Cauchy Fast Multipole Method for General Analytic Kernels

Abstract

The fast multipole method (FMM) is a technique allowing the fast calculation of long-range interactions between $N$ points in ${\mathcal{O}}(N)$ or ${\mathcal{O}}(N \log N)$ steps with some prescribed error tolerance. The FMM has found many applications in the field of integral equations and boundary element methods, in particular by accelerating the solution of dense linear systems arising from such formulations. Standard FMMs are derived from analytical expansions of the kernel, for example using spherical harmonics or Taylor expansions. In recent years, the range of applicability and the ease of use of FMMs have been extended by the introduction of black-box and kernel independent techniques. In these approaches, the user provides only a subroutine to numerically calculate the interaction kernel. This allows changing the definition of the kernel with minimal changes to the computer program. This paper presents a novel kernel independent FMM, which leads to diagonal multipole-to-local operators. The result is a significant reduction in the computational cost, particularly when high accuracy is needed. The approach is based on Cauchy's integral formula and the Laplace transform. We will present a numerical analysis of the convergence and numerical results in the case of a multilevel one-dimensional FMM.