Solving this integral equation

In summary, the conversation discusses an expression involving a measured value y_E and a function f(r), which is rewritten in the form of a Fredholm integral equation of the first kind. The goal is to find a parametrization for f(r) with four parameters. One approach is to use a Gauss-Laguerre quadrature form, but it may not be suitable due to the presence of exponential integrals. Another suggestion is to use polynomial interpolation or approximation to find the four parameters.
  • #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 !
 
Last edited:
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  • #2
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.
 
  • #3
imsolost said:
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.
 
  • #4
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 :-/
 
  • #5
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.
 

Related to Solving this integral equation

1. How do you determine the limits of integration for an integral equation?

The limits of integration for an integral equation are determined by the range of values that the variable of integration can take on. This is usually specified in the given problem or can be determined by examining the function being integrated.

2. What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration and gives a numerical value as the result. An indefinite integral, on the other hand, does not have limits of integration and gives a function as the result.

3. How do you solve a complex integral equation?

Solving a complex integral equation involves breaking it down into simpler parts and applying various integration techniques such as substitution, integration by parts, or partial fractions. It also requires a good understanding of the properties of integrals and advanced mathematical concepts.

4. Can you use a calculator to solve an integral equation?

Yes, certain calculators have the capability to solve basic integrals using numerical integration techniques. However, for more complex integrals, it is recommended to use mathematical software or solve them by hand.

5. What are some common mistakes to avoid when solving an integral equation?

Some common mistakes to avoid when solving an integral equation include forgetting to apply the appropriate substitution, misinterpreting the limits of integration, and making calculation errors. It is important to double-check all steps and be careful with notation and calculations.

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