Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences

Jinqiang Chen; Vandana Dwarka; Cornelis Vuik

doi:10.1007/978-3-031-54517-7_7

Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences

Jinqiang Chen^*, Vandana Dwarka, Cornelis Vuik

^*Corresponding author for this work

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

Abstract

We present a matrix-free parallel iterative solver for the Helmholtz equation related to applications in seismic problems and study its parallel performance. We apply Krylov subspace methods, GMRES, Bi-CGSTAB and IDR(s), to solve the linear system obtained from a second-order finite difference discretization. The Complex Shifted Laplace Preconditioner (CSLP) is employed to improve the convergence of Krylov solvers. The preconditioner is approximately inverted by multigrid iterations. For parallel computing, the global domain is partitioned blockwise. The standard MPI library is employed for data communication. The matrix-vector multiplication and preconditioning operator are implemented in a matrix-free way instead of constructing large, memory-consuming coefficient matrices. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that the matrix-free parallel solution method has satisfactory parallel performance and weak scalability. It allows us to solve larger problems in parallel to obtain more accurate numerical solutions.

Original language	English
Title of host publication	Mathematics in Industry
Subtitle of host publication	SCEE 2022, Amsterdam, The Netherlands, July 2022
Editors	M. van Beurden, N.V. Budko, G. Ciuprina, W. Schilders, H. Bansal, R. Barbulescu
Publisher	Springer
Pages	61-68
Number of pages	8
Edition	1
ISBN (Electronic)	978-3-031-54517-7
ISBN (Print)	978-3-031-54516-0
DOIs	https://doi.org/10.1007/978-3-031-54517-7_7
Publication status	Published - 2024
Event	Scientific Computing in Electrical Engineering - Amsterdam, Netherlands Duration: 11 Jul 2022 → 14 Jul 2022

Publication series

Name	Mathematics in Industry
Volume	43
ISSN (Print)	1612-3956
ISSN (Electronic)	2198-3283

Conference

Conference	Scientific Computing in Electrical Engineering
Abbreviated title	SCEE 2022
Country/Territory	Netherlands
City	Amsterdam
Period	11/07/22 → 14/07/22

Bibliographical note

Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

Access to Document

10.1007/978-3-031-54517-7_7

Cite this

Chen, J., Dwarka, V., & Vuik, C. (2024). Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences. In M. van Beurden, N. V. Budko, G. Ciuprina, W. Schilders, H. Bansal, & R. Barbulescu (Eds.), Mathematics in Industry: SCEE 2022, Amsterdam, The Netherlands, July 2022 (1 ed., pp. 61-68). (Mathematics in Industry; Vol. 43). Springer. https://doi.org/10.1007/978-3-031-54517-7_7

Chen, Jinqiang ; Dwarka, Vandana ; Vuik, Cornelis. / Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences. Mathematics in Industry: SCEE 2022, Amsterdam, The Netherlands, July 2022. editor / M. van Beurden ; N.V. Budko ; G. Ciuprina ; W. Schilders ; H. Bansal ; R. Barbulescu. 1. ed. Springer, 2024. pp. 61-68 (Mathematics in Industry).

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title = "Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences",

abstract = "We present a matrix-free parallel iterative solver for the Helmholtz equation related to applications in seismic problems and study its parallel performance. We apply Krylov subspace methods, GMRES, Bi-CGSTAB and IDR(s), to solve the linear system obtained from a second-order finite difference discretization. The Complex Shifted Laplace Preconditioner (CSLP) is employed to improve the convergence of Krylov solvers. The preconditioner is approximately inverted by multigrid iterations. For parallel computing, the global domain is partitioned blockwise. The standard MPI library is employed for data communication. The matrix-vector multiplication and preconditioning operator are implemented in a matrix-free way instead of constructing large, memory-consuming coefficient matrices. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that the matrix-free parallel solution method has satisfactory parallel performance and weak scalability. It allows us to solve larger problems in parallel to obtain more accurate numerical solutions.",

author = "Jinqiang Chen and Vandana Dwarka and Cornelis Vuik",

note = "Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.; Scientific Computing in Electrical Engineering, SCEE 2022 ; Conference date: 11-07-2022 Through 14-07-2022",

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Chen, J , Dwarka, V & Vuik, C 2024, Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences. in M van Beurden, NV Budko, G Ciuprina, W Schilders, H Bansal & R Barbulescu (eds), Mathematics in Industry: SCEE 2022, Amsterdam, The Netherlands, July 2022. 1 edn, Mathematics in Industry, vol. 43, Springer, pp. 61-68, Scientific Computing in Electrical Engineering, Amsterdam, Netherlands, 11/07/22. https://doi.org/10.1007/978-3-031-54517-7_7

Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences. / Chen, Jinqiang ; Dwarka, Vandana ; Vuik, Cornelis.
Mathematics in Industry: SCEE 2022, Amsterdam, The Netherlands, July 2022. ed. / M. van Beurden; N.V. Budko; G. Ciuprina; W. Schilders; H. Bansal; R. Barbulescu. 1. ed. Springer, 2024. p. 61-68 (Mathematics in Industry; Vol. 43).

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

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N1 - Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

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N2 - We present a matrix-free parallel iterative solver for the Helmholtz equation related to applications in seismic problems and study its parallel performance. We apply Krylov subspace methods, GMRES, Bi-CGSTAB and IDR(s), to solve the linear system obtained from a second-order finite difference discretization. The Complex Shifted Laplace Preconditioner (CSLP) is employed to improve the convergence of Krylov solvers. The preconditioner is approximately inverted by multigrid iterations. For parallel computing, the global domain is partitioned blockwise. The standard MPI library is employed for data communication. The matrix-vector multiplication and preconditioning operator are implemented in a matrix-free way instead of constructing large, memory-consuming coefficient matrices. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that the matrix-free parallel solution method has satisfactory parallel performance and weak scalability. It allows us to solve larger problems in parallel to obtain more accurate numerical solutions.

AB - We present a matrix-free parallel iterative solver for the Helmholtz equation related to applications in seismic problems and study its parallel performance. We apply Krylov subspace methods, GMRES, Bi-CGSTAB and IDR(s), to solve the linear system obtained from a second-order finite difference discretization. The Complex Shifted Laplace Preconditioner (CSLP) is employed to improve the convergence of Krylov solvers. The preconditioner is approximately inverted by multigrid iterations. For parallel computing, the global domain is partitioned blockwise. The standard MPI library is employed for data communication. The matrix-vector multiplication and preconditioning operator are implemented in a matrix-free way instead of constructing large, memory-consuming coefficient matrices. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that the matrix-free parallel solution method has satisfactory parallel performance and weak scalability. It allows us to solve larger problems in parallel to obtain more accurate numerical solutions.

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Chen J , Dwarka V , Vuik C. Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences. In van Beurden M, Budko NV, Ciuprina G, Schilders W, Bansal H, Barbulescu R, editors, Mathematics in Industry: SCEE 2022, Amsterdam, The Netherlands, July 2022. 1 ed. Springer. 2024. p. 61-68. (Mathematics in Industry). doi: 10.1007/978-3-031-54517-7_7