| The Radial Basis Function (RBF) method is one of the primary tools for interpolating multidimensional scattered data. The methods' ability to handle arbitrarily scattered data, to easily generalize to several space dimensions, and to provide spectral accuracy have made it particularly popular in several different types of applications. Some of the more recent of these applications include cartography, neural networks, medical imaging, and the numerical solution of partial differential equations (PDEs). In this thesis we study three issues with the RBF method that have received very little attention in the literature.; First, we focus on the behavior of RBF interpolants near boundaries. Like most interpolation methods, a common feature of the RBF method is how relatively inaccurate the interpolants are near boundaries. Such boundary induced errors can severely limit the utility of the RBF method for numerically solving certain PDEs. With that as motivation, we investigate the behavior of RBF interpolants near boundaries and propose the first practical techniques for ameliorating the errors there.; We next focus on some numerical developments for the RBF method based on infinitely smooth RBFs. Most infinitely smooth RBFs feature a free “shape” parameter ϵ such that, as the magnitude of ϵ decreases, the RBFs become increasingly flat. While small values of ϵ typically result in more accurate interpolants, the direct method of computing the interpolants suffers from severe numerical ill-conditioning as ϵ → 0. Until recently, this ill-conditioning has severely limited the range of ϵ that could be considered in the RBF method. We present a novel numerical approach that largely overcomes the numerical ill-conditioning and allows for the stable computation of RBF interpolants for all values of ϵ, including the limiting ϵ = 0 case. This new method provides the first tool for the numerical exploration of RBF interpolants as ϵ → 0.; The third focus of the thesis is on the behavior of RBF interpolants as ϵ → 0. In most cases the interpolants converge to a finite degree multivariate polynomial interpolant as ϵ → 0. However, in rare situations the interpolants may diverge. We investigate this phenomenon in great detail both numerically and analytically, and link it directly to the failure of a condition known as “polynomial unisolvency”. We also find that the Gaussian RBF is inherently different from the other standard infinitely smooth RBFs in that it appears to result in an interpolant that never diverges as ϵ → 0.; We conclude with a brief overview of two future research opportunities related to the topics of the thesis. The first involves using RBF interpolants to generate scattered-node finite difference formulas. The second involves using RBF interpolants to generate linear multistep methods for solving ordinary differential equations. |
