" No, that is definitely not true. The correlations are expressed by the
eigenvectors associated with the eigenvalues. J^T J is symmetric and
positive definite, so all of the eigenvectors are mutually orthogonal.
But they will generally not lie along the coordinate axes, and so
express correlations between the parameters."
I see. My point pertains to the following:
If J has full rank, this means that none of the 8 columns is linearly
dependent
on another column or, equivalently, no parameter is linearly dependent on
another parameter.
" 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. "
Is it reasonable to choose \sum_{i=1}^8 ev_i (ev_i are the normalized
eigenvectors)
as the new set of parameters?
" Of course, all of this requires computing the eigenvalues and
eigenvectors of the current J^T J once. "
If I get the gist correctly, I first run my dealii program to compute J
like I do it right now,
thence compute the eigenvectors, and based on them the new set of
parameters.
Finally, I run my dealii program again to compute the new J with the new
parameters.
Correct?
I did not read about such a scaling technique in the context of parameter
estimation so far.
Do you have any reference where the procedure is described?
I have to check whether the above scaling works for my problem or not:
For instance, if there are three old parameters {0, 20,000, 300,000} and
your proposed method
produces the new set of parameters {5,000, 40,000, 90,000},
it may happen that my pde solver does not converge anymore.
Also, the old ranking (0 < 20,000 < 300,000) has to be retained during
scaling.
Best
Simon
Am Do., 13. Okt. 2022 um 22:03 Uhr schrieb Wolfgang Bangerth <
bange...@colostate.edu>:
> On 10/13/22 11:55, Simon Wiesheier wrote:
> >
> > " and if it is poorly
> > conditioned, you need to choose the 8 parameters differently, for
> > example (i) scaled in a different way, and (ii) as linear combinations
> > of what you use right now. "
> >
> > I double-checked that my J has full rank.
> > If I interpret this correctly, there are no correlation between my
> > parameters
> > and what describe as your point (ii) is not neccessary in my case.
>
> No, that is definitely not true. The correlations are expressed by the
> eigenvectors associated with the eigenvalues. J^T J is symmetric and
> positive definite, so all of the eigenvectors are mutually orthogonal.
> But they will generally not lie along the coordinate axes, and so
> express correlations between the parameters.
>
>
> > I definitely have to scale my paramters somehow.
> > Do you have a recommendation how to scale the parameters?
>
> 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. But if you
> scale the eigenvectors by something like the square root of the
> eigenvalues, you'll get J^T J to be the identity matrix.
>
> Of course, all of this requires computing the eigenvalues and
> eigenvectors of the current J^T J once.
>
> Best
> W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bange...@colostate.edu
> www: http://www.math.colostate.edu/~bangerth/
>
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