I would like to find a numerical solution of a system of $N$ equations of the form:
$A^i = w_1 F(x^i_k) + w_2 F(x^i_l) + w_3 F(x^i_m) +...$
$A^j = \,\,\,\,\,\,0 \,\,\,\,\ \,\,\ + w_2 F(x^j_l) + w_3 F(x^j_m) +...$
$A^k = w_1 F(x^k_k) + w_2 F(x^k_l) +\,\,\,0\,\,\,\, +...$,
where the unknowns are the $w_i$ and the function $F$, while my data are the $A$ and $x_i$s. I have a system of those equations for different $A$s and different points $x_i$ where the function $F$ is evaluated.
The plan is to expand the function $F$ on some basis.
I am wondering whether there is some numerical method specifically designed for problems with this kind of structure, like an iterative method that will alternate between optimizing the $w_i$s while holding the $F$ fixed and then optimising the function approximation for fixed $w_i$s.
Is anyone of you aware of something along these lines?
UPDATE: In essence, I observe a process that gets potentially influenced by a bunch of other processes. So $w_i$s determine whether there is a direct influence between the process of interest and the $i$th process, and the relative strength from this influence. The function $F$ is related to some internal dynamics of the process of interest.
I do not explain more since this is a resulting equation after several approximations and simplifications of my initial problem that amount to 2 pages of handwritten equations. But I think that at this stage my equation should be solvable/ the system is identifiable.
The function $F$ is assumed to be continuous and smooth but we do not want to include further knowledge on it.
Furthermore, not all terms $w_i F(x_j)$ appear in all equations in my system, but in general, in all equations, I have more than one term appearing, so solution through elimination is not possible.