An Adaptive Fast Gauss Transform in Two Dimensions

Abstract

Abstract.A variety of problems in computational physics and engineering require the convolution of the heat kernel (a Gaussian) with either discrete sources, densities supported on boundaries, or continuous volume distributions. We present a unified fast Gauss transform for this purpose in two dimensions, making use of an adaptive quad-tree discretization on a unit square which is assumed to contain all sources. Our implementation permits either free-space or periodic boundary conditions to be imposed, and is efficient for any choice of variance in the Gaussian.

Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform

“…Unfortunately, it is difficult to develop an efficient sweeping method that can be used on an adaptive data structure. For this, hierarchical versions of the FGT were developed [21,38], using quad-tree or oct-tree subdivisions of space.…”

“…With a workload proportional to the number of boxes, N and M would have to be much larger before linear scaling is more evident. In the context of solving diffusion problems, it is natural to assume that the number of points per leaf node is in the range 16 to 64, corresponding to a 4th or 8th order discretization (as described, for example, in [38] in the context of volume integral versions of the FGT). Fig.…”

“…The main advantage of exponential functions in this context follows from the fact that they are eigenfunctions of the translation operator, which leads to a simple "sweeping" algorithm in one dimension, whose performance is entirely independent of the variance δ. In higher dimensions, the SOE approximation can be used to accelerate existing FGTs [13,14,29,38]. The SOE-based scheme shares some feature with the "plane wave" versions of the FGT [14,29,38], with an important difference.…”

“…In higher dimensions, the SOE approximation can be used to accelerate existing FGTs [13,14,29,38]. The SOE-based scheme shares some feature with the "plane wave" versions of the FGT [14,29,38], with an important difference. The earlier plane-wave schemes use the Fourier transform to develop an approximation of the Gaussian in terms of oscillatory exponentials with a restricted range of validity.…”

Explicit unconditionally stable methods for the heat equation via potential theory

“…Recent advances in fast algorithms for heat potentials, however, have removed this obstacle. We refer the reader to the papers [8,9,10,15,24,25,37,39,43,44,45] and the references therein for further discussion.…”