A hierarchical butterfly LU preconditioner for two-dimensional electromagnetic scattering problems involving open surfaces

Liu, Yang; Yang, Haizhao

doi:10.1016/j.jcp.2019.109014

Title: A hierarchical butterfly LU preconditioner for two-dimensional electromagnetic scattering problems involving open surfaces

Abstract

This paper introduces a hierarchical interpolative decomposition butterfly-LU factorization (H-IDBF-LU) preconditioner for solving two-dimensional electric-field integral equations (EFIEs) in electromagnetic scattering problems of perfect electrically conducting objects with open surfaces. H-IDBF-LU leverages the interpolative decomposition butterfly factorization (IDBF) to compress dense blocks of the discretized EFIE operator to expedite its application; this compressed operator also serves as an approximate LU factorization of the EFIE operator leading to an efficient preconditioner in iterative solvers. Both the memory requirement and computational cost of the H-IDBF-LU solver scale as $$O$$($$N$$ log² $$N$$) in one iteration; the total number of iterations required for a reasonably good accuracy scales as $$O$$(1) to O(log²$$N$$) in all of our numerical tests. The efficacy and accuracy of the proposed preconditioned iterative solver are demonstrated via its application to a broad range of scatterers involving up to 100 million unknowns.

Authors:

Liu, Yang ^[1];

^[2]

Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
National Univ. of Singapore (Singapore). Dept. of Mathematics

Publication Date:: Thu Oct 10 00:00:00 EDT 2019

Research Org.:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1564075

Alternate Identifier(s):: OSTI ID: 1576807

Grant/Contract Number:: AC02-05CH11231

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Name: Journal of Computational Physics; Journal ID: ISSN 0021-9991

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Citation Formats


                    Liu, Yang, and Yang, Haizhao. A hierarchical butterfly LU preconditioner for two-dimensional electromagnetic scattering problems involving open surfaces.  United States: N. p., 2019. 
Web.  doi:10.1016/j.jcp.2019.109014.

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                    Liu, Yang, & Yang, Haizhao. A hierarchical butterfly LU preconditioner for two-dimensional electromagnetic scattering problems involving open surfaces.  United States.  https://doi.org/10.1016/j.jcp.2019.109014

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                    Liu, Yang, and Yang, Haizhao. Thu .  
"A hierarchical butterfly LU preconditioner for two-dimensional electromagnetic scattering problems involving open surfaces".  United States.  https://doi.org/10.1016/j.jcp.2019.109014.  https://www.osti.gov/servlets/purl/1564075.

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@article{osti_1564075,

  title        = {A hierarchical butterfly LU preconditioner for two-dimensional electromagnetic scattering problems involving open surfaces},

  author       = {Liu, Yang and Yang, Haizhao},

  abstractNote = {This paper introduces a hierarchical interpolative decomposition butterfly-LU factorization (H-IDBF-LU) preconditioner for solving two-dimensional electric-field integral equations (EFIEs) in electromagnetic scattering problems of perfect electrically conducting objects with open surfaces. H-IDBF-LU leverages the interpolative decomposition butterfly factorization (IDBF) to compress dense blocks of the discretized EFIE operator to expedite its application; this compressed operator also serves as an approximate LU factorization of the EFIE operator leading to an efficient preconditioner in iterative solvers. Both the memory requirement and computational cost of the H-IDBF-LU solver scale as $O$($N$ log2 $N$) in one iteration; the total number of iterations required for a reasonably good accuracy scales as $O$(1) to O(log2$N$) in all of our numerical tests. The efficacy and accuracy of the proposed preconditioned iterative solver are demonstrated via its application to a broad range of scatterers involving up to 100 million unknowns.},

  doi          = {10.1016/j.jcp.2019.109014},

  journal      = {Journal of Computational Physics},

  number       = ,

  volume       = ,

  place        = {United States},

  year         = {Thu Oct 10 00:00:00 EDT 2019},

  month        = {Thu Oct 10 00:00:00 EDT 2019}

}

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Title: A hierarchical butterfly LU preconditioner for two-dimensional electromagnetic scattering problems involving open surfaces

Abstract

Citation Formats

Rapid solution of integral equations of scattering theory in two dimensions journal, February 1990

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

A wideband fast multipole method for the Helmholtz equation in three dimensions journal, July 2006

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

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

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

Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects journal, January 1997

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Fast Directional Multilevel Algorithms for Oscillatory Kernels journal, January 2007

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

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Fast Numerical Computations of Oscillatory Integrals Related to Acoustic Scattering, I journal, December 1993

Sparse Approximation for Solving Integral Equations with Oscillatory Kernels journal, January 1992

Wave atoms and sparsity of oscillatory patterns journal, November 2007

A multilevel matrix decomposition algorithm for analyzing scattering from large structures journal, January 1996

An algorithm for the rapid evaluation of special function transforms journal, March 2010

Butterfly Factorization journal, January 2015

Analytic preconditioners for the electric field integral equation journal, October 2004

A Preconditioner for the Electric Field Integral Equation Based on Calderon Formulas journal, January 2002

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Directional Preconditioner for 2D High Frequency Obstacle Scattering journal, January 2015

Fast direct solvers for integral equations in complex three-dimensional domains journal, May 2009

Multiscale Compressed Block Decomposition for Fast Direct Solution of Method of Moments Linear System journal, February 2011

Direct Solve of Electrically Large Integral Equations for Problem Sizes to 1 M Unknowns journal, August 2008

An ${\cal H}^{2}$-Matrix-Based Integral-Equation Solver of Reduced Complexity and Controlled Accuracy for Solving Electrodynamic Problems journal, October 2009

Hierarchical LU Decomposition-based Preconditioners for BEM journal, November 2004

A Recursive Skeletonization Factorization Based on Strong Admissibility journal, January 2017

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

A fast direct solver for scattering problems involving elongated structures journal, January 2007

Scattering from elongated objects: direct solution in O(N log2 N) operations journal, January 1996

Fast Direct Solver for Essentially Convex Scatterers Using Multilevel Non-Uniform Grids journal, August 2014

An HSS Matrix-Inspired Butterfly-Based Direct Solver for Analyzing Scattering From Two-Dimensional Objects journal, January 2017

A Butterfly-Based Direct Integral-Equation Solver Using Hierarchical LU Factorization for Analyzing Scattering From Electrically Large Conducting Objects journal, September 2017

A butterfly-based direct solver using hierarchical LU factorization for Poggio-Miller-Chang-Harrington-Wu-Tsai equations journal, April 2018

Construction and Arithmetics of H -Matrices journal, August 2003

Fast construction of hierarchical matrix representation from matrix–vector multiplication journal, May 2011

A Fast Randomized Algorithm for Computing a Hierarchically Semiseparable Representation of a Matrix journal, October 2011

Volterra approximation for low grazing angle shadowing on smooth ocean-like surfaces journal, November 1995

Forward-backward: a new method for computing low-grazing angle scattering journal, May 1996

Forward-backward iterative physical optics algorithm for computing the RCS of open-ended cavities journal, February 2005

Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation journal, February 2011

On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime journal, June 2015

An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments journal, May 2018

Numerical Differentiation by High Order Interpolation journal, November 1987

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Rapid solution of integral equations of scattering theory in two dimensions
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Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions
journal, December 1993

A wideband fast multipole method for the Helmholtz equation in three dimensions
journal, July 2006

The fast multipole method for the wave equation: a pedestrian prescription
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The Fast Multipole Method I: Error Analysis and Asymptotic Complexity
journal, January 2000

Multipole Translation Theory for the Three-Dimensional Laplace and Helmholtz Equations
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Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects
journal, January 1997

Fast Directional Computation of High Frequency Boundary Integrals via Local FFTs
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A fast directional algorithm for high frequency acoustic scattering in two dimensions
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Fast Directional Multilevel Algorithms for Oscillatory Kernels
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Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation
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Efficient Computation of Oscillatory Integrals via Adaptive Multiscale Local Fourier Bases
journal, July 2000

Fast Numerical Computations of Oscillatory Integrals Related to Acoustic Scattering, I
journal, December 1993

Sparse Approximation for Solving Integral Equations with Oscillatory Kernels
journal, January 1992

Wave atoms and sparsity of oscillatory patterns
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A multilevel matrix decomposition algorithm for analyzing scattering from large structures
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An algorithm for the rapid evaluation of special function transforms
journal, March 2010

Butterfly Factorization
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Analytic preconditioners for the electric field integral equation
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A Preconditioner for the Electric Field Integral Equation Based on Calderon Formulas
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Second-kind integral solvers for TE and TM problems of diffraction by open arcs: SECOND-KIND OPEN ARC INTEGRAL SOLVERS
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Direct Solve of Electrically Large Integral Equations for Problem Sizes to 1 M Unknowns
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An ${\cal H}^{2}$-Matrix-Based Integral-Equation Solver of Reduced Complexity and Controlled Accuracy for Solving Electrodynamic Problems
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Hierarchical LU Decomposition-based Preconditioners for BEM
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A Recursive Skeletonization Factorization Based on Strong Admissibility
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An $O (N)$ direct solver for integral equations on the plane
journal, March 2015

A fast direct solver for scattering problems involving elongated structures
journal, January 2007

Scattering from elongated objects: direct solution in O(N log2 N) operations
journal, January 1996

Fast Direct Solver for Essentially Convex Scatterers Using Multilevel Non-Uniform Grids
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Construction and Arithmetics of H -Matrices
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Fast construction of hierarchical matrix representation from matrix–vector multiplication
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A Fast Randomized Algorithm for Computing a Hierarchically Semiseparable Representation of a Matrix
journal, October 2011

Volterra approximation for low grazing angle shadowing on smooth ocean-like surfaces
journal, November 1995

Forward-backward: a new method for computing low-grazing angle scattering
journal, May 1996

Forward-backward iterative physical optics algorithm for computing the RCS of open-ended cavities
journal, February 2005

Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation
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On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime
journal, June 2015

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