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.
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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