Fast convolution with free-space Green's functions

TL;DR

This paper presents a fast, FFT-based algorithm for computing volume potentials involving free-space Green's functions, achieving superalgebraic convergence for smooth data and enabling efficient derivatives computation.

Contribution

The authors introduce a novel FFT-based method that regularizes the Green's function Fourier transform, allowing rapid and accurate volume potential calculations with explicit matrix entries.

Findings

Achieves superalgebraic convergence for smooth data

Enables fast computation of derivatives of potentials

Provides explicit matrix entries for integral operators

Abstract

We introduce a fast algorithm for computing volume potentials - that is, the convolution of a translation invariant, free-space Green's function with a compactly supported source distribution defined on a uniform grid. The algorithm relies on regularizing the Fourier transform of the Green's function by cutting off the interaction in physical space beyond the domain of interest. This permits the straightforward application of trapezoidal quadrature and the standard FFT, with superalgebraic convergence for smooth data. Moreover, the method can be interpreted as employing a Nystrom discretization of the corresponding integral operator, with matrix entries which can be obtained explicitly and rapidly. This is of use in the design of preconditioners or fast direct solvers for a variety of volume integral equations. The method proposed permits the computation of any derivative of the potential, at the cost of an additional FFT.