A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations

Conclusions

A parallelization of the moving PML sweeping preconditioner has been presented which has allowed us to efficiently solve 3D Helmholtz equations in parallel with essentially $O(1)$ iterations, with the only observed frequency-dependence arising from a moderate growth in the PML size with increasing frequency. This size of the PML, $\gamma(\omega)$ was explained to result in a linear growth in the memory requirements of the preconditioner and a quadratic growth in the setup cost. Results were then presented for a variety of models, one of which had a velocity field which varied by ten orders of magnitude, and convergence was shown to be essentially independent of frequency for the challenging Overthrust model.

Also, despite the requirement that each panel must be solved against one at a time when applying the preconditioner, a custom approach was introduced and implemented which eliminates most of the communication associated with performing traditional black-box sparse-direct factorizations and solves over each subdomain. These implementations are now released as part of the open-source packages Clique and Parallel Sweeping Preconditioner (PSP). There are at least five important directions for future work:

• developing a heuristic for tailoring the PML profile to the velocity field,
• extending the preconditioner to more general discretizations and time-harmonic wave equations,
• finding a fast preconditioner for problems with large cavities (perhaps through more general local auxiliary problems),
• testing the performance improvements resulting from simultaneously factoring the subdomain problems and then redistributing the frontal trees, as well as a trsm approach to solving many right-hand sides, and
• carefully studying the spectrum of the preconditioned operator for various classes of velocity models.

A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations