A fast, direct algorithm for the Lippmann-Schwinger integral equation in two dimensions

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The state-of-the-art, large-scale numerical simulations of the scattering problem for the Helmholtz equation in two dimensions rely on iterative solvers for the Lippmann-Schwinger integral equition, with an optimal CPU time O(m 3 log(m)) for an m-by-m wavelength problem. We present a method to solve the same problem directly, as opposed to iteratively, with the obvious advantage in efficiency for multiple right-hand sides corresponding to distinct incident waves. Analytically, this direct method is a hierarchical, recursive scheme consisting of the so-called splitting and merging processes. Algebraically, it amounts to a recursive matrix decomposition, for a cost of O(m3), of the discretized Lippmann-Schwinger operator. With this matrix decomposition, each back substitution requires only O(m2 log(m)); therefore, a scattering problem with m incident waves can be solved, altogether, in O(m3 log(m)) flops.

Original languageEnglish (US)
Pages (from-to)175-190
Number of pages16
JournalAdvances in Computational Mathematics
Issue number2-3
StatePublished - 2002


  • Fast algorithm
  • Lippmann-Schwinger-Helmholtz
  • Scattering matrix

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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