An adaptive fast Gauss transform in two dimensions

TL;DR

This paper introduces an adaptive fast Gauss transform in two dimensions that efficiently computes Gaussian convolutions for various sources and boundary conditions using an adaptive quad-tree approach.

Contribution

It presents a unified, efficient algorithm for 2D Gaussian convolutions with adaptive discretization, supporting different boundary conditions and Gaussian variances.

Findings

Efficient computation of Gaussian convolutions in 2D.

Supports free-space and periodic boundary conditions.

Handles arbitrary Gaussian variances efficiently.

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.