Generalized fast multipole method

Abstract

The fast multipole method (FMM) is a technique allowing the fast calculation of long-range interactions between N points in O(N) or O(N ln 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. Original FMMs required 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 has been extended by the introduction of black box [1] or kernel independent techniques [2]. In these approaches, the user only provides a subroutine to numerically calculate the interaction kernel. This allows changing the definition of the kernel with minimal change to the computer program. In this talk we will present a novel kernel independent FMM, which leads to diagonal multipole-to-local operators. This results in a significant reduction in the computational cost [1], in particular 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, methods to choose the parameters in the FMM given some tolerance, and the steps required to build a multilevel scheme from the single level formulation. Numerical results are given for benchmark calculations to demonstrate the accuracy as a function of the number of multipole coefficients, and the computational cost of the different steps in the method.