- #1

imsolost

- 18

- 1

I have the following expression :

$$ y_{E} = \int_{0}^{\infty} 0.5 * [E_{1}(µ(E)*r) - E_{1}(\frac{µ(E)*r}{cos \alpha})] * f(r) dr $$

where :

- $y_{E}$ has been measured for some E (something like 5 different $E_{i}$, to give you an idea)

- µ(E) is retrieved from a table in the litterature (basically that means I have no analytical expression for µ(E) but I can easily get access for its value). If you're curious about the shape of that thing in function of E, here is the link : https://physics.nist.gov/PhysRefData/XrayMassCoef/ElemTab/z82.html

- E1 is the exponential integral function

My goal is to find f(r).

This can be rewritten in the following form :

$$ y_{E} = \int_{0}^{\infty} K(E;r) * f(r) dr $$

which (thx wikipedia) is called a Fredholm integral equation of the first kind. And which is very ugly to solve from what I've read. :'(

I would like to introduce a parametrization of f(r) as the following :

$$f(r) = C*e^{-\lambda_{1}*(r+\lambda_{3})} * [1-e^{-\lambda_{2}*(r+\lambda_{3})}]$$

The justification for this form is that i know f(r) usually has a profile where it increases then exponentially descreases. So I think such an expression is okay. Of course if there was a way to solve this without postulating such a hard thing, that would be better.

Anyway, so basically an expression of 4 parameters : lambdas 1, 2, 3 and C. If I can find a decent approximation of these 4 parameters, I will be a happy man.

From there, if you have any suggestion how to solve this, my ears are wide open.

I have tried the following :

First I introduced a quadrature form. I choosed a Gauss-Laguerre one, because of the limits on the integral from 0 to infinity, but I have no idea if that is a good choice. I've read Gauss-Laguerre works great to estimate an integral with a polynomial function multiplied by an exponential, but here I have no polynomial function but rather 2 ugly exponential integral E1 which probably behaves very differently, particulary close to 0+. So is it a wise idea?

Anyway, by doing so, I can get some kind of linear system :

$$\hat{K} \hat{f} = \hat{g}$$

where :

Kij = K'(Ei;rj)

fj=f(rj) with j going from 1 to n

gi=y(Ei) with i going from 1 to m

n is something like 15 or more (to get enough points in the G-L quadrature for a decent fit)

m is, as previously stated, around 5 or so.

So K is not a square matrix, and the system is clearly undertermined because n>m.

But since all the fj only depend on 4 parameters only, this looks like some kind of 5 equations with 4 unknown, but non-linear (because of the expression of f). Am I correct on this ?

And if so, how to solve this to find my 4 parameters ?

As already said, any idea would be greatly appreciated, and thank you for reading this !

$$ y_{E} = \int_{0}^{\infty} 0.5 * [E_{1}(µ(E)*r) - E_{1}(\frac{µ(E)*r}{cos \alpha})] * f(r) dr $$

where :

- $y_{E}$ has been measured for some E (something like 5 different $E_{i}$, to give you an idea)

- µ(E) is retrieved from a table in the litterature (basically that means I have no analytical expression for µ(E) but I can easily get access for its value). If you're curious about the shape of that thing in function of E, here is the link : https://physics.nist.gov/PhysRefData/XrayMassCoef/ElemTab/z82.html

- E1 is the exponential integral function

My goal is to find f(r).

This can be rewritten in the following form :

$$ y_{E} = \int_{0}^{\infty} K(E;r) * f(r) dr $$

which (thx wikipedia) is called a Fredholm integral equation of the first kind. And which is very ugly to solve from what I've read. :'(

I would like to introduce a parametrization of f(r) as the following :

$$f(r) = C*e^{-\lambda_{1}*(r+\lambda_{3})} * [1-e^{-\lambda_{2}*(r+\lambda_{3})}]$$

The justification for this form is that i know f(r) usually has a profile where it increases then exponentially descreases. So I think such an expression is okay. Of course if there was a way to solve this without postulating such a hard thing, that would be better.

Anyway, so basically an expression of 4 parameters : lambdas 1, 2, 3 and C. If I can find a decent approximation of these 4 parameters, I will be a happy man.

From there, if you have any suggestion how to solve this, my ears are wide open.

I have tried the following :

First I introduced a quadrature form. I choosed a Gauss-Laguerre one, because of the limits on the integral from 0 to infinity, but I have no idea if that is a good choice. I've read Gauss-Laguerre works great to estimate an integral with a polynomial function multiplied by an exponential, but here I have no polynomial function but rather 2 ugly exponential integral E1 which probably behaves very differently, particulary close to 0+. So is it a wise idea?

Anyway, by doing so, I can get some kind of linear system :

$$\hat{K} \hat{f} = \hat{g}$$

where :

Kij = K'(Ei;rj)

fj=f(rj) with j going from 1 to n

gi=y(Ei) with i going from 1 to m

n is something like 15 or more (to get enough points in the G-L quadrature for a decent fit)

m is, as previously stated, around 5 or so.

So K is not a square matrix, and the system is clearly undertermined because n>m.

But since all the fj only depend on 4 parameters only, this looks like some kind of 5 equations with 4 unknown, but non-linear (because of the expression of f). Am I correct on this ?

And if so, how to solve this to find my 4 parameters ?

As already said, any idea would be greatly appreciated, and thank you for reading this !

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