I would like to find the solution of a system of equations of the form:
$A = w_1 F(x_k) + w_2 F(x_l) + w_3 F(x_m) +...$,
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.
One possible solution would involve iterations with an alternating least-squares between optimizing the $w_i$s while holding the $F$ fixed and then optimising the function approximation for fixed $w_i$s. Yet I am not sure whether this approach would converge.
Is anyone of you aware of something along these lines?