### 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) |
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Pages (from-to) | 175-190 |

Number of pages | 16 |

Journal | Advances in Computational Mathematics |

Volume | 16 |

Issue number | 2-3 |

DOIs | |

State | Published - 2002 |

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### Keywords

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

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

**A fast, direct algorithm for the Lippmann-Schwinger integral equation in two dimensions.** / Chen, Yu.

Research output: Contribution to journal › Article

*Advances in Computational Mathematics*, vol. 16, no. 2-3, pp. 175-190. https://doi.org/10.1023/A:1014450116300

}

TY - JOUR

T1 - A fast, direct algorithm for the Lippmann-Schwinger integral equation in two dimensions

AU - Chen, Yu

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Fast algorithm

KW - Lippmann-Schwinger-Helmholtz

KW - Scattering matrix

UR - http://www.scopus.com/inward/record.url?scp=0141625907&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141625907&partnerID=8YFLogxK

U2 - 10.1023/A:1014450116300

DO - 10.1023/A:1014450116300

M3 - Article

VL - 16

SP - 175

EP - 190

JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

SN - 1019-7168

IS - 2-3

ER -