High performance BLAS formulation of the multipole-to-local operator in the fast multipole method

  • Authors:

  • O. Coulaud;P. Fortin;J. Roman

  • Affiliations:

  • ScAlApplix Project, INRIA Futurs, LaBRI and IMB, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France;ScAlApplix Project, INRIA Futurs, LaBRI and IMB, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France and Laboratoire d'Astrophysique de Marseille, 2 place Leve ...;ScAlApplix Project, INRIA Futurs, LaBRI and IMB, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2008

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Abstract

The multipole-to-local (M2L) operator is the most time-consuming part of the far field computation in the fast multipole method for Laplace equation. Its natural expression, though commonly used, does not respect a sharp error bound: we here first prove the correctness of a second expression. We then propose a matrix formulation implemented with basic linear algebra subprograms (BLAS) routines in order to speed up its computation for these two expressions. We also introduce special data storages in memory to gain greater computational efficiency. This BLAS scheme is finally compared, for uniform distributions, to other M2L improvements such as block FFT, FFT with polynomial scaling, rotations and plane wave expansions. When considering runtime, extra memory storage, numerical stability and common precisions for Laplace equation, the BLAS version appears as the best one.