Title
A matrix-free parallel solution method for the three-dimensional heterogeneous Helmholtz equation
Author
Chen, J. (TU Delft Numerical Analysis) 
Dwarka, V.N.S.R. (TU Delft Numerical Analysis)
Vuik, Cornelis (TU Delft Delft Institute of Applied Mathematics)
Date
2024
Abstract
The Helmholtz equation is related to seismic exploration, sonar, antennas, and medical imaging applications. It is one of the most challenging problems to solve in terms of accuracy and convergence due to the scalability issues of the numerical solvers. For 3D large-scale applications, high-performance parallel solvers are also needed. In this paper, a matrix-free parallel iterative solver is presented for the three-dimensional (3D) heterogeneous Helmholtz equation. We consider the preconditioned Krylov subspace methods for solving the linear system obtained from finite-difference discretization. The Complex Shifted Laplace Preconditioner (CSLP) is employed since it results in a linear increase in the number of iterations as a function of the wavenumber. The preconditioner is approximately inverted using one parallel 3D multigrid cycle. For parallel computing, the global domain is partitioned blockwise. The matrix-vector multiplication and preconditioning operator are implemented in a matrix-free way instead of constructing large, memory-consuming coefficient matrices. Numerical experiments of 3D model problems demonstrate the robustness and outstanding strong scaling of our matrix-free parallel solution method. Moreover, the weak parallel scalability indicates our approach is suitable for realistic 3D heterogeneous Helmholtz problems with minimized pollution error.
Subject
Helmholtz equation
parallel computation
matrix-free
geometric multigrid
preconditioner
scalability
To reference this document use:
http://resolver.tudelft.nl/uuid:01b73100-531d-4923-8c36-bcbc4b4b1855
DOI
https://doi.org/10.1553/etna_vol59s270
Embargo date
2024-07-15
ISSN
1068-9613
Source
Electronic Transactions on Numerical Analysis, 59, 270 - 294
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.
Part of collection
Institutional Repository
Document type
journal article
Rights
© 2024 J. Chen, V.N.S.R. Dwarka, Cornelis Vuik