Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


A convergence theorem for the fast multipole method for 2 dimensional scattering problems
HTML articles powered by AMS MathViewer

by Christophe Labreuche PDF
Math. Comp. 67 (1998), 553-591 Request permission


The Fast Multipole Method (FMM) designed by V. Rokhlin rapidly computes the field scattered from an obstacle. This computation consists of solving an integral equation on the boundary of the obstacle. The main result of this paper shows the convergence of the FMM for the two dimensional Helmholtz equation. Before giving the theorem, we give an overview of the main ideas of the FMM. This is done following the papers of V. Rokhlin. Nevertheless, the way we present the FMM is slightly different. The FMM is finally applied to an acoustic problem with an impedance boundary condition. The moment method is used to discretize this continuous problem.
  • Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
  • F. Canning, The Impedance Matrix Localization Method for Moment-Method Calculations, IEEE antenna propag.,Vol 32, pp 18-30, oct 1990.
  • F. Canning, Improved Matrix Localization, IEEE antenna propag., Vol 41,No 5, pp 659-667, may 1993.
  • G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms. I, Comm. Pure Appl. Math. 44 (1991), no. 2, 141–183. MR 1085827, DOI 10.1002/cpa.3160440202
  • David L. Colton and Rainer Kress, Integral equation methods in scattering theory, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 700400
  • L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), no. 2, 325–348. MR 918448, DOI 10.1016/0021-9991(87)90140-9
  • Mohamed Ali Hamdi, Une formulation variationnelle par équations intégrales pour la résolution de l’équation de Helmholtz avec des conditions aux limites mixtes, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 292 (1981), no. 1, 17–20 (French, with English summary). MR 637242
  • Ami Harten and Itai Yad-Shalom, Fast multiresolution algorithms for matrix-vector multiplication, SIAM J. Numer. Anal. 31 (1994), no. 4, 1191–1218. MR 1286223, DOI 10.1137/0731062
  • Ami Harten, Discrete multi-resolution analysis and generalized wavelets, Appl. Numer. Math. 12 (1993), no. 1-3, 153–192. Special issue to honor Professor Saul Abarbanel on his sixtieth birthday (Neveh, 1992). MR 1227185, DOI 10.1016/0168-9274(93)90117-A
  • J. Jin, The Finite Element Method in Electromagnetics, Wiley interscience, New York, 1993.
  • P. Martin, Multipole Scattering: an Invitation, in The third international conference on mathematical and numerical aspects of wave propagation, G. Cohen, 1995.
  • H. Petersen, D. Soelvason, J. Perran, E. Smith, Error Estimates for the Fast Multipole Method. I. The Two-Dimensional Case, Proceedings of the Royal Society of London, serie A, Vol 448, pp 389-400, march 1995.
  • V. Rokhlin, Rapid solution of integral equations of classical potential theory, J. Comput. Phys. 60 (1985), no. 2, 187–207. MR 805870, DOI 10.1016/0021-9991(85)90002-6
  • V. Rokhlin, Rapid solution of integral equations of scattering theory in two dimensions, J. Comput. Phys. 86 (1990), no. 2, 414–439. MR 1036660, DOI 10.1016/0021-9991(90)90107-C
  • Bradley K. Alpert and Vladimir Rokhlin, A fast algorithm for the evaluation of Legendre expansions, SIAM J. Sci. Statist. Comput. 12 (1991), no. 1, 158–179. MR 1078802, DOI 10.1137/0912009
  • V. Rokhlin, N. Engheta, W. Murphy, M. Vassiliu, The Fast Multipole Method for Electromagnetic Scattering Problems, IEEE transactions on antenna and propagation, Vol 40, No 6, pp 634-641, june 1992.
  • V. Rokhlin, R. Coifman, S. Wandzura The Fast Multipole Method for Wave Equation: a Pedestrian Prescription, IEEE anten. and propag. mag., Vol 35, No 3, pp 7-12, june 1993.
  • V. Rokhlin, Diagonal forms of translation operators for the Helmholtz equation in three dimensions, Appl. Comput. Harmon. Anal. 1 (1993), no. 1, 82–93. MR 1256528, DOI 10.1006/acha.1993.1006
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC (1991): 41A58, 35J05, 65N30
  • Retrieve articles in all journals with MSC (1991): 41A58, 35J05, 65N30
Additional Information
  • Christophe Labreuche
  • Affiliation: Thomson CSF-LCR, Domaine de Corbeville, 91404 Orsay cedex, France
  • Email:
  • Received by editor(s): December 11, 1995
  • Received by editor(s) in revised form: October 7, 1996
  • © Copyright 1998 American Mathematical Society
  • Journal: Math. Comp. 67 (1998), 553-591
  • MSC (1991): Primary 41A58, 35J05, 65N30
  • DOI:
  • MathSciNet review: 1458223