A Matlab code for stable differentiation of radial basis function approximations

Abstract

Radial basis function approximation allows for fitting of scattered data and for solving partial differential equations in non-trivial geometries. This requires solving potentially ill-conditioned linear systems of equations. The RBF-QR class of methods provides a change of basis that results in well-conditioned matrices in the same approximation space. In this paper, we derive improved algorithms for differentiation in the RBF-QR basis that solves some accuracy issues of the original implementation. We provide all second derivatives in three spatial dimensions, while previously only the Laplacian was implemented. We also provide a high level interface to compute differentiation matrices using either RBF-QR or the direct evaluation method.