Need for translation theorem in Fast Multipole Method

Question

I have rephrased my original question to be clearer:

From my understanding, a multipole expansion is used to approximate the potential from a cluster of points, and can be evaluated at any point sufficiently far away. Suppose you have a cluster of points in box A which is centered at some point in space Q. The potential from these points can be approximated by the multipole potential evaluated at some point P using:

Where the primed variables are the coordinates P - Q.

Now suppose we want to find the multipole potential for the parent of box A. Of course, we could start over by finding the cluster of points in the parent box and recomputing the multipole expansion. But surely, we want to reduce computation time so it is better to form the parent multipole expansion from the parent’s children.

If we find the multipole expansion given earlier for all the children, the potential at some point P due to the parent will of course be the sum of the children’s multipole expansions evaluated at P. (note: the math is correct for this, and it makes complete sense because potentials follow the superposition principle)

However, the translation theorem commonly used to convert children multipole expansions to their parent is given by:

It seems obvious one could just add the multipole expansions to form the parent expansion. Why is a different translation theorem used when converting children expansions to their parent expansion? Is this translation theorem applied in a different setting?

source: A short course on fast multipole methods

Answer 1

The FMM has been exhaustively studied, so if people aren't doing something that seems obvious, then there's likely a fairly obvious reason they aren't doing it. This isn't to say there's nothing new to discover, but it's well-trodden ground.

I think the main thing you're missing here is the fact that the multipole expansion for each box is written in a local coordinate system, e.g. relative to the center of each box. Suppose your box A has eight children, each centered at $Q_i,i=1,\ldots,8$. You are correct that one could evaluate the eight multipole expansions at the point $P$ and obtain the correct result, but realize that you must express $P$ in coordinates relative to each $Q_i$ to get the right answer. Simply adding up the coefficients and evaluating the multipole expansion using $Q$ will not produce the correct answer. Instead, you must re-center these expansions about the parent point $Q$ using the above theorem and then add them together. For reference, see the Wikipedia page on the solid harmonics addition theorem.

The following pertains to a question posed in your original post. I'm leaving it here.

On the point of computing multipole-to-local (outer-to-inner) translations between boxes at different levels, let's consider the worst-case scenario.

Imagine you've got two particles located at positions $\mathbf{r}$ and $\mathbf{r}'$. Box $A$ with diameter $d$ contains the point $\mathbf{r}$ and box $B$ with diameter $2d$ contains the point $\mathbf{r}'$, and let the boxes be separated by one box of diameter $d$. Define $\mathbf{r}-\mathbf{r}'=\mathbf{t}+\mathbf{v}$, where $\mathbf{t}$ is the vector from the center of box $B$ to the center of box $A$.

The convergence of the FMM series expansion relies on the ratio $v/t<1$ being small. It can easily be shown that when you place the points $\mathbf{r},\mathbf{r}'$ in the opposite corners of their respective boxes, the ratio $v/t=1$, invalidating the addition theorem. For configurations close to this one, the ratio will still be close to 1 and a huge number of terms is required to achieve any useful accuracy.