A Parameter optimization for the eignevalues of a matrix

AI Thread Summary
The discussion focuses on optimizing parameters for a 20x20 matrix representing a Hamiltonian, aiming to match its eigenvalues to experimentally measured energies. The user is considering gradient descent for optimization but is encountering difficulties due to the large gradient of the function with respect to the small parameter b, which complicates the diagonalization process. The diagonal terms are primarily quadratic and quartic functions of x, with expected values for a and b based on theoretical calculations. The nonlinearity introduced by the off-diagonal terms further complicates the gradient behavior. The user seeks advice on effectively managing these challenges in the context of fitting rotational parameters to a molecular spectrum.
kelly0303
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Hello! I have a matrix (about 20 x 20), which corresponds to a given Hamiltonian. I would like to write an optimization code that matches the eigenvalues of this matrix to some experimentally measured energies. I wanted to use gradient descent, but that seems to not work in a straightforward manner and I was wondering if someone has any advice on how to proceed. In my case, the diagonal term are mainly of the form ##ax^2+bx^4##, where a and b are the values I want to fit for, and in my case x is around 20. I expect (based on some theoretical calculations) that a is around 5000 and b is around 0.005, so the first term is on the order of ##5000 \times 20^2 = 2000000## and the second term is on the order ##0.005\times 20^4 = 800##. The off diagonal terms are much smaller on the order ~1. The main problem is that the gradient of the function with respect to b is huge i.e. ##x^4##, while b itself is very small. Moreover, when doing the diagonalization the ##bx^4## term gets mixed nonlinearly with the other terms of the matrix so in the end the gradient is not just simply ##x^4## and for example going from 0.0055 to 0.0056 changes the gradient of the eigenvalues with respect to b by almost 5 orders of magnitude. Is there a way to deal with this (for context this is for fitting rotational parameters to a molecular spectrum). Thank you!
 
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