On Local RBF Approximation

TL;DR

This paper investigates radial basis function (RBF) approximation with nonuniform centers, introducing an alternative local density characterization to improve center placement and error estimation.

Contribution

It provides an equivalent, practical characterization of local density for RBF centers, enabling better placement at varying resolutions and improved error analysis.

Findings

New local density characterization for RBF centers

Effective placement strategies for nonuniform centers

Comparison of error estimates with previous works

Abstract

The purpose of this paper is to investigate RBF approximation with highly nonuniform centers. Recently, DeVore and Ron have developed a notion of the local density of a set of centers -- a notion that permits precise pointwise error estimates for surface spline approximation. We give an equivalent, alternative characterization of local density, one that allows effective placement of centers at different resolutions. We compare, also, the pointwise results of DeVore--Ron to previously works of Wu and Schaback and of Duchon.