A note on radial basis function computing

TL;DR

This paper introduces a new linear boundary knot method scheme for nonlinear convection-diffusion problems using RBFs, explores distance functions, and discusses matrix structures to improve computational efficiency and stability.

Contribution

It presents a novel linear RBF-based scheme for nonlinear problems, extends distance function concepts, and analyzes matrix structures to enhance RBF computation.

Findings

High accuracy and efficiency demonstrated in numerical results.

Symmetric node placement leads to structured matrices reducing computation.

A method to mitigate ill-conditioning in RBF matrices is proposed.

Abstract

This note carries three purposes involving our latest advances on the radial basis function (RBF) approach. First, we will introduce a new scheme employing the boundary knot method (BKM) to nonlinear convection-diffusion problem. It is stressed that the new scheme directly results in a linear BKM formulation of nonlinear problems by using response point-dependent RBFs, which can be solved by any linear solver. Then we only need to solve a single nonlinear algebraic equation for desirable unknown at some inner node of interest. The numerical results demonstrate high accuracy and efficiency of this nonlinear BKM strategy. Second, we extend the concepts of distance function, which include time-space and variable transformation distance functions. Finally, we demonstrate that if the nodes are symmetrically placed, the RBF coefficient matrices have either centrosymmetric or skew centrosymmetric structures. The factorization features of such matrices lead to a considerable reduction in the RBF computing effort. A simple approach is also presented to reduce the ill-conditioning of RBF interpolation matrices in general cases.