Stabilized interpolation using radial basis functions augmented with selected radial polynomials

Abstract

Infinitely smooth radial basis functions (RBFs) have a shape parameter that controls their shapes. When using these RBFs (e.g., the Gaussian RBF) for interpolation problems, we have ill-conditioning when the shape parameter is very small, while in some cases small shape parameters lead to high accuracy. In this study, we are going to reduce the effect of the ill-conditioning of the infinitely smooth RBFs. We propose a new basis augmenting the infinitely smooth RBFs at different locations with radial polynomials of different even powers. Numerical experiments show that the new basis is stable for all values of the shape parameter.