2$ spaces weighted by a probability density function. Our procedure is a particular weighted version of the approximate Fekete points method, with the weight function chosen as the (inverse) Christoffel function. Our procedure has theoretical advantages: when linear systems with optimal condition number exist, the procedure finds them. In the one-dimensional setting with any density function, our greedy procedure almost always generates optimally-conditioned linear systems. Our method also has practical advantages: our procedure is impartial to compactness of the domain of approximation, and uses only pivoted linear algebraic routines. We show through numerous examples that our sampling design outperforms competing randomized and deterministic designs when the domain is both low and high dimensional.