An experiment yields a functional dependence on two variables $F(x,y)$, can be imagined as a 2D map. It is known that the function can be factored as follows $F(x,y)=fx(x) gy(y) +gx(x) fy(y)$. Given the tabulated $F(x,y)$ I would like to numerically find the four unknown functions $fx$, $fy$, $gx$, $gy$. It is clear that in full generality the problem is ill posed and has no unique solution. There are however several constraints that are known a priori:
i) $fx(x)$, $fy(y)$ are non-zero only on a finite interval
ii) $gy(y)$ has a gaussian profile, $gx(x)$ decay exponentially for $x>0$.
Just to give an example let us use the following definitions:
fx[x_] := 1/(Exp[-x - 7] + 1) + 1/(Exp[x - 7] + 1) - 1
gy[y_] := Exp[-y^2]
fy[y_] := (1/(Exp[-y - 10] + 1) + 1/(Exp[y - 10] + 1) - 1) (5 + 0.5 Sin[y])
gx[x_] := Exp[-0.4 x]/(Exp[-5 x] + 5)
F[x_, y_] := fx[x] gy[y] + gx[x] fy[y]
looking as
The resulting function has a cross shape
Plot3D[F[x, y], {x, -15, 15}, {y, -15, 15}, PlotRange -> All, PlotPoints -> {50, 50}]
I do not have many ideas how to find $fx,fy,gx,gy$ numerically. I was trying to use a fitting procedure, however, it works like you get what you put in. I am seeking for a more general solution, perhaps with the help of Fourier or wavelet analysis. I would be grateful for any ideas.
gx[x_] := gx[x_]
$\endgroup$ – chris May 1 '14 at 18:10{x_i,y_i}
s ? $\endgroup$ – chris May 1 '14 at 18:12