A Data-Parallel Implementation of the Adaptive Fast Multipole Algorithm 1

Given an ensemble of n bodies in space whose interaction is governed by a potential function, the N-body problem is to calculate the force on each body in the ensemble that results from its interaction with all other bodies. An efficient algorithm for this problem is critical in the simulation of molecular dynamics, turbulent fluid flow, intergalactic matter and other problems. The fast multipole algorithm (FMA) developed by Greengard approximates the solution with bounded error in time O(n ). For non-uniform distributions of bodies, an adaptive variation of the algorithm is required to maintain this time complexity. The parallel execution of the FMA poses complex implementation issues in the decomposition of the problem over processors to reduce communication. As a result the 3D Adaptive FMA has, to our knowledge, never been implemented on a scalable parallel computer. This paper describes several variations on the parallel adaptive 3D FMA algorithm that are expressed using the data- parallel subset of the high-level parallel prototyping language Proteus. These formulations have implicit parallelism that is executed sequentially using the current Proteus execution system to yield some insight into the performance of the variations. Efforts underway will make it possible to directly generate vector code from the formulations, rendering them executable on a broad class of parallel computers.