An adaptive residual sub-sampling algorithm for kernel interpolation based on maximum likelihood estimations

TL;DR

This paper introduces MPLE-RSM, an adaptive kernel interpolation method that automatically selects the RBF shape parameter using maximum likelihood estimation, ensuring unique interpolants and high accuracy across various functions.

Contribution

It presents a novel automatic shape parameter selection technique for RBF interpolation based on maximum likelihood, improving reliability and uniqueness over previous methods.

Findings

High accuracy in 1D and 2D benchmarks

Guaranteed unique RBF interpolants

Automatic parameter selection enhances reliability

Abstract

In this paper we propose an enhanced version of the residual sub-sampling method (RSM) in [9] for adaptive interpolation by radial basis functions (RBFs). More precisely, we introduce in the context of sub-sampling methods a maximum profile likelihood estimation (MPLE) criterion for the optimal selection of the RBF shape parameter. This choice is completely automatic, provides highly reliable and accurate results for any RBFs, and, unlike the original RSM, guarantees that the RBF interpolant exists uniquely. The efficacy of this new method, called MPLE-RSM, is tested by numerical experiments on some 1D and 2D benchmark target functions.