Ill-posedness and integral equations of 1st kind

[sarrah1]

Hi
I know why integral equations like Fredholm of first kind necessitates that the forcing function $f$ must lie in the space of eigenfunctions of the operator $K$ otherwise no solution exists. The equation is

$f(x)=\int_{0}^{1} \,k(x,s) \psi(s) ds$

Now I am confused because I know that this equation is usually also ill-conditionned in the sense that small changes in the data lead to large change in the result, OK. But this has nothing to do with having or not having a solution.

Again does the fact that "NOT having a solution" means to be ill-posed and need regularization like the case of least-squares problem in linear equations, or that these are two separate issues. Suppose the equation which is $f=K\psi$ that I write it like in the least-squares problem ${K}^{*}f={K}^{*}K\psi$ will it then have a solution ?

I will be grateful should one define for me the exact meanings of

1. ill-posed
2- difference between ill-posed and not having a solution
3. can the equation have a least-squares solution if the original equation doesn't have one. In linear equations it does since $rank( {K}^{*}K)=rank ({K}^{*}K, {K}^{*}f)$

thanks
Sarrah

 

[jvicens]

,Hello Sarrah,

Great question! Ill-posedness and the existence of a solution in integral equations are definitely related, but they are not the same thing. Let me try to explain in more detail.

1. Ill-posedness refers to the sensitivity of a mathematical problem to small changes in the input data. In other words, small errors or uncertainties in the data can lead to large errors in the solution. This can make it difficult to obtain a reliable solution, and can also make it challenging to analyze the problem mathematically. In the context of integral equations, ill-posedness can arise when the operator $K$ is not well-behaved, for example when it is not invertible or when it maps a smooth function to a less smooth one.

2. The existence of a solution in an integral equation is related to the properties of the operator $K$ and the forcing function $f$. For a Fredholm integral equation of the first kind, as you mentioned, the forcing function must lie in the space of eigenfunctions of the operator $K$ in order for a solution to exist. This is because the eigenfunctions of $K$ form a basis for the space of functions that $K$ maps to itself. If the forcing function does not lie in this space, then there is no solution to the integral equation.

3. In some cases, it is possible to find a least-squares solution to an integral equation even if the original equation does not have a solution. This can happen when the operator $K$ is not well-behaved, but it still has a pseudo-inverse that can be used to find a least-squares solution. The pseudo-inverse is a generalization of the inverse of a matrix, and it can be used to find a solution to an equation even when the equation does not have a unique solution.

I hope this helps clarify the differences between ill-posedness and the existence of a solution in integral equations. Please let me know if you have any further questions or if you would like me to explain anything in more detail.