STABLE AND ACCURATE LEAST SQUARES RADIAL BASIS FUNCTION APPROXIMATIONS ON BOUNDED DOMAINS
Document Type
Article
Publication Date
2024
Department
Department of Mathematical Sciences
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 neighboring 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.
Publication Title
SIAM Journal on Numerical Analysis
Recommended Citation
Adcock, B.,
Huybrechs, D.,
&
Piret, C. M.
(2024).
STABLE AND ACCURATE LEAST SQUARES RADIAL BASIS FUNCTION APPROXIMATIONS ON BOUNDED DOMAINS.
SIAM Journal on Numerical Analysis,
62(6), 2698-2718.
http://doi.org/10.1137/23M1593243
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p2/1303
