Improving error bounds for multipole-based treecodes

Rapid evaluation of potentials in particle systems is an important and time-consuming step in many physical simulations. Over the past decade (1988-98), the development of treecodes such as the Fast Multipole Method (FMM) and the Barnes-Hut method has enabled large scale simulations in domains such as astrophysics, molecular dynamics, and material science. FMM and related methods rely on fixed degree polynomial (p) approximations of the potential of a set of points in a hierarchy. We present a sequence of results to illustrate that keeping the multipole degree constant can lead to large aggregate errors. An alternate strategy based on a careful selection of the multipole degree leads to asymptotically lower errors; while incurring minimal computation overhead for practical problem sizes. The paper presents theoretical results for computing the degree of a particle cluster interaction, the error associated with the interaction, the error associated with a particle for all of its interactions, and the computational complexity of the new method. These results show that it is possible to reduce the simulation error asymptotically while incurring minimal computational overhead. The paper also presents experimental validation of these results on a 32 processor Origin 2000 in the context of problems ranging from astrophysics to boundary element solvers. In addition to verifying theoretical results, we also show that it is possible to achieve excellent parallel speedup for the treecode.