[petsc-users] Condition Number and GMRES iteration

[Fande Kong]

Hi All,

MOOSE team, Alex and I are working on some variable scaling techniques to
improve the condition number of the matrix of linear systems. The goal of
variable scaling is to make the diagonal of matrix as close to unity as
possible. After scaling (for certain example), the condition number of the
linear system is actually reduced, but the GMRES iteration does not
decrease at all.

>From my understanding, the condition number is the worst estimation for
GMRES convergence. That is, the GMRES iteration should not increases when
the condition number decreases. This actually could example what we saw:
the improved condition number does not necessary lead to a decrease in
GMRES iteration. We try to understand this a bit more, and we guess that
the number of eigenvalue clusters of the matrix of the linear system
may/might be related to the convergence rate of GMRES.  We plot eigenvalues
of scaled system and unscaled system, and the clusters look different from
each other, but the GMRRES iterations are the same.

Anyone know what is the right relationship between the condition number and
GMRES iteration? How does the number of eigenvalue clusters affect GMRES
iteration?  How to count eigenvalue clusters? For example, how many
eigenvalue clusters we have in the attach image respectively?

If you need more details, please let us know. Alex and I are happy to
provide any details you are interested in.


Thanks,

Fande Kong,
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