Stable computations with Gaussian radial basis functions

A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned 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 been formulated only for the surface of the sphere. This paper focuses on an RBF-QR formulation for node sets in one, two, and three dimensions. The algorithm is stable for arbitrarily small shape parameters.