A rescaled method for RBF approximation

TL;DR

This paper analyzes a rescaled kernel interpolation method that improves stability and smoothness of RBF approximations by interpreting it as standard interpolation with a rescaled kernel, providing theoretical insights.

Contribution

It offers a systematic theoretical analysis of the rescaled RBF interpolation method, clarifying its stability and error properties.

Findings

Rescaled interpolation can be viewed as standard kernel interpolation with a rescaled kernel.

The method enhances stability and smoothness of RBF approximations.

Theoretical error and stability bounds are established.

Abstract

In the recent paper [8], a new method to compute stable kernel-based interpolants has been presented. This \textit{rescaled interpolation} method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allow us to consider its error and stability properties.