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