On 10/15/22 03:15, Simon Wiesheier wrote:
This makes sense.
So, given the scaled eigenvectors E_1,...,E_8, how can I find the coefficients
A^*,...,H^* ?
Is it just a matrix multiplication
P* = (E_1; ... ; E_8) \times p* ,
where P* = (A^*,...,H^*) are the new parameters and p* = (a^*,...,h^*) are the
old parameters?
Something of the sort. It's the same point in 8-space, you're just expressing
it with regard to different bases.
Assuming that my pde solver still converges for the new parameters, the
overall procedure would be as follows:
1. run dealii program to compute J with old parameters p*
2. compute the new basis (EV_i) and the new parameters P*
3. run dealii program to compute the new J with the new parameters P*
4. compute p* = (E_1; ... ; E_8)^-1 \times P*
Repeat 1-4 for all iterations of the optimsation algorithm
(Levenberg-Marquardt).
Is that correct?
Conceptually, this is correct. In practice, it may not be necessary to
actually do it that way: All you're looking for is a well-conditioned basis.
You don't need the exact vectors E_i. In some applications you can guess such
vectors (like in the model case I drew up), in others you compute them once
for one case and re-use for other cases where maybe they are not exactly the
eigenvalues of the matrix J^T J. Or you live with the ill-conditionedness.
At the end, the ensuing parameters have to be the same, no matter
wheter I use the above scaling or not. The sole difference is that
the scaled version improves (amongs others) the condition number of J and may
lead to
a better convergence of the optimsation algorithm, right?
Yes.
Best
W.
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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