Solving the three-dimensional high-frequency Helmholtz equation using contour integration and polynomial preconditioning

Xiao Liu; Yuanzhe Xi; Yousef Saad; Maarten V. de Hoop

doi:10.1137/18M1228128

Solving the three-dimensional high-frequency Helmholtz equation using contour integration and polynomial preconditioning

Xiao Liu, Yuanzhe Xi, Yousef Saad, Maarten V. de Hoop

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10 Scopus citations

Abstract

We propose an iterative solution method for the three-dimensional high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solution in certain invariant subspaces is approximated by solving complex-shifted linear systems, resulting in faster GMRES iterations due to the restricted spectrum. The shifted systems are solved by exploiting a polynomial fixed-point iteration, which is a robust scheme even if the magnitude of the shift is small. Numerical tests in three dimensions indicate that O(n¹/³) matrix-vector products are needed to solve a high-frequency problem with a matrix size n with high accuracy. The method has a small storage requirement, can be applied to both dense and sparse linear systems, and is highly parallelizable.

Original language	English (US)
Pages (from-to)	58-82
Number of pages	25
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	41
Issue number	1
DOIs	https://doi.org/10.1137/18M1228128
State	Published - 2020
Externally published	Yes

Bibliographical note

Funding Information:
∗Received by the editors November 21, 2018; accepted for publication (in revised form) by L. Giraud November 4, 2019; published electronically January 9, 2020. https://doi.org/10.1137/18M1228128 Funding: The work of the second and third authors was supported by National Science Foundation grant DMS-1521573 and the Minnesota Supercomputing Institute. The work of the fourth author was supported by the Simons Foundation under the MATH + X program, National Science Foundation grant DMS-1559587, the corporate members of the Geo-Mathematical Group at Rice University, and Total. †Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005 ([email protected], [email protected]). ‡Department of Mathematics, Emory University, Atlanta, GA 30322 ([email protected]). §Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455 ([email protected]).

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics

Keywords

Cauchy integral
Helmholtz preconditioner
Polynomial iteration
Shifted Laplacian

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10.1137/18M1228128

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abstract = "We propose an iterative solution method for the three-dimensional high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solution in certain invariant subspaces is approximated by solving complex-shifted linear systems, resulting in faster GMRES iterations due to the restricted spectrum. The shifted systems are solved by exploiting a polynomial fixed-point iteration, which is a robust scheme even if the magnitude of the shift is small. Numerical tests in three dimensions indicate that O(n1/3) matrix-vector products are needed to solve a high-frequency problem with a matrix size n with high accuracy. The method has a small storage requirement, can be applied to both dense and sparse linear systems, and is highly parallelizable.",

keywords = "Cauchy integral, Helmholtz preconditioner, Polynomial iteration, Shifted Laplacian",

author = "Xiao Liu and Yuanzhe Xi and Yousef Saad and {de Hoop}, {Maarten V.}",

note = "Funding Information: ∗Received by the editors November 21, 2018; accepted for publication (in revised form) by L. Giraud November 4, 2019; published electronically January 9, 2020. https://doi.org/10.1137/18M1228128 Funding: The work of the second and third authors was supported by National Science Foundation grant DMS-1521573 and the Minnesota Supercomputing Institute. The work of the fourth author was supported by the Simons Foundation under the MATH + X program, National Science Foundation grant DMS-1559587, the corporate members of the Geo-Mathematical Group at Rice University, and Total. †Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005 ([email protected], [email protected]). ‡Department of Mathematics, Emory University, Atlanta, GA 30322 ([email protected]). §Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455 ([email protected]). Publisher Copyright: {\textcopyright} 2020 Society for Industrial and Applied Mathematics",

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AU - Xi, Yuanzhe

AU - Saad, Yousef

AU - de Hoop, Maarten V.

N1 - Funding Information: ∗Received by the editors November 21, 2018; accepted for publication (in revised form) by L. Giraud November 4, 2019; published electronically January 9, 2020. https://doi.org/10.1137/18M1228128 Funding: The work of the second and third authors was supported by National Science Foundation grant DMS-1521573 and the Minnesota Supercomputing Institute. The work of the fourth author was supported by the Simons Foundation under the MATH + X program, National Science Foundation grant DMS-1559587, the corporate members of the Geo-Mathematical Group at Rice University, and Total. †Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005 ([email protected], [email protected]). ‡Department of Mathematics, Emory University, Atlanta, GA 30322 ([email protected]). §Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455 ([email protected]). Publisher Copyright: © 2020 Society for Industrial and Applied Mathematics

PY - 2020

Y1 - 2020

N2 - We propose an iterative solution method for the three-dimensional high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solution in certain invariant subspaces is approximated by solving complex-shifted linear systems, resulting in faster GMRES iterations due to the restricted spectrum. The shifted systems are solved by exploiting a polynomial fixed-point iteration, which is a robust scheme even if the magnitude of the shift is small. Numerical tests in three dimensions indicate that O(n1/3) matrix-vector products are needed to solve a high-frequency problem with a matrix size n with high accuracy. The method has a small storage requirement, can be applied to both dense and sparse linear systems, and is highly parallelizable.

AB - We propose an iterative solution method for the three-dimensional high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solution in certain invariant subspaces is approximated by solving complex-shifted linear systems, resulting in faster GMRES iterations due to the restricted spectrum. The shifted systems are solved by exploiting a polynomial fixed-point iteration, which is a robust scheme even if the magnitude of the shift is small. Numerical tests in three dimensions indicate that O(n1/3) matrix-vector products are needed to solve a high-frequency problem with a matrix size n with high accuracy. The method has a small storage requirement, can be applied to both dense and sparse linear systems, and is highly parallelizable.

KW - Cauchy integral

KW - Helmholtz preconditioner

KW - Polynomial iteration

KW - Shifted Laplacian

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Solving the three-dimensional high-frequency Helmholtz equation using contour integration and polynomial preconditioning

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