Fast-multipole method: a mathematical study

Abstract

Realistic treatment of scattering from an object involves the use of a huge number of points; thus using "fast methods" is a crucial issue. The fast multipole method is one of them. We are here interested in solving the Helmholtz equation by a boundary element method. The algorithm, first introduced by Vladimir Rokhlin, consists in approximating the kernel of the second order integral equation. In 3D, we here prove, in a rigorous way, that the complexity of the method is n 3/2. The main object of this paper is to find the optimal number of "poles" that has to be taken, and the optimal number of "directions" to consider. From this result, we obtain the complexity of the algorithm. We will also give precise estimations of the error. In a forthcoming paper we will present a modified multilevel version of the same algorithm, with which we can reduce the complexity to n · log2(n).