Comparison of the Efficiency of Translation Operators Used in the Fast Multipole Method for the 3D Laplace Equation

We examine the practical implementation of a fast multipole method algorithm for the rapid summation of Laplace multipoles. Several translation operators with different asymptotic computational and memory complexities have been proposed for this problem. These algorithms include: Method 0 — the originally proposed matrix based translations due to Greengard andRokhlin (1987), Method 1 — the rotation, axial translation and rotation algorithm due to White and Martin Head-Gordon (1993), and Method 3 — the plane-wave version of the multipole-to local translation operator due to Greengard and Rokhlin (1997). We compare the algorithms on data sets of varying size and with varying imposed accuracy requirements. While from the
literature it would have been expected that method 2 would always be the method of choice, at least as far as computational speed is concerned, we find that this is not always the case. We find that as far as speed is concerned the choice between methods 1 and 2 depends on problem
size and error requirements. Method 2 is the algorithm of choice for large problems where high accuracy is required, though the advantage is not clear cut, especially if memory requirements are an issue. If memory is an issue, Method 1 is the method of choice for most problems. A new analysis of the computational complexities of the algorithms is provided, which explains
the observed results. We provide guidelines for choosing parameters for FMM algorithms.