Stable Computations with Gaussian Radial Basis Functions

Radial basis function (RBF) approximation is an extremely powerful tool for representing smooth functions in non-trivial geometries, since the method is meshfree and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly illconditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. Two stable approaches that overcome this problem exist, the Contour-Padé method and the RBF-QR method. However, the former is limited to small node sets and the latter has until now only been formulated for the surface of the sphere. This paper focuses on an RBF-QR formulation for node sets in 1-D, 2-D, and 3-D. The algorithm is stable for arbitrarily small shape parameters. It can be used for thousands of node points in 2-D and more still in 3-D. A sample matlab code for the 2-D case is provided.