Stable computations with flat radial basis functions using vector-valued rational approximations

TL;DR

This paper introduces a new rational approximation algorithm for stable and accurate computation with flat radial basis functions, applicable to various kernels and tasks, including solving PDEs.

Contribution

The paper presents RBF-RA, a novel rational approximation method that overcomes ill-conditioning in flat RBF computations, outperforming existing methods in accuracy and versatility.

Findings

RBF-RA achieves higher accuracy than traditional methods.

The method is robust and applicable to various smooth kernels.

Demonstrated effectiveness in solving 3D Poisson problems.

Abstract

One commonly finds in applications of smooth radial basis functions (RBFs) that scaling the kernels so they are `flat' leads to smaller discretization errors. However, the direct numerical approach for computing with flat RBFs (RBF-Direct) is severely ill-conditioned. We present an algorithm for bypassing this ill-conditioning that is based on a new method for rational approximation (RA) of vector-valued analytic functions with the property that all components of the vector share the same singularities. This new algorithm (RBF-RA) is more accurate, robust, and easier to implement than the Contour-Pad\'e method, which is similarly based on vector-valued rational approximation. In contrast to the stable RBF-QR and RBF-GA algorithms, which are based on finding a better conditioned base in the same RBF-space, the new algorithm can be used with any type of smooth radial kernel, and it is also applicable to a wider range of tasks (including calculating Hermite type implicit RBF-FD stencils). We present a series of numerical experiments demonstrating the effectiveness of this new method for computing RBF interpolants in the flat regime. We also demonstrate the flexibility of the method by using it to compute implicit RBF-FD formulas in the flat regime and then using these for solving Poisson's equation in a 3D spherical shell.