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

Research output: Contribution to journalArticle

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(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
Volume16
Issue number2-3
DOIs
StatePublished - 2002

Fingerprint

Matrix Decomposition
Scattering Problems
Integral equations
Integral Equations
Two Dimensions
Scattering
Decomposition
Iterative Solvers
Helmholtz equation
CPU Time
Helmholtz Equation
Direct Method
Merging
Program processors
Substitution
Substitution reactions
Wavelength
Distinct
Numerical Simulation
Computer simulation

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.

In: Advances in Computational Mathematics, Vol. 16, No. 2-3, 2002, p. 175-190.

Research output: Contribution to journalArticle

@article{8e838bf65d9944439bfe505f2e7a1dc3,
title = "A fast, direct algorithm for the Lippmann-Schwinger integral equation in two dimensions",
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(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.",
keywords = "Fast algorithm, Lippmann-Schwinger-Helmholtz, Scattering matrix",
author = "Yu Chen",
year = "2002",
doi = "10.1023/A:1014450116300",
language = "English (US)",
volume = "16",
pages = "175--190",
journal = "Advances in Computational Mathematics",
issn = "1019-7168",
publisher = "Springer Netherlands",
number = "2-3",

}

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

AN - SCOPUS:0141625907

VL - 16

SP - 175

EP - 190

JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

SN - 1019-7168

IS - 2-3

ER -