Solidfmm: A highly optimised library of operations on the solid harmonics for use in fast multipole methods

TL;DR

Solidfmm is a highly optimized C++ library that accelerates solid harmonic operations in fast multipole methods, significantly improving practical performance over naive implementations.

Contribution

The paper introduces solidfmm, a vectorized, optimized library for solid harmonic operations in fast multipole methods, with empirical performance improvements.

Findings

Translation operations exhibit empirical complexity of O(P^2)

Library outperforms naive implementations by orders of magnitude

Provides efficient, vectorized implementations of M2M, M2L, L2L

Abstract

We present solidfmm, a highly optimised C++ library for the solid harmonics as they are needed in fast multipole methods. The library provides efficient, vectorised implementations of the translation operations M2M, M2L, and L2L, and is available as free software. While asymptotically of complexity $O(P^3)$, for all practically relevant expansion orders, the translation operators display an empirical complexity of $O(P^2)$, outperforming the na\"ive implementation by orders of magnitude.