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. 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)
  2. Chávez, Gustavo; Turkiyyah, George; Zampini, Stefano; Keyes, David: Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients (2018)
  3. Liu, Fei; Ying, Lexing: Sparsify and sweep: an efficient preconditioner for the Lippmann-Schwinger equation (2018)
  4. Safin, Artur; Minkoff, Susan; Zweck, John: A preconditioned finite element solution of the coupled pressure-temperature equations used to model trace gas sensors (2018)
  5. Vion, Alexandre; Geuzaine, Christophe: Improved sweeping preconditioners for domain decomposition algorithms applied to time-harmonic Helmholtz and Maxwell problems (2018)
  6. Xu, Yingxiang: The influence of domain truncation on the performance of optimized Schwarz methods (2018)
  7. Zepeda-Núñez, Leonardo; Demanet, Laurent: Nested domain decomposition with polarized traces for the 2D Helmholtz equation (2018)
  8. 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)
  9. Erlangga, Yogi A.; García Ramos, Luis; Nabben, Reinhard: The multilevel Krylov-multigrid method for the Helmholtz equation preconditioned by the shifted Laplacian (2017)
  10. Lahaye, D.; Vuik, C.: How to choose the shift in the shifted Laplace preconditioner for the Helmholtz equation combined with deflation (2017)
  11. Stolk, Christiaan C.: An improved sweeping domain decomposition preconditioner for the Helmholtz equation (2017)
  12. Treister, Eran; Haber, Eldad: Full waveform inversion guided by travel time tomography (2017)
  13. Eslaminia, Mehran; Guddati, Murthy N.: A double-sweeping preconditioner for the Helmholtz equation (2016)
  14. Liu, Fei; Ying, Lexing: Recursive sweeping preconditioner for the three-dimensional Helmholtz equation (2016)
  15. Liu, Fei; Ying, Lexing: Additive sweeping preconditioner for the Helmholtz equation (2016)
  16. Stolk, Christiaan C.: A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory (2016)
  17. Zepeda-Núñez, Leonardo; Demanet, Laurent: The method of polarized traces for the 2D Helmholtz equation (2016)
  18. Tsuji, P.; Tuminaro, R.: Augmented AMG-shifted Laplacian preconditioners for indefinite Helmholtz problems. (2015)
  19. Vion, A.; Geuzaine, C.: Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem (2014)
  20. C. Stolk, Christiaan: A rapidly converging domain decomposition method for the Helmholtz equation (2013)

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