Hi all,
Is there anyone who have the experience of solving nonlinear eigenvalue problem? I meet a special question.
The question is to solving the eigenfunction:
H(x) x=a x
where a is a real number, x is an eigenvector while H(x) is a matrix depended by x. The aim is to find a vector x and H(x) satisfying the equation which make the eigenvalue a minimum. In this question, H is far from a Hermitian in most case. And this question is not similar to conventional Kohn-Sham equation where most of the eigenvectors contribute to the construction of H. In contrast, in present question, H(x) may have many eigenvectors but, it depends on one of its eigenvector (x) only.
The method I try is self-consistent field method, i.e. determine a "temporary" H by a random or guess x, and by solving this temporary H a new x is obtained, and from new x a new H can be determined, etc. Currently, a major problem is that the matrix is not Hermitian, and therefore no real eigenvalue and corresponding eigenvector can be selected to define new "H". Since the target is to find a real number a and vector with real number elements, I hope that any temporary H can be composed by real number as well.
Does anyone have some idea about solving such a question? Any suggestion will be appreciated.
Thanks!
Yixian Liu You will find many papers on NEPs
Here are some references
Article Matrix iteration method for nonlinear eigenvalue problems wi...
https://pure.manchester.ac.uk/ws/files/55114004/GT17_final.pdf
Article Nonlinear Eigenvalue Problems: A Challenge for Modern Eigenv...
Article An Arnoldi Method for Nonlinear Eigenvalue Problems
Yixian Liu
One suggestion is to explore the use of iterative methods that can accommodate non-Hermitian matrices. For instance, the Arnoldi iteration or the Lanczos algorithm can be adapted to handle non-Hermitian cases. These methods can help approximate the eigenvalues and eigenvectors of H(x) even when it is not Hermitian. Additionally, consider employing a regularization technique to ensure that the temporary H remains well-defined and composed of real numbers.
Another approach could be to reformulate the problem in terms of a minimization strategy, where you define a cost function that quantifies the deviation from the desired properties of H(x) and its eigenvalues. By minimizing this cost function, you may be able to converge to a suitable H(x) and corresponding eigenvector x that yields a real eigenvalue a.
Lastly, it may be beneficial to analyze the structure of H(x) further. If there are specific properties or symmetries in your problem, leveraging them could simplify the calculations and lead to more straightforward solutions.
Selase A. K. Kpo
Thank you for your suggestions!
I have tried the Arnoldi method and modified Lanczos method before, and as you said, the methods can solve non-Hermitian. When I tried to use self-consistence field method to solve the nonlinear eigenvalue problem, I used Arnoldi method for each iteration. And yes, the trouble is that the solutions of non-Hermitian H are all complex (with large imaginary part), and thus I give up the method. I hope that the iterations can be carried out by real numbers entirely. You have mentioned "regularization technique", can you give some details of that?
Recently I tried to apply Newton-type method to solve this question, and the eigenvalue problem becomes a minimization problem in this method. I don't know details of "eigenvalue problem version" of this method and I am still working on it.
Thank you for giving further details Yixian Liu
I agree that solutions of non-Hermitian H are mostly complex with large imaginary parts.
Concerning the regularization techniques, they could help ensure that the matrix H(x) remains well-defined and produces real eigenvalues and eigenvectors.
You mentioned the use of the temporary H in your earlier submission, so you may want to also take a look at the "Real-Symmetric Part" technique. For this, you extract the real-symmetric part of H(x), denoted as (H(x) + H(x)^T)/2, and use this as the temporary H. This ensures that the matrix is real and symmetric, with real eigenvalues.
Another technique is the “Tikhonov regularization” that involves adding a small multiple of the identity matrix to H(x), i.e., H(x) + αI, where α is a regularization parameter. This addition ensures that H(x) is more stable, potentially reducing the likelihood of non-real eigenvalues or ill-defined eigenvectors.
There is also the “Perturbation method” where you introduce a small perturbation to H(x), such as H(x) + εN, where N is a random or fixed matrix and ε is a small parameter. This can help break any degeneracies and ensure real eigenvalues.
You could also try the “Shift-and-Invert approach”. With this one, you may apply a shift-and-invert strategy, where you solve the eigenvalue problem for (H(x) - σI)^(-1), with σ being a real shift parameter. This can help improve the conditioning of the matrix and yield real eigenvalues.
While at that, kindly remember to carefully choose the regularization parameter or perturbation size to balance stability and accuracy. You may experiment with different techniques to find the most suitable approach for your specific problem.
Thank you!
Selase A. K. Kpo
Thank you so much for these information!
In my case the Non-symmetric part of H is not trivial (In fact I haven't figured out how important the Non-symmetric part in my case is, because everything of this H is new to me. Nevertheless, I have to assume that it is not trivial at this stage), therefore "Real-Symmetric Part" technique may not suitable currently. However, I will try other methods you recommended for me.
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