Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method

Takahashi, Toru; Coulier, Pieter; Darve, Eric

doi:10.1016/j.jcp.2017.04.016

Title: Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method

Journal Article · Fri Apr 07 00:00:00 EDT 2017 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2017.04.016· OSTI ID:1533958

Takahashi, Toru ^[1]; Coulier, Pieter ^[2]; Darve, Eric ^[3]

Nagoya University (Japan)
Katholieke University Leuven, Heverlee (Belgium); Stanford University, CA (United States)
Stanford University, CA (United States)

Here we investigate an efficient preconditioning of iterative methods (such as GMRES) for solving dense linear systems Ax = b that follow from a boundary element method (BEM) for the 3D Helmholtz equation, focusing on the low-frequency regime. While matrix–vector products in GMRES can be accelerated through the low-frequency fast multipole method (LFFMM), the BEM often remains computationally expensive due to the large number of GMRES iterations. We propose the application of the inverse fast multipole method (IFMM) as a preconditioner to accelerate the convergence of GMRES. The IFMM is in essence an approximate direct solver that uses a multilevel hierarchical decomposition and low-rank approximations. The proposed IFMM-based preconditioning has a tunable parameter ε that balances the cost to construct a preconditioner M, which is an approximation of A^-1, and the cost to perform the iterative process by means of M. Namely, using a small (respectively, large) value of ε takes a long (respectively, short) time to construct M, while the number of iterations can be small (respectively, large). A comprehensive set of numerical examples involving various boundary value problems with complicated geometries and mixed boundary conditions is presented to validate the efficiency of the proposed method. We show that the IFMM preconditioner (with a nearly optimal ε of 10^-2) clearly outperforms some common preconditioners for the BEM, achieving 1.2–10.8 times speed-up of the computations, in particular when the scale of the underlying scatterer is about five wavelengths or more. In addition, the IFMM preconditioner is capable of solving complicated problems (in a reasonable amount of time) that BD preconditioner can not.

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Research Organization:: Stanford Univ., CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); US Army Research Laboratory (USARL); Japan Society for the Promotion of Science (JSPS)

Grant/Contract Number:: NA0002373; 15K06683; W911NF-07-0027; NA0002373-1

OSTI ID:: 1533958

Alternate ID(s):: OSTI ID: 1397802

Journal Information:: Journal of Computational Physics, Vol. 341, Issue C; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 9 works

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Title: Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method

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