Fast Multipole Method: Explained by Derivator

[Derivator]

hi,

barnes hut method approximates the interaction by treating a bunch of far away particles as one big particle located in the center of mass of the bunch of particles.

My lecture notes say, that the fast multipole methode not only does the above 'barnes hut' approximation, but also assumes, that the potential varies only slowly in the region where the particle for which the interactions should be computed is located.

In the fast multipole method, there is a multipole expansion and a taylor expansion. Is the taylor expansion modelling the slow variation of the potential, or is the taylor expansion responsible for the fact that far away bunches of particles are treated as one big single particle? Or is this for what the multipole expnsion is responsible for?

Could somone please explain?

best,
derivator

 

[cgk]

The taylor expansion /is/ the multipole expansion. In this case you have two: One for the far away bunch of particles, and one for the region for which the interaction is computed. Both are in principle diffuse electron distributions, but here you approximate both as point multipoles.

But I would actually recommend not caring about such implementation details very much. I'm not aware of even a single program which uses the "fast multipole method" for anything; mainly because it turns out to be actually slower and less inaccurate than good implementations of the standard ways of treating the electron interaction in Hartree-Fock/Kohn-Sham programs (density fitting/resolution of the identity approximation (those two are the same thing) or screened 4index integrals). The only related implementation I can think of is Turbomole's MARIJ (multipole assisted resolution of the identity) approximation, but this is something rather different in practice, and becomes only relevant in very specific combinations of molecule size basis sets, and molecule geometry... combinations most computational chemists don't care much about.

 

[Derivator]

ok, in the upward pass, all the multipole coefficients for the cells of different resolution levels are created. In the downward pass, the multipole coefficients in interactive cells are converted to taylor expansion coefficients, the taylor coefficients of far away cells are taken from the next lower resolution level (on this lower resolution level the far away cells of the higher resolution level are interactive cells and thus the taylor coefficients are obtained from the multipole coefficients). The interaction of particles in near cells is computed directly on the finest resolution level.

It is easy to see, that far away cells are approximated by coarse cells, simply beacuse only the taylor expansion coefficients from low resolutions cells are taken for the high resolution far cells.

But the lecture notes say, that the potential of far away mass distributions is only slowly varying in the neighborhood of an observation point. I can't see, where this is used in the Far Multipole Method.

 

[Derivator]

i don't want to be rude, but... anybody?
*push*

 

[Derivator]

hi,
My lecture notes say, that the fast multipole methode not only does the above 'barnes hut' approximation, but also assumes, that the potential varies only slowly in the region where the particle for which the interactions should be computed is located.

i've got an idea:

is it possible, that the intention was the following:

"slowly varying functions are approximated quite reasonable by a comparatively short taylor expansion"
Is this statement true (in any sense), at all?