A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations. A parallelization of a sweeping preconditioner for three-dimensional Helmholtz equations without large cavities is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are shown to be O(γ 2 N 4/3 ) and O(γNlogN), where γ(ω) denotes the modestly frequency-dependent number of grid points per perfectly matched layer. Several computational and memory improvements are introduced relative to using black-box sparse-direct solvers for the auxiliary problems, and competitive runtimes and iteration counts are reported for high-frequency problems distributed over thousands of cores. Two open-source packages are released along with this paper: parallel sweeping preconditioner (PSP) and the underlying distributed multifrontal solver, clique.

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  1. Buchatsky, Sagi; Treister, Eran: Full waveform inversion using extended and simultaneous sources (2021)
  2. Dastour, Hatef; Liao, Wenyuan: An optimal 13-point finite difference scheme for a 2D Helmholtz equation with a perfectly matched layer boundary condition (2021)
  3. Diwan, Ganesh C.; Mohamed, M. Shadi: Iterative solution with shifted Laplace preconditioner for plane wave enriched isogeometric analysis and finite element discretization for high-frequency acoustics (2021)
  4. Kouri, Drew P.; Ridzal, Denis; Tuminaro, Ray: KKT preconditioners for PDE-constrained optimization with the Helmholtz equation (2021)
  5. Kouri, Drew P.; Ridzal, Denis; Tuminaro, Ray: KKT preconditioners for PDE-constrained optimization with the Helmholtz equation (2021)
  6. Yavich, Nikolay; Khokhlov, Nikolay; Malovichko, Mikhail; Zhdanov, Michael S.: Contraction operator transformation for the complex heterogeneous Helmholtz equation (2021)
  7. Diwan, Ganesh C.; Mohamed, M. Shadi: Iterative solution of Helmholtz problem with high-order isogeometric analysis and finite element method at mid-range frequencies (2020)
  8. Graham, Ivan G.; Spence, Euan A.; Zou, Jun: Domain decomposition with local impedance conditions for the Helmholtz equation with absorption (2020)
  9. Fu, Shubin; Gao, Kai; Gibson, Richard L. jun.; Chung, Eric T.: An efficient high-order multiscale finite element method for frequency-domain elastic wave modeling (2019)
  10. Gander, Martin J.; Zhang, Hui: A class of iterative solvers for the Helmholtz equation: factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods (2019)
  11. Xiang, Xueshuang: Double source transfer domain decomposition method for Helmholtz problems (2019)
  12. Chávez, Gustavo; Turkiyyah, George; Zampini, Stefano; Keyes, David: Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients (2018)
  13. Liu, Fei; Ying, Lexing: Sparsify and sweep: an efficient preconditioner for the Lippmann-Schwinger equation (2018)
  14. Safin, Artur; Minkoff, Susan; Zweck, John: A preconditioned finite element solution of the coupled pressure-temperature equations used to model trace gas sensors (2018)
  15. Vion, Alexandre; Geuzaine, Christophe: Improved sweeping preconditioners for domain decomposition algorithms applied to time-harmonic Helmholtz and Maxwell problems (2018)
  16. Xu, Yingxiang: The influence of domain truncation on the performance of optimized Schwarz methods (2018)
  17. Zepeda-Núñez, Leonardo; Demanet, Laurent: Nested domain decomposition with polarized traces for the 2D Helmholtz equation (2018)
  18. Calandra, H.; Gratton, S.; Vasseur, X.: A geometric multigrid preconditioner for the solution of the Helmholtz equation in three-dimensional heterogeneous media on massively parallel computers (2017)
  19. Erlangga, Yogi A.; García Ramos, Luis; Nabben, Reinhard: The multilevel Krylov-multigrid method for the Helmholtz equation preconditioned by the shifted Laplacian (2017)
  20. Lahaye, D.; Vuik, C.: How to choose the shift in the shifted Laplace preconditioner for the Helmholtz equation combined with deflation (2017)

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