Solving this integral equation

[imsolost]

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 !

 

[mfb]

Don’t use your values of f as unknowns, use the lambdas and C. Then your system is overdetermined and you can do a least square fit or something similar. It also means you can use much more than 15 points to evaluate the integral.
The downside: The system is not linear any more. But it should behave well enough for a fit.

 

[eys_physics]

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 !

One alternative could be to try to approximate $y(E_i)$ by polynomial interpolation, or as a polynomial times an asymptotic function. Whether or not this is possible depends of course on how the measured $y(E_i)$ looks like. In that they to can approximately compute y(E_i) at all n points, and you get a system of $$n\times n$$ equations to be solved.

 

[imsolost]

Thank you for these answers !

Don’t use your values of f as unknowns, use the lambdas and C. Then your system is overdetermined and you can do a least square fit or something similar. It also means you can use much more than 15 points to evaluate the integral.
The downside: The system is not linear any more. But it should behave well enough for a fit.

Yes, this is what I was saying in the last part of my post. About using more than 15 points, yes I can do that but this doesn't give me more data. It just approximates the integral better, so the error is on the 4 unknown parameters should be smaller. The problem is that each of these 15 points costs me some measurement time so I think 15 is a good number.

One alternative could be to try to approximate y(Ei)" role="presentation">y(Ei) by polynomial interpolation, or as a polynomial times an asymptotic function. Whether or not this is possible depends of course on how the measured y(Ei)" role="presentation">y(Ei) looks like. In that they to can approximately compute y(E_i) at all n points, and you get a system of n×n" role="presentation">n×n equations to be solved.

Interesting alternative indeed. I don't think I can reasonnably find a good interpolation when I see how y(Ei) looks like. But that's definitively an idea.

Now, by using the method suggested by mfb, I can indeed get some values. The next question I am dealing with now is :

How can I estimate the uncertainty on these 4 parameters value that i find using the method described above ?

So I have like 10 equations, non-linear in my 4 parameters. Some of the coefficients in these equations have a value with an associated standard deviation. I use a least-square method to kind the best value of my 4 parameters but i'd like to know their standard deviation.

I'm kinda blocked here :-/

 

[mfb]

Every fitting package will be able to do both minimization and an error estimate- typically both together as they come from the same procedure.

Using more points for the integral shouldn’t be related to the number of measurements you make - that doesn’t enter there.