Evaluation of pair potentials is critical in a number of areas of physics. The classical$N$-body problem has its root in evaluating the Laplace potential, and has spawned tree-algorithms, the fast multipole method (FMM), as well as kernel independent approaches. Over the years, FMM for Laplace potential has had a profound impact on a number of disciplines as it has been possible to develop highly scalable parallel versions of these algorithms. This is in stark contrast to parallel algorithms for oscillatory potentials such as the Helmholtz potential. The principal bottlenecks to scalable parallelism are the computation and communication costs of operations necessary to traverse up, across, and down the tree. In this article, we analyze asymptotic costs for both computation and communication in a parallel implementation, and describe techniques to overcome bottlenecks and achieve high performance evaluation of the Helmholtz potential for different distributions of particles. We demonstrate that the resulting implementation has a load balancing effect that significantly reduces the time-to-solution and enhances the scale of problems that can be treated using full wave physics.

The Multi-Level Fast Multipole Algorithm (MLFMA), a variant of the fast multiple method (FMM) for problems with oscillatory potentials, significantly accelerates the solution of problems based on wave physics, such as those in electromagnetics and acoustics. Existing shared memory parallel approaches for MLFMA have adopted the bulk synchronous parallel (BSP) model. While the BSP approach has served well so far, it is prone to significant thread synchronization overheads, but more importantly fails to leverage the communication/computation overlap opportunities due to complicated data dependencies in MLFMA. In this paper, we develop a task parallel MLFMA implementation for shared memory architectures, and discuss optimizations to improve its performance. We then evaluate the new task parallel MLFMA implementation against a BSP implementation for a number of geometries. Our findings suggest that task parallelism is generally superior to the BSP model, and considering its potential advantages over the BSP model in a hybrid parallel setting, we see it to be a promising approach in addressing the scalability issues of MLFMA in large scale computations.

The fast multipole method (FMM) provides a fast and accurate method of solving large N-body problems with application in a broad range of physics problems. While there exists a large body of work for efficient implementation of the Laplace variant of the FMM algorithm, an optimized implementation of the Helmholtz variant of FMM which is used to study wave physics related phenomena is more complicated. In this study, we focus on the computationally expensive tree traversal operations of the Helmholtz FMM algorithm and describe a series of techniques to improve their performance. We demonstrate that the described techniques yield significant speedups (up to a factor of 4.71x) on a realistic surface geometry commonly encountered in Helmholtz problems.