Abstract
The Fast Multipole Method was introduced by Greengard and Rokhlin in a seminal paper appeared in 1987 for studying large systems of particle interactions with reduced algorithmic and memory complexity [60]. Developments of the original idea are successfully applied to the analysis of many scientific and engineering problems of practical interest. In scattering analysis, multipole techniques may enable to reduce the computational complexity of iterative solution procedures involving dense matrices arising from the discretization of integral operators from O(n2) to O(n log n) arithmetic operations. In this paper we discuss recent algorithmic developments of algebraic preconditioning techniques for the Fast Multipole Method for 2D and 3D scattering problems. We focus on design aspects, implementation details, numerical scalability, parallel performance on emerging computer systems, and give some minor emphasis to theoretical aspects as well. Thanks to the use of iterative techniques and efficient parallel preconditioners, fast integral solvers involving tens of million unknowns are nowadays feasible and can be integrated in the design processes. Keywords: algebraic preconditioners, Fast Multipole Method, Krylov solvers, electromagnetic scattering applications, Maxwell's equations.