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. 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)
  2. Erlangga, Yogi A.; García Ramos, Luis; Nabben, Reinhard: The multilevel Krylov-multigrid method for the Helmholtz equation preconditioned by the shifted Laplacian (2017)
  3. Lahaye, D.; Vuik, C.: How to choose the shift in the shifted Laplace preconditioner for the Helmholtz equation combined with deflation (2017)
  4. Stolk, Christiaan C.: An improved sweeping domain decomposition preconditioner for the Helmholtz equation (2017)
  5. Eslaminia, Mehran; Guddati, Murthy N.: A double-sweeping preconditioner for the Helmholtz equation (2016)
  6. Liu, Fei; Ying, Lexing: Additive sweeping preconditioner for the Helmholtz equation (2016)
  7. Liu, Fei; Ying, Lexing: Recursive sweeping preconditioner for the three-dimensional Helmholtz equation (2016)
  8. Stolk, Christiaan C.: A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory (2016)
  9. Zepeda-Núñez, Leonardo; Demanet, Laurent: The method of polarized traces for the 2D Helmholtz equation (2016)
  10. Tsuji, P.; Tuminaro, R.: Augmented AMG-shifted Laplacian preconditioners for indefinite Helmholtz problems. (2015)
  11. Vion, A.; Geuzaine, C.: Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem (2014)
  12. C.Stolk, Christiaan: A rapidly converging domain decomposition method for the Helmholtz equation (2013)
  13. Poulson, Jack; Engquist, Björn; Li, Siwei; Ying, Lexing: A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations (2013)