[petsc-users] Condition Number and GMRES iteration

[Alexander Lindsay]

It looks like Fande has attached the eigenvalue plots with the real axis
having a logarithmic scale. The same plots with a linear scale are attached
here.

The system has 306 degrees of freedom. 12 eigenvalues are unity for both
scaled and unscaled cases; this number corresponds to the number of mesh
nodes with Dirichlet boundary conditions (just a 1 on the diagonal for the
corresponding rows). The rest of the eigenvalues are orders of magnitude
smaller for the unscaled case; using scaling these eigenvalues are brought
much closer 1.

This particular problem is linear but we solve it with SNES, so constant
Jacobian. We run with options '-pc_type none -ksp_gmres_restart 1000
-snes_rtol 1e-8 -ksp_rtol 1e-5` so for this linear problem it takes two
non-linear iterations to solve.

Unscaled:

first nonlinear iteration takes 2 linear iterations
second nonlinear iteration takes 99 linear iterations

Scaled:

first nonlinear iteration takes 94 linear iterations
second nonlinear iteration takes 100 linear iterations

Running with `-pc_type svd` the condition number for the unscaled
simulation is 4e9 while it is 2e3 for the scaled simulation.



On Thu, Feb 6, 2020 at 4:36 PM Fande Kong <fdkong.jd at gmail.com> wrote:

> 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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