Abstract

This paper introduces a parallel directional fast multipole method (FMM) for solving $N$-body problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a more restrictive low-rank criterion than that of the low-frequency regime, and thus effective parallelizations must adapt to the modified data dependencies. We propose a simple partition at a fixed level of the octree and show that, if the partitions are properly balanced between $p$ processes, the overall runtime is essentially $\mathcal{O}{N \log N/p + p}$. By the structure of the low-rank criterion, we are able to avoid communication at the top of the octree. We demonstrate the effectiveness of our parallelization on several challenging models.

Keywords

  1. parallel
  2. fast multipole methods
  3. $N$-body problems
  4. scattering problems
  5. Helmholtz equation
  6. oscillatory kernels
  7. directional
  8. multilevel

MSC codes

  1. 65Y05
  2. 65Y20
  3. 78A45

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Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: C335 - C352
ISSN (online): 1095-7197

History

Submitted: 18 November 2013
Accepted: 5 May 2014
Published online: 14 August 2014

Keywords

  1. parallel
  2. fast multipole methods
  3. $N$-body problems
  4. scattering problems
  5. Helmholtz equation
  6. oscillatory kernels
  7. directional
  8. multilevel

MSC codes

  1. 65Y05
  2. 65Y20
  3. 78A45

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