## Abstract

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(m^{3}), of the discretized Lippmann-Schwinger operator. With this matrix decomposition, each back substitution requires only O(m^{2} log(m)); therefore, a scattering problem with m incident waves can be solved, altogether, in O(m^{3} log(m)) flops.

Original language | English (US) |
---|---|

Pages (from-to) | 175-190 |

Number of pages | 16 |

Journal | Advances in Computational Mathematics |

Volume | 16 |

Issue number | 2-3 |

DOIs | |

State | Published - 2002 |

## Keywords

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

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics