Fourier-Based Fast Multipole Method for the Helmholtz Equation

Handle URI:
http://hdl.handle.net/10754/598360
Title:
Fourier-Based Fast Multipole Method for the Helmholtz Equation
Authors:
Cecka, Cris; Darve, Eric
Abstract:
The fast multipole method (FMM) has had great success in reducing the computational complexity of solving the boundary integral form of the Helmholtz equation. We present a formulation of the Helmholtz FMM that uses Fourier basis functions rather than spherical harmonics. By modifying the transfer function in the precomputation stage of the FMM, time-critical stages of the algorithm are accelerated by causing the interpolation operators to become straightforward applications of fast Fourier transforms, retaining the diagonality of the transfer function, and providing a simplified error analysis. Using Fourier analysis, constructive algorithms are derived to a priori determine an integration quadrature for a given error tolerance. Sharp error bounds are derived and verified numerically. Various optimizations are considered to reduce the number of quadrature points and reduce the cost of computing the transfer function. © 2013 Society for Industrial and Applied Mathematics.
Citation:
Cecka C, Darve E (2013) Fourier-Based Fast Multipole Method for the Helmholtz Equation. SIAM Journal on Scientific Computing 35: A79–A103. Available: http://dx.doi.org/10.1137/11085774X.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Scientific Computing
Issue Date:
Jan-2013
DOI:
10.1137/11085774X
Type:
Article
ISSN:
1064-8275; 1095-7197
Sponsors:
This research was supported by the U.S. Army Research Laboratory, through the Army HighPerformance Computing Research Center, Cooperative Agreement W911NF-07-0027, the StanfordSchool of Engineering, and the King Abdullah University of Science and Technology.
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Full metadata record

DC FieldValue Language
dc.contributor.authorCecka, Crisen
dc.contributor.authorDarve, Ericen
dc.date.accessioned2016-02-25T13:19:22Zen
dc.date.available2016-02-25T13:19:22Zen
dc.date.issued2013-01en
dc.identifier.citationCecka C, Darve E (2013) Fourier-Based Fast Multipole Method for the Helmholtz Equation. SIAM Journal on Scientific Computing 35: A79–A103. Available: http://dx.doi.org/10.1137/11085774X.en
dc.identifier.issn1064-8275en
dc.identifier.issn1095-7197en
dc.identifier.doi10.1137/11085774Xen
dc.identifier.urihttp://hdl.handle.net/10754/598360en
dc.description.abstractThe fast multipole method (FMM) has had great success in reducing the computational complexity of solving the boundary integral form of the Helmholtz equation. We present a formulation of the Helmholtz FMM that uses Fourier basis functions rather than spherical harmonics. By modifying the transfer function in the precomputation stage of the FMM, time-critical stages of the algorithm are accelerated by causing the interpolation operators to become straightforward applications of fast Fourier transforms, retaining the diagonality of the transfer function, and providing a simplified error analysis. Using Fourier analysis, constructive algorithms are derived to a priori determine an integration quadrature for a given error tolerance. Sharp error bounds are derived and verified numerically. Various optimizations are considered to reduce the number of quadrature points and reduce the cost of computing the transfer function. © 2013 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipThis research was supported by the U.S. Army Research Laboratory, through the Army HighPerformance Computing Research Center, Cooperative Agreement W911NF-07-0027, the StanfordSchool of Engineering, and the King Abdullah University of Science and Technology.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectAnterpolationen
dc.subjectBoundary element methoden
dc.subjectFast Fourier transformen
dc.subjectFast multipole methoden
dc.subjectFourier basisen
dc.subjectHelmholtzen
dc.subjectIntegral equationsen
dc.subjectInterpolationen
dc.subjectMaxwellen
dc.titleFourier-Based Fast Multipole Method for the Helmholtz Equationen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Scientific Computingen
dc.contributor.institutionHarvard University, Cambridge, United Statesen
dc.contributor.institutionStanford University, Palo Alto, United Statesen
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