Static Fast Multipole Algorithm

[chelsea_avb]

Hey, Ive just started to work on the fast multipole algorithm and from what I get by reading about it is that the static case is the easiest but I am not able to find any simple albeit comprehensive source who talk about the formulation but rather talk about it in terms of a mixed form FMA. Any one know of any sources I could tap?

 

[Biftheunderstudy]

Can you explain what context you are using FMA in? And is this another way of saying fast multipole method for use in say, N-body calculations?

I don't have any direct experience, but I could probably dig up some sources where it's used in.

 

[Born2bwire]

I've never worked with it in the static case, just in terms of the Helmholtz wave equation. But the original papers were for static cases. I think the obvious place to start in that context would be the Rokhlin and Greengard papers. "Fast and Efficient Algorithms in Computational Electromagnetics" is an excellent resource on multilevel fast multipole algorithm for the Helmholtz equation. We do use the static case when the frequency is DC but I don't recall the text actually dealing with the static case specifically (which is the 1/r potential for us). But when we do the static case, we still use the low-frequency multilevel fast multipole algorithm that is discussed in that text. May not be the most accessible text though, I think your best bet is still to hunt through Rokhlin's papers. I'm sure you can find a suitable review paper.

 

[chelsea_avb]

Ohh.. thanks a lot, its just that, would translation be possible for the static case, Ive seen formulas where there is a direct translation that happens for the dynamic case, but havnt been able to find one for the static case...
Im trying to look for formlations that allow us to perform translations in the DC case... coz, from what I understand, translation would be the first step for the FMM right?

 

[Born2bwire]

Ohh.. thanks a lot, its just that, would translation be possible for the static case, Ive seen formulas where there is a direct translation that happens for the dynamic case, but havnt been able to find one for the static case...
Im trying to look for formlations that allow us to perform translations in the DC case... coz, from what I understand, translation would be the first step for the FMM right?

Well yeah, I mean if you are just talking about a single level FMA then translation is pretty much the only step. The aggregation and disaggregation processes are pretty much built off of the translation. Basically just taking the net contribution of a translation of potentials to a common local coordinate system is the process of aggregation. You could probably get a better understanding of the static case by working through the constituents of the theory and understanding how that fits into FMM and how DC changes it. For example, derive the addition theory at DC and that would be a strong indication of how the translators would work. Another thing you should think about searching for is the FMA for Laplace. That would be the differential equation for the static case while Helmholtz would be associated with the dynamic. Most of the early work was for the Laplacian since it was originally thought of as a way for describing static potentials like gravitational bodies. The dynamic case was developed later on. It's probably going to be more work for you to try and generalize the dynamic formulation back to a static one than starting over and deriving the theory based on static potential alone.