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Hierarchical matrix techniques for low- and high-frequency Helmholtz problems


Banjai, L; Hackbusch, W (2008). Hierarchical matrix techniques for low- and high-frequency Helmholtz problems. IMA Journal of Numerical Analysis, 28(1):46-79.

Abstract

In this paper, we discuss the application of hierarchical matrix techniques to the solution of Helmholtz problems with large wave number {kappa} in 2D. We consider the Brakhage–Werner integral formulation of the problem discretized by the Galerkin boundary-element method. The dense n x n Galerkin matrix arising from this approach is represented by a sum of an Formula -matrix and an Formula 2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the Formula 2-matrix. We present a new approach to dealing with the numerical instability problems of this expansion: the parts of the matrix that can cause problems are approximated in a stable way by an Formula -matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the Formula -matrix. Further, an approximate LU decomposition of such a recompressed Formula -matrix is an effective preconditioner. We prove that the construction of the matrices as well as the matrix-vector product can be performed in almost linear time in the number of unknowns. Numerical experiments for scattering problems in 2D are presented, where the linear systems are solved by a preconditioned iterative method.

Abstract

In this paper, we discuss the application of hierarchical matrix techniques to the solution of Helmholtz problems with large wave number {kappa} in 2D. We consider the Brakhage–Werner integral formulation of the problem discretized by the Galerkin boundary-element method. The dense n x n Galerkin matrix arising from this approach is represented by a sum of an Formula -matrix and an Formula 2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the Formula 2-matrix. We present a new approach to dealing with the numerical instability problems of this expansion: the parts of the matrix that can cause problems are approximated in a stable way by an Formula -matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the Formula -matrix. Further, an approximate LU decomposition of such a recompressed Formula -matrix is an effective preconditioner. We prove that the construction of the matrices as well as the matrix-vector product can be performed in almost linear time in the number of unknowns. Numerical experiments for scattering problems in 2D are presented, where the linear systems are solved by a preconditioned iterative method.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Helmholtz equation; boundary element method; hierarchical matrices
Language:English
Date:2008
Deposited On:09 Nov 2009 00:01
Last Modified:05 Apr 2016 13:23
Publisher:Oxford University Press
ISSN:0272-4979
Publisher DOI:https://doi.org/10.1093/imanum/drm001

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