Stable Evaluation of Gaussian Radial Basis Function Interpolants

We provide a new way to compute and evaluate Gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter, i.e., for “flat” kernels. This work is motivated by the fundamental ideas proposed earlier by Bengt Fornberg and his coworkers. However, following Mercer's theorem, an L₂(Rᵈ, ρ)-orthonormal expansion of the Gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by Fornberg, Larsson, and Flyer and that is applicable in arbitrary space dimensions d. In addition to obtaining an accurate approximation of the radial basis function interpolant (using many terms in the series expansion of the kernel), we also propose and investigate a highly accurate least-squares approximation based on early truncation of the kernel expansion.