Stable interpolation with isotropic and anisotropic Gaussians using Hermite generating function

Abstract: Gaussian kernels can be an efficient and accurate tool for multivariate interpolation. In practice, high accuracies are often achieved in the flat limit where the interpolation matrix becomes increasingly ill-conditioned. Stable evaluation algorithms for isotropic Gaussians (Gaussian radial basis functions) have been proposed based on a Chebyshev expansion of the Gaussians by Fornberg, Larsson & Flyer and based on a Mercer expansion with Hermite polynomials by Fasshauer & McCourt. In this paper, we propose a new stabilization algorithm for the multivariate interpolation with isotropic or anisotropic Gaussians derived from the generating function of the Hermite polynomials. We also derive and analyse a new analytic cut-off criterion for the generating function expansion that allows to automatically adjust the number of the stabilizing basis functions.