Fast multipole method: help on tutorial

Question

I have some hard time with the FMM (Fast multipole method). I try to understand the basics with: "http://www.umiacs.umd.edu/labs/cvl/pirl/vikas/publications/FMM_tutorial.pdf". So here are my questions:

1. [Solved: we need to think locally in order to not include source point inside the sphere] p7: why the middleman method only work for regular potential ? (and not for singular ones)

2. [Solved: we need accuracy too] p7 what is the interest in multiple center ? We can easily see on that the figure (b) is simpler that the (c)! One center seem to give less calculus wich is the objective ]2

3. What does :S-expansion and R-expansion stand for ? Is it "Regular-Expansion" and "Singular-Expansion" ?

4.[Solved]I don't understand the graphic. Why the branch come from source to expznsion center in case of R expansion and from expansion center to target in S expansion. I can't make the link between the image/graphic and the formula.

5. I am completely lost with translation: Is there an easy way to compute the matrix $(F|G)$ ? I don't understand what is done with theses opérator.Here is an example and i have no idea of what the author is doing... If you could give more details please :)

Answer 1

On your first question: That sentence is false; the text itself goes on to state in the very next paragraph under which circumstances the method can be used for singular potentials. The case in which it can't be used is with targets and sources spatially intermingled and a singular potential, because in this case there's no single expansion that will work for all source-target pairs.

On your second question: The aim is not to minimize the computational effort. (If it were, we'd choose $p=0$ and have $0$ computational effort and an estimate of $0$ for the total interaction energy). The aim is to minimize the computational effort given a target accuracy. Depending on circumstances, it may be that using three expansion centres is more efficient in boosting the accuracy than increasing $p$ in the expansion. This will be the case in particular if the sources and/or the targets are located in clusters.

Answer 2

3.[solved]

I can imagine (even if i am not sure it is defined the same way by all specialists) that

• R-expansion stands for Receiver (cause we focused on the configuration where target/receivers are in the target box)
• S-expansion stands for Source (the sources should be in the box!)