A Parallel Directional Fast Multipole Method

TL;DR: In this article, a parallel directional fast multipole method (FMM) was proposed for solving the Helmholtz kernel in three dimensions with a more restrictive low-rank criterion than that of the low-frequency regime.

Abstract: This paper introduces a parallel directional fast multipole method (FMM) for solving $N$-body problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a more restrictive low-rank criterion than that of the low-frequency regime, and thus effective parallelizations must adapt to the modified data dependencies. We propose a simple partition at a fixed level of the octree and show that, if the partitions are properly balanced between $p$ processes, the overall runtime is essentially $\mathcal{O}{N \log N/p + p}$. By the structure of the low-rank criterion, we are able to avoid communication at the top of the octree. We demonstrate the effectiveness of our parallelization on several challenging models.