The black-box fast multipole method

  • Authors:

  • William Fong;Eric Darve

  • Affiliations:

  • Institute for Computational and Mathematical Engineering, Stanford University, 496 Lomita Mall, Durand Building, Stanford, CA 94305-4042, USA;Institute for Computational and Mathematical Engineering, Stanford University, 496 Lomita Mall, Durand Building, Stanford, CA 94305-4042, USA and Department of Mechanical Engineering, Stanford Uni ...

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2009

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Abstract

A new O(N) fast multipole formulation is proposed for non-oscillatory kernels. This algorithm is applicable to kernels K(x,y) which are only known numerically, that is their numerical value can be obtained for any (x,y). This is quite different from many fast multipole methods which depend on analytical expansions of the far-field behavior of K, for |x-y| large. Other ''black-box'' or ''kernel-independent'' fast multipole methods have been devised. Our approach has the advantage of requiring a small pre-computation time even for very large systems, and uses the minimal number of coefficients to represent the far-field, for a given L^2 tolerance error in the approximation. This technique can be very useful for problems where the kernel is known analytically but is quite complicated, or for kernels which are defined purely numerically.