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

Abstract

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 thatmore »« less

Authors:

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)

Publication Date:: Fri Apr 07 00:00:00 EDT 2017

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

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

OSTI Identifier:: 1533958

Alternate Identifier(s):: OSTI ID: 1397802

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

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 341; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; fast multipole method; boundary element method; inverse fast multipole method; iterative solver; preconditioning; low rank compression

Citation Formats


                    Takahashi, Toru, Coulier, Pieter, and Darve, Eric. Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method.  United States: N. p., 2017. 
Web.  doi:10.1016/j.jcp.2017.04.016.

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                    Takahashi, Toru, Coulier, Pieter, & Darve, Eric. Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method.  United States.  https://doi.org/10.1016/j.jcp.2017.04.016

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                    Takahashi, Toru, Coulier, Pieter, and Darve, Eric. Fri .  
"Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method".  United States.  https://doi.org/10.1016/j.jcp.2017.04.016.  https://www.osti.gov/servlets/purl/1533958.

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@article{osti_1533958,

  title        = {Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method},

  author       = {Takahashi, Toru and Coulier, Pieter and Darve, Eric},

  abstractNote = {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.},

  doi          = {10.1016/j.jcp.2017.04.016},

  journal      = {Journal of Computational Physics},

  number       = C,

  volume       = 341,

  place        = {United States},

  year         = {Fri Apr 07 00:00:00 EDT 2017},

  month        = {Fri Apr 07 00:00:00 EDT 2017}

}

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

Abstract

Citation Formats

Parallel Hierarchical Solvers and Preconditioners for Boundary Element Methods journal, January 1998

An application of fast multipole method to isogeometric boundary element method for Laplace equation in two dimensions journal, December 2012

A fast boundary element method for the scattering analysis of high-intensity focused ultrasound journal, November 2015

The Fast Multipole Method: Numerical Implementation journal, May 2000

Rapid solution of integral equations of classical potential theory journal, September 1985

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Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions journal, December 1993

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The fast multipole method for the wave equation: a pedestrian prescription journal, June 1993

Adaptive Low-Rank Approximation of Collocation Matrices journal, February 2003

A fast multipole boundary element method based on the improved Burton–Miller formulation for three-dimensional acoustic problems journal, May 2011

GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems journal, July 1986

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Hierarchical matrix techniques for low- and high-frequency Helmholtz problems journal, March 2007

Advances in Iterative Methods and Preconditioners for the Helmholtz Equation journal, December 2007

A wideband fast multipole accelerated boundary integral equation method for time-harmonic elastodynamics in two dimensions: A WIDEBAND FAST MULTIPOLE ACCELERATED BIEM FOR ELASTODYNAMICS journal, March 2012

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The black-box fast multipole method journal, December 2009

Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions journal, January 2011

A low frequency elastodynamic fast multipole boundary element method in three dimensions journal, September 2015

Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems journal, February 2003

A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization journal, January 2012

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Preconditioning based on Calderon’s formulae for periodic fast multipole methods for Helmholtz’ equation journal, January 2012

A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation journal, January 2009

A high-order accurate accelerated direct solver for acoustic scattering from surfaces journal, July 2014

The fast multipole method on parallel clusters, multicore processors, and graphics processing units journal, February 2011

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A fast multipole method for Maxwell equations stable at all frequencies journal, January 2004

An O ( N ) direct solver for integral equations on the plane journal, March 2015

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Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations journal, May 2015

Efficient fast multipole method for low-frequency scattering journal, June 2004

A Flexible Inner-Outer Preconditioned GMRES Algorithm journal, March 1993

Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation journal, February 2012

Fourier-Based Fast Multipole Method for the Helmholtz Equation journal, January 2013

Regularized integral equations and fast high-order solvers for sound-hard acoustic scattering problems: REGULARIZED EQUATIONS AND EFFICIENT SOLVERS FOR SOUND-HARD SCATTERING journal, June 2012

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Parallel Hierarchical Solvers and Preconditioners for Boundary Element Methods
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A fast boundary element method for the scattering analysis of high-intensity focused ultrasound
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journal, May 2000

Rapid solution of integral equations of classical potential theory
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Electromagnetic scattering by surfaces of arbitrary shape
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Geometry Based Preconditioner for Radiation Problems Involving wire and Surface Basis Functions
journal, January 2009

Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions
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Recent Advances and Emerging Applications of the Boundary Element Method
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The fast multipole method for the wave equation: a pedestrian prescription
journal, June 1993

Adaptive Low-Rank Approximation of Collocation Matrices
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A fast multipole boundary element method based on the improved Burton–Miller formulation for three-dimensional acoustic problems
journal, May 2011

GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986

The Fast Multipole Method I: Error Analysis and Asymptotic Complexity
journal, January 2000

Hierarchical matrix techniques for low- and high-frequency Helmholtz problems
journal, March 2007

Advances in Iterative Methods and Preconditioners for the Helmholtz Equation
journal, December 2007

A wideband fast multipole accelerated boundary integral equation method for time-harmonic elastodynamics in two dimensions: A WIDEBAND FAST MULTIPOLE ACCELERATED BIEM FOR ELASTODYNAMICS
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A Preconditioned 3-D Multi-Region Fast Multipole Solver for Seismic Wave Propagation in Complex Geometries
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journal, January 1985

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journal, January 2007

Controlling sound with acoustic metamaterials
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journal, February 2003

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A fast direct solver for boundary integral equations in two dimensions
journal, May 2005

Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations
journal, May 2015

Efficient fast multipole method for low-frequency scattering
journal, June 2004

A Flexible Inner-Outer Preconditioned GMRES Algorithm
journal, March 1993

Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation
journal, February 2012

Fourier-Based Fast Multipole Method for the Helmholtz Equation
journal, January 2013

Regularized integral equations and fast high-order solvers for sound-hard acoustic scattering problems: REGULARIZED EQUATIONS AND EFFICIENT SOLVERS FOR SOUND-HARD SCATTERING
journal, June 2012

On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem
journal, July 2003

Efficient mesh deformation based on radial basis function interpolation by means of the inverse fast multipole method
journal, August 2016

Combining Fast Multipole Techniques and an Approximate Inverse Preconditioner for Large Electromagnetism Calculations
journal, January 2005

Multipole Translation Theory for the Three-Dimensional Laplace and Helmholtz Equations
journal, July 1995

An efficient blocking M2L translation for low-frequency fast multipole method in three dimensions
journal, May 2016

A fast algorithm for particle simulations
journal, December 1987

Combining analytic preconditioner and Fast Multipole Method for the 3-D Helmholtz equation
journal, March 2013