Sorry that I come back to this again, but I would appreciate your feedback
to the issue I am confronted with:
" You would generally choose the (normalizes) eigenvectors of the J^T J
you have as the unit directions and the new set of parameters are then
multipliers for these directions. (So far your parameters are
multipliers for the unit vectors e_i in your 8-dimensional parameter
space.) The matrix J^T J using this basis for your parameter space will
then already be diagonal, though still poorly conditioned. "
I tried to implement this for only 3 parameters and the 3x3 Hessian
evaluated at the start vector reads
H = J^T J = [4.91e-12 1.07e-12 -5.98e-12
6.50e-13 -1.72e-12
sym 7.70e-12 ]
The eigenvectors are
EV_1 = [0.58; 0.58; 0.58]
EV_2 = [0.54; -0.80; 0.26]
EV_3 = [-0.61; -0.16; 0.77]
with associated eigenvalues {-8.72e-28, 4.80e-13; 1.28e-11}.
Clearly, I can not scale the eigenvectors by 1/sqrt(eigenvalue) as the
eigenvalues are numerically negative or very small.
Thus, I transformed the
old parameters [500000;0;500000]
using the transformation matrix [EV_1; EV_2; EV_3] and
obtained the
new parameters [-16778; 206381; 675972].
The set of new parameters is inadmissible in my case as the first and last
parameter have to be greater than the second parameter.
There is probably no straightforward way to incorporate this constraint in
the transformation, is it?
But anyway, it seems that I have to scale my parameters already before I
compute the J^T J.
The best alternative to linearly scale the parameters p_i is to normalize
them according to
-1 < p_i < +1 .
Right?
Best,
Simon
Am Mo., 17. Okt. 2022 um 16:20 Uhr schrieb Wolfgang Bangerth <
[email protected]>:
> On 10/17/22 08:12, Simon Wiesheier wrote:
> >
> > I do not plan to do anything else than computing (J^t J)^-1
> > and some matrix multiplications with it.
> > Based on what you said, this should not cause any undue problems, right?
> >
>
> Probably not.
> W.
>
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