double-check Taylor-expansion FMM equation

[labarba]

The equation for Taylor-expanded FMM representation of an N-body problem seems to have typos both in Simon's and Christopher's thesis. I want the version I wrote in the paper double-checked.

In Simon's thesis:

What's psi? The N-body equation (2.5) uses phi for the potential and K for the kernel. There's no psi anywhere, so this must be a copy-paste error. And where are the multipoles? And the weights, q_i?

In Christopher's thesis:

His N-body equation is (2.119) and uses V for the potential and psi for the kernel. The expression above is supposed to be a Taylor expansion approximating the kernel, so I think there is a psi missing as the derivative operator appears to be applied to the multipoles here.

Also, both formulas are inconsistent with their use of the vector bold for indices of the derivative operator; I used the bolded ones.

@slayton58 @cdcooper84 I need extra eyes checking my equations in the paper because clearly many errors creep up in this.

[cdcooper84]

More than happy to help!

You are right, Lorena, there is a typo in both versions.

We got this notation from Li, Johnston and Krasny's 2009 JCP paper (http://www.sciencedirect.com/science/article/pii/S0021999109000916), equation (7). That equation is supposed to be the Taylor expansion of $\phi$ in SL's version ($\psi$ in CC's version). In SL, there is a $\phi$ missing between $D^k_y$ and $(\mathbf{x)_i,...$, and $\psi$ shouldn't be there. In CC, there is a $\psi$ missing between $D^k_y$ and $(\mathbf{x)_i,...$.

Another typo: in the arguments of the differential operator SL's $\mathbf{y}$ should be $\mathbf{y}_c$, and CC's $\mathbf{y}_j$ should be $\mathbf{y)_c$.

The k and y in the differential operator should be bold.

In my thesis I have the typos in Equation (2.120), but then I continue the derivation correctly.

Chris

[labarba]

OK, I suppose you don't feel like cloning the repo and processing the latex, so here's a screenshot of the version I currently have on the paper. Please more eyes on this to catch typos! Is this good? Thanks.

[cdcooper84]

Sorry, didn't realize you wanted me to edit the text!

That equation looks good. I'll read through the rest of the equations and let you know if I catch any more typos.

One thing (I don't know if you want to get to this level of detail, but in case you want to mention something about it): It's true that $\phi_i$ in Equation (2.13) is the solution of an elliptic equation, but only when the source term ($\mathbf{F}_i$) is a collection of delta functions, in which case the kernel is the free-space Green's function.

[cdcooper84]

Didn't find any more typos.

I see that Equation (2.14) is in a section entitled FMM, but it is the derivation of the treecode only. For the FMM you need to do another expansion in $\mathbf{x}_i$ about the local expansion center to show where the M2L, L2L and L2P come from, in case you want to add that later or something.