Improving the FMM performance using optimal group size on heterogeneous system architectures

Abstract

The fast multipole method (FMM) is commonly used to speed-up the time to solution of a wide diversity of N-body type problems. To use the FMM, the elements that constitute the geometry of the problem are clustered into groups of a given size (D) that may deeply vary the time to solution of the FMM. The optimal value of D, in the sense of minimizing the time cost, is unknown beforehand and it depends on several factors. Nevertheless, during the solver setup, it is possible to produce clusters of varying size D and to estimate the time to solution associated with each D. In this paper, we use octree structures to efficiently perform clustering and time cost estimation to find the optimal group size for the FMM implementations on a heterogeneous architecture. In addition, two different frameworks have been analyzed: single-level FMM and fast Fourier transform FMM (FMM-FFT). We found that the sensitivity of the time cost to the parameter D depends on factors, such as the problem size or the implementation of the FMM framework. Moreover, we observed that the time cost may be conspicuously reduced if a proper D is employed.