Analyzing the Error Bounds of Multipole-Based Treecodes

Abstract: The problem of evaluating the potential due to a set of particles is an important and time- consuming one. The development of fast treecodes such as the Barnes-Hut and Fast Multipole Methods for n-body systems has enabled large scale simulations in astrophysics [9, 10, 13] and molecular dynamics [1]. Coupled with efficient parallel processing, these treecodes are capable of yielding several orders of magnitude improvement in performance [6, 14, 15]. In addition, treecodes have applications in the solution of dense linear systems arising from boundary element methods [3, 4, 5, 11, 12]. Using a p-term multipole expansion, the FMM reduces the complexity of a single timestep from O(n2) to O(p2n) and Barnes-Hut method reduces it to O(p2log n) for a uniform distribution. In this paper, we analyze the approximations introduced by these methods. We describe an algorithm that reduces the error significantly by selecting the multipole degree appropriately for different clusters. Furthermore, we show that for practical problem sizes, this increases the computational complexity marginally. We support our theoretical result with experiments in the context of particle simulations as well as boundary element methods. Our POSIX threads-based treecode yields excellent speedups on a 32 processor SGI Origin 2000, even for relatively small problems.