Fast, adaptive, high-order accurate discretization of the Lippmann-Schwinger equation in two dimensions

Sivaram Ambikasaran, Carlos Borges, Lise Marie Imbert-Gerard, Leslie Greengard

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


We present a fast direct solver for two-dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of Lippmann-Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad-tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require O(N3/2) work, where N denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both low and high frequency regimes.

Original languageEnglish (US)
Pages (from-to)A1770-A1787
JournalSIAM Journal on Scientific Computing
Issue number3
StatePublished - 2016
Externally publishedYes


  • Acoustic scattering
  • Adaptivity
  • Electromagnetic scattering
  • Fast direct solver
  • High-order accuracy
  • Integral equation
  • Lippmann-schwinger equation
  • Penetrable media

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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