# Hierarchical Particle-Mesh: an FFT-accelerated Fast Multipole Method

## TL;DR

This paper introduces a modified Fast Multipole Method that uses FFT-accelerated gridlets to efficiently compute gravitational fields with preserved accuracy, applicable to various Green functions.

## Contribution

It presents a novel FFT-accelerated FMM variant using gridlets to improve computational efficiency while maintaining accuracy.

## Key findings

- Reduces computational cost of FMM calculations.
- Preserves multipole moment accuracy with gridlet approximation.
- Applicable to any Green function in gravitational simulations.

## Abstract

I describe a modification to the original Fast Multipole Method (FMM) of Greengard & Rokhlin that approximates the gravitation field of an FMM cell as a small uniform grid (a "gridlet") of effective masses. The effective masses on a gridlet are set from the requirement that the multipole moments of the FMM cells are reproduced exactly, hence preserving the accuracy of the gravitational field representation. The calculation of the gravitational field from a multipole expansion can then be computed for all multipole orders simultaneously, with a single Fast Fourier Transform, significantly reducing the computational cost at a given value of the required accuracy. The described approach belongs to the class of "kernel independent" variants of the FMM method and works with any Green function.