A New Version of the Adaptive Fast Gauss Transform for Discrete and Continuous Sources

SIAM Review, Volume 66, Issue 2, Page 287-315, May 2024.
We present a new version of the fast Gauss transform (FGT) for discrete and continuous sources. Classical Hermite expansions are avoided entirely, making use only of the plane-wave representation of the Gaussian kernel and a new hierarchical merging scheme. For continuous source distributions sampled on adaptive tensor product grids, we exploit the separable structure of the Gaussian kernel to accelerate the computation. For discrete sources, the scheme relies on the nonuniform fast Fourier transform (NUFFT) to construct near field plane-wave representations. The scheme has been implemented for either free-space or periodic boundary conditions. In many regimes, the speed is comparable to or better than that of the conventional FFT in work per grid point, despite being fully adaptive.