The Fast Generalized Gauss Transform

Abstract

The fast Gauss transform allows for the calculation of the sum of $N$ Gaussians at $M$ points in $\mathcal{O}(N+M)$ time. Here, we extend the algorithm to a wider class of kernels, motivated by quadrature issues that arise in using integral equation methods for solving the heat equation on moving domains. In particular, robust high-order product integration methods require convolution with $\mathcal{O}(q)$ distinct Gaussian-type kernels in order to obtain $q$th-order accuracy in time. The generalized Gauss transform permits the computation of each of these kernels and, thus, the construction of fast solvers with optimal computational complexity. We also develop plane-wave representations of these Gaussian-type fields, permitting the “diagonal translation” version of the Gauss transform to be applied. When the sources and targets lie on a uniform grid, or a hierarchy of uniform grids, we show that the curse of dimensionality (the exponential growth of complexity constants with dimension) can be mitigated. Under these conditions, the algorithm has a lower operation count than the fast Fourier transform even for modest values of $N$ and $M$.