Characterization of the errors of the FMM in particle simulations

TL;DR

This paper provides an extensive experimental analysis of the error characteristics of the Fast Multipole Method (FMM) in particle simulations, offering insights into how various parameters affect accuracy.

Contribution

It presents the first large-scale empirical study of FMM errors, correlating accuracy with different algorithmic parameters and visualizing error distributions.

Findings

Error distribution varies spatially within simulations.

Multipole and local expansion errors significantly impact overall accuracy.

Parameter choices influence the balance between speed and precision.

Abstract

The Fast Multipole Method (FMM) offers an acceleration for pairwise interaction calculation, known as $N$-body problems, from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ with $N$ particles. This has brought dramatic increase in the capability of particle simulations in many application areas, such as electrostatics, particle formulations of fluid mechanics, and others. Although the literature on the subject provides theoretical error bounds for the FMM approximation, there are not many reports of the measured errors in a suite of computational experiments. We have performed such an experimental investigation, and summarized the results of about 1000 calculations using the FMM algorithm, to characterize the accuracy of the method in relation with the different parameters available to the user. In addition to the more standard diagnostic of the maximum error, we supply illustrations of the spatial distribution of the errors, which offers visual evidence of all the contributing factors to the overall approximation accuracy: multipole expansion, local expansion, hierarchical spatial decomposition (interaction lists, local domain, far domain). This presentation is a contribution to any researcher wishing to incorporate the FMM acceleration to their application code, as it aids in understanding where accuracy is gained or compromised.