Research Spotlights

Abstract

SIAM Review, Volume 66, Issue 2, Page 285-285, May 2024.
The Gauss transform---convolution with a Gaussian in the continuous case and the sum of $N$ Gaussians at $M$ points in the discrete case---is ubiquitous in applied mathematics, from solving ordinary and partial differential equations to probability density estimation to science applications in astrophysics, image processing, quantum mechanics, and beyond. For the discrete case, the fast Gauss transform (FGT) enables the approximate calculation of the sum of $N$ Gaussians at $M$ points in order $N + M$ (instead of $NM$) operations by a fast summation strategy, which shares work between the sums at different points, similarly to the fast multipole method. In this issue's Research Spotlights section, “A New Version of the Adaptive Fast Gauss Transform for Discrete and Continuous Sources,” authors Leslie F. Greengard, Shidong Jiang, Manas Rachh, and Jun Wang present a new FGT technique that avoids the use of Hermite and local expansions. The new technique employs Fourier spectral approximations, which are accelerated by nonuniform fast Fourier transforms, and results in a considerably more efficient adaptive implementation. Adaptivity is especially vital for realizing the acceleration from a fast transform when points are highly nonuniform. The paper presents compelling illustrations and examples of the computational approach and the adaptive tree-based hierarchy employed. This hierarchy is used to resolve point distributions down to a refinement level determined by accuracy demands; this results in significantly better work per grid point than conventional FGT techniques. Consequently, the authors note that there are potential key benefits in parallelization of the proposed technique. In addition to the technique's clever composition of a broad variety of advanced computing paradigms and exploitation of mathematical structure to facilitate such fast transforms, the authors present several pathways of future research. For example, the analysis is readily accessible from dimensions larger than the illustrative examples illuminate, and univariate sum-of-exponentials structure also may be exploited; the computing techniques detailed by the authors could be tailored to such regimes. These future directions have broad application in scientific computing.