$\mathcal{H}$-inverses for RBF interpolation

TL;DR

This paper demonstrates that the inverse of the system matrix for RBF interpolation, including polyharmonic splines, can be efficiently approximated using $ ext{H}$-matrices with exponential convergence, enhancing computational efficiency.

Contribution

It introduces a method to approximate the inverse of RBF interpolation matrices in the $ ext{H}$-matrix format with exponential accuracy, applicable to classical and new RBFs.

Findings

Exponential convergence of inverse approximation in $ ext{H}$-matrix format.

Applicability to classical polyharmonic splines and other RBFs.

Efficient computational approach for RBF interpolation matrices.

Abstract

We consider the interpolation problem for a class of radial basis functions (RBFs) that includes the classical polyharmonic splines (PHS). We show that the inverse of the system matrix for this interpolation problem can be approximated at an exponential rate in the block rank in the $\mathcal{H}$-matrix format if the block structure of the $\mathcal{H}$-matrix arises from a standard clustering algorithm.