The Fast Multipole Method: Numerical Implementation

Abstract

We study integral methods applied to the resolution of the Maxwell equations where the linear system is solved using an iterative method which requires only matrix-vector products. The fast multipole method (FMM) is one of the most efficient methods used to perform matrix-vector products and accelerate the resolution of the linear system. A problem involving N degrees of freedom may be solved in CN iter N log N floating operations, where C is a constant depending on the implementation of the method. In this article several techniques allowing one to reduce the constant C are analyzed. This reduction implies a lower total CPU time and a larger range of application of the FMM. In particular, new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms. Several numerical tests are also described. These confirm the efficiency and the theoretical complexity of the FMM.

A Fast Multipole Method formulation for 3D elastodynamics in the frequency domain

“…The latter are seen to be linear combinations of the Green's function for the Helmholtz equation (3), for which such a reformulation is known from earlier works, e.g. [3,4,8]…”

“…The present Note improves on the methodology of [5] by incorporating recent advances of FMM implementations for Maxwell equations (e.g. [8]), which allow to run BEM models of much larger size.…”

“…Semblat, C. R. Mecanique 335 (2007 La BEM pour l'élastodynamique consisteà exploiter la formule de représentation intégrale (1) et l'équation intégrale (4). Les tenseurs de Greenélastodynamiques (2) s'exprimantà l'aide de la solutionélémentaire (3) de l'équation de Helmholtz (pour les nombres d'onde de compression et de cisaillement), ils peuventêtre reformulés en termes du développement en série de multipôles (5), déjà utilisé enélectromagnétisme [4,8]. La séparation des variables x et y ainsi introduite permet la réutilisation d'intégrations par rapportà y dans (1) ou (4) quand le point de collocation x change, ce qui est impossible en BEM classique.…”

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

“…The original version of the fast multipole method relied on analytic expressions for the multipole expansion of the kernels but several kernel independent versions have been proposed [36,38]. For more details, the reader is referred to [60][61][62].…”

“…Various fast summation schemes have been devised to provide matrix-vector products in O(N log β N) for such problems, where N is the number of unknowns and β ≥ 0 an integer depending on the method. They broadly fall under three different categories: -FFT (Fast Fourier Transform) based methods, -fast multipole methods [36][37][38][60][61][62], -hierarchical matrices [21,22,24].…”