Error Controlled Multipole Methods for Arbitrary Particle Distributions

Ananth Grama
Department of Computer Sciences
Purdue University
W. Lafayette, IN 47907.

Hierarchical multipole methods such as the Fast Multipole Method (FMM) and Barnes-Hut (BH) method find extensive application in particle dynamics problems and boundary element solvers. The computational complexity of the dense matrix-vector product in these solvers is effectively reduced by hierarchical multipole approximations. It has been shown that the computational complexity of FMM can be bounded by O(n log n) for an arbitrarily distributed set of n particles using box-collapsing and fair-split techniques. The overall error introduced can be shown to grow linearly in the magnitude of total charge in the system. This error behavior can be reduced to logarithmic in total charge magnitude by using variable degree multipoles for uniform distributions. In this paper, we demonstrate the use of alternate multipole acceptance criteria to reduce the error in a single matrix-vector product for arbitrarily unstructured domains. These results are particularly important for boundary element solvers that typically mesh only surfaces of 3-D objects, resulting in highly unstructured domain models. Experimental results are provided on a variety of distributions to illustrate the error and computational characteristics of our method.