The Interpolation Theory of Radial Basis Functions

TL;DR

This work develops a comprehensive theory for radial basis function interpolation, exploring conditions for solvability, spectral properties of associated matrices, and efficient solution methods, with applications to multiquadric functions and band-limited approximation.

Contribution

It introduces new results on the interpolation capabilities for different p-norms, spectral bounds for Toeplitz matrices, and fast algorithms for solving interpolation systems.

Findings

Interpolation is always possible for 1<p<2 with distinct points.

Spectral bounds for Toeplitz matrices enable efficient preconditioning.

Large shape parameters in multiquadrics relate to sinc functions, aiding band-limited function approximation.

Abstract

In this dissertation, it is first shown that, when the radial basis function is a $p$-norm and $1 < p < 2$, interpolation is always possible when the points are all different and there are at least two of them. We then show that interpolation is not always possible when $p > 2$. Specifically, for every $p > 2$, we construct a set of different points in some $\Rd$ for which the interpolation matrix is singular. The greater part of this work investigates the sensitivity of radial basis function interpolants to changes in the function values at the interpolation points. Our early results show that it is possible to recast the work of Ball, Narcowich and Ward in the language of distributional Fourier transforms in an elegant way. We then use this language to study the interpolation matrices generated by subsets of regular grids. In particular, we are able to extend the classical theory of Toeplitz operators to calculate sharp bounds on the spectra of such matrices. Applying our understanding of these spectra, we construct preconditioners for the conjugate gradient solution of the interpolation equations. Our main result is that the number of steps required to achieve solution of the linear system to within a required tolerance can be independent of the number of interpolation points. The Toeplitz structure allows us to use fast Fourier transform techniques, which imp lies that the total number of operations is a multiple of $n \log n$, where $n$ is the number of interpolation points. Finally, we use some of our methods to study the behaviour of the multiquadric when its shape parameter increases to infinity. We find a surprising link with the {\it sinus cardinalis} or {\it sinc} function of Whittaker. Consequently, it can be highly useful to use a large shape parameter when approximating band-limited functions.