Stable evaluation of Gaussian radial basis functions using Hermite polynomials
For researchers in multivariate interpolation, this provides a simpler and more general stabilization method compared to existing approaches.
The paper presents a new stable algorithm for evaluating Gaussian radial basis functions using Hermite polynomials, achieving high accuracy without complex parameter tuning and extending naturally to high-dimensional tensor grids and anisotropic bases.
Gaussian radial basis functions can be an accurate basis for multivariate interpolation. In practise, high accuracies are often achieved in the flat limit where the interpolation matrix becomes increasingly ill-conditioned. Stable evaluation algorithms have been proposed by Fornberg, Larsson & Flyer based on a Chebyshev expansion of the Gaussian basis and by Fasshauer & McCourt based on a Mercer expansion with Hermite polynomials. In this paper, we propose another stabilization algorithm based on Hermite polynomials but derived from the generating function of Hermite polynomials. The new expansion does not require a complicated choice of parameters and offers a simple extension to high-dimensional tensor grids as well as a generalization for anisotropic multivariate basis functions using the Hagedorn generating function.