A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations

TL;DR

This paper introduces a parallel sweeping preconditioner for 3D Helmholtz equations that significantly improves computational efficiency and scalability, enabling high-frequency problem solving on large distributed systems.

Contribution

It presents a novel parallel implementation of a sweeping preconditioner for heterogeneous 3D Helmholtz equations, with detailed complexity analysis and open-source software.

Findings

Achieves O(γ^2 N^{4/3}) setup and O(γ N log N) application costs

Demonstrates competitive runtimes on thousands of cores for high-frequency problems

Provides open-source packages for the community

Abstract

A parallelization of a sweeping preconditioner for 3D 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({\gamma}^2 N^{4/3}) and O({\gamma} N log N), where {\gamma}({\omega}) 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".