Stable and accurate least squares radial basis function approximations on bounded domains

TL;DR

This paper investigates stable and accurate least squares RBF approximation methods on bounded domains, focusing on the Gaussian RBF and optimal shape parameter scaling to achieve high precision even with ill-conditioned systems.

Contribution

It introduces a linear scaling of the shape parameter with degrees of freedom, providing explicit constants and extending the approach to elliptic boundary value problems.

Findings

Linear shape parameter scaling yields stable approximations.

High accuracy close to machine precision is achievable.

Method extends to solving elliptic boundary value problems.

Abstract

The computation of global radial basis function (RBF) approximations requires the solution of a linear system which, depending on the choice of RBF parameters, may be ill-conditioned. We study the stability and accuracy of approximation methods using the Gaussian RBF in all scaling regimes of the associated shape parameter. The approximation is based on discrete least squares with function samples on a bounded domain, using RBF centers both inside and outside the domain. This results in a rectangular linear system. We show for one-dimensional approximations that linear scaling of the shape parameter with the degrees of freedom is optimal, resulting in constant overlap between neighbouring RBF's regardless of their number, and we propose an explicit suitable choice of the proportionality constant. We show numerically that highly accurate approximations to smooth functions can also be obtained on bounded domains in several dimensions, using a linear scaling with the degrees of freedom per dimension. We extend the least squares approach to a collocation-based method for the solution of elliptic boundary value problems and illustrate that the combination of centers outside the domain, oversampling and optimal scaling can result in accuracy close to machine precision in spite of having to solve very ill-conditioned linear systems.