Stable interpolation with isotropic and anisotropic Gaussians using Hermite generating function
For practitioners using Gaussian kernels in interpolation, this method improves numerical stability without sacrificing accuracy, though it is an incremental improvement over existing stabilization techniques.
The paper proposes a new stabilization algorithm for multivariate interpolation with isotropic or anisotropic Gaussians, derived from the Hermite polynomial generating function, and introduces an analytic cut-off criterion to automatically adjust the number of basis functions.
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.