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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.

can't stop me now
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    For any specified $F$ you will have a linear equation on the $w_i$ which typically has infinitely many solutions. What are you trying to optimize? – GReyes Aug 22 '20 at 00:39
  • But it is a system of equations and let's say I could have as many equations as I want. So for a fixed $F$ I will have an overdetermined system of equations, that can be solved with least squares. I was wondering whether there is any more involved method for these kind of problems, since I am not sure whether alternating least squares (once for $w$s and once for basis coefficients/$F$) would converge. – can't stop me now Aug 22 '20 at 00:47
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    What's about the number of $x_i$ per one equation? Is it fixed? Or one may choose as many $x_i$s as one like? Do you have any a-priori knowledge about $F$? Is it smooth / continuous? – uranix Aug 23 '20 at 21:07
  • The number of $x_i$s is not fixed and not controllable. They depend on the observations. So in my system of equations for some equations the term $w_i F(x_j)$ is set to zero. – can't stop me now Aug 23 '20 at 23:08
  • @uranix See also the update that I wrote in the initial question and the comment above this one comment – can't stop me now Aug 23 '20 at 23:13
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    The problem now is that you have two separate unknown functions $w(x)$ and $F(x)$ and your equations are basically $\sum_{i \in \text{some set}}w(x_i) F(x_i) = A$. It could be solved, but you need somehow separate function $w$ from the function $F$. In other words you may reconstruct the product $w F$, but not the individual functions themselves. – uranix Aug 24 '20 at 14:19
  • @uranix Not exactly. The $w$s are not a function. They are some scalar each, but they don't necessary arise all simultaneously in all equations. Let's say Eq. 1 has $w_1, w_2$ and $ w_3$, and Eq. 2 only $w_2, w_3$. So in this case I would have to identify 3 $w$s plus the function $F$ that is the same in all terms but evaluated at different points – can't stop me now Aug 25 '20 at 01:28
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    It's only a subject of notation, you may have used $w(x_k) F(x_k)$ instead with no difference in equations. I'm talking about $w_i \to \alpha w_i, F(x_i) \to F(x_i) / \alpha$ substitution does not change the equations. So you may take all $w_i = 1$ or any other arbitrary values and "merge" weights into the $F(x)$ function. The system of equations will remain the same. – uranix Aug 25 '20 at 10:53
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    Also, why are you using numbers to enumerate weights, but letters to enumerate $x$? Maybe I don't quite understand your notation – uranix Aug 25 '20 at 10:56
  • @uranix Yes that is true. I am still exploring whether I can add some type of normalization on F, but from the physical problem pe se I cannot think of any such constrain. But I am still exploring. The choice of numbers/letters is random. – can't stop me now Aug 28 '20 at 10:09

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