What does Matlab have to offer for ill-conditioned inverse problems?

When solving for x in the linear problem Ax = b, there are regularization methods for solving (like Tikhonov for example) in the case when A is an ill conditioned matrix. In my case, pinv, lsqnonneg, deconvlucy as well as the naive solution with A\b all of them do not work. In python there are a module that has an iterative solution for Tikhonov regularization but in Matlab I wasn't able to find a similar function. Any help will be appreciated! :)

[Jayant Gangwar]

You can take a look at the following example of Tikhonov regularization test for hilbert matrix to get some idea of how to do it in MATLAB.

Regularized Least Square: Tikhonov Regularization test for Hilbert matrix. - File Exchange - MATLAB Central (mathworks.com)

[Tanouir Aloui]

Thanks!

[Rondall]

The automatically regularized solver, ARLS, has been available for nearly a year.

Answer 1

[Bjorn Gustavsson]

Have a look at the excellent regtools toolbox. It contains all sorts of regularization functions Tikhonov, damped sdv, maximum-entropy etc. It has been very useful to me - making it possible to solve small and medium-sized inverse problems of all sorts without having to write my own versions of these (1: large-size I've taken to mean too large to be solved with this tool, 2 zeroth-order Tikhonov is rather easy to solve when the problem is small enough to do an svd, but...).

HTH

[ayoub bouayaben]

Does this toolbox contain also a nonlinear ill-posed problem ?. If not, could you please suggest me references where i can find a them ?

Thank you.

[Tanouir Aloui]

@Bjorn Gustavsson That is what I have been using since then! Thanks for sharing. However, professor Hansen mentioned that regtools is more suitable for large scale probmems.

@ayoub bouayaben It is for linear problems but you can find more intresting infos on his publications.

Answer 2

[Rondall]

Try ARLS, a completely automatic regularizing solver.

It was added to MATLAB in 2023.

[John D'Errico]

This is absolutely incorrect. It was NOT added to MATLAB.

It appears you posted it on the file exchange, a very different thing.

[Rondall]

Sorry... I meant AVAILABLE in MATLAB.

Try it.

Ron