Take a field $\phi(\bf{x})$ created from a charge distribution contained within a radius $R$. The multipole expansion in spherical harmonics $Y_{\ell,m}$ outside of $R$ is approximated by:
$$ \phi({\bf x}) \approx \frac{1}{4\pi \epsilon_0} \sum_{\ell=0}^{\ell_{MAX}} \sum_{m=-\ell}^{\ell} \frac{4\pi}{2\ell +1} \alpha_{\ell, m} \frac{Y_{\ell,m}(\theta, \phi)}{r^{\ell +1}} $$
Given only the finite set of the multipole moments $\alpha_{\ell,m}$, is it possible to find a charge distribution of $N$ discrete charges $(q_i, r_i, \theta_i, \phi_i)$ that "best-fit" this potential? Right now I'm using a simple minima finder over the $3^N$ variables (see note 1), but I'm wondering if there is any prior work/observations on inverting the multipole moments.
Note 1
If the positions of the charges are fixed, the charge magnitudes $q_i$ are simply linear combinations:
$$ \alpha_{\ell, m} = \sum_i^N q_i Y^*_{\ell,m}(\theta_i, \phi_i) r_i^\ell $$
thus simple linear algebra gives the best-fit charge magnitudes.
Note 2
It's easy to see that this charge distribution need not be unique, indeed if $\ell_{MAX}=0$ any combination such that $\sum_i^N q_i \propto \alpha_{0,0}$ should work. This is ok, I'm more concerned with finding good approximations than the absolute best fit.
Note 3
If the charge magnitudes are real (which is a given for a physical problem), the number of unique moments are reduced by roughly a factor of 2 since:
$$ \alpha_{\ell,-m} = (-1)^m \alpha_{\ell,m}^* $$