[petsc-users] The condition number of the ill-conditioned matrix

[Matthew Knepley]

On Mon, Aug 20, 2012 at 8:10 AM, Barry Smith <bsmith at mcs.anl.gov> wrote:

>
>    I think it is even simpler than this.   Let   your coupled matrix look
> like
>
>       (    A    C12    )
>       (   C21  B       )
>
>    and the uncoupled be
>
>       (    A       0   )
>       (    0       B   )
>
>
>       Now solving the uncoupled with GMRES, CG etc is actually solving at
> the same time two independent systems and the relative size of the
> eigenvalues of A and B don't really matter.
>
>      Jed is right that the first thing you need to do is
>
> > 1. non-dimensionalize: scale your model to use some representative units
> so that the solution and residual are well-scaled
>
>
> you want the A part and the B part in the Jacobian and function
> evaluations to have similar scales of values.
>
>
> BTW: the FAQ has one sentence "Change the units (nondimensionalization),
> boundary condition scaling, or formulation so that the Jacobian is better
> conditioned."
> How about putting some references there and some good examples on how to
> do this? What this means? This is hard shit and we dismiss it in one
> sentence like a kindergartener can do it.


http://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem

  Matt


>
>     Barry
>
>
> On Aug 20, 2012, at 7:39 AM, Jed Brown <jedbrown at mcs.anl.gov> wrote:
>
> > On Mon, Aug 20, 2012 at 7:07 AM, w_ang_temp <w_ang_temp at 163.com> wrote:
> > Hello,
> >     I have a problem about the ill-conditioned matrix. I deal with the
> soil-water coupled problem in
> > geotechnical engineering. It can be demonstrated that the stiffness
> matrix of soil-water coupled problem is
> > ill-conditioned due to the coupled effects of large relative differences
> in stiffnesses and permeabilities
> > of materials. But it should not be ill-conditioned for soil-water
> uncoupled problem, although there is no derivation
> > in papers.
> >     In my three models, the condition numbers are 3.5595E+05,
> 1.8849E+06, 9.0206E+07 for soil-water
> > coupled models and 5.2885E+04, 3.3826 E+05, 3.4565E+05 for soil-water
> uncoupled ones, respectively.
> > Therefore, in my opinion, the uncoupled models are also ill-conditioned
> due to the large condition
> > numbers.
> >     However, when using GMRES, CG and CGS, the three models of uncoupled
> ones show rapid convergent rate
> > while coupled models have bad convergence performance.
> >
> > This indicates that the spectrum of the simpler model has a small number
> of outliers. It could be as simple as boundary condition scaling or it
> could be more meaningful. The coupled model does not have that nice
> property, so convergence is slower. You should start with:
> >
> > 1. non-dimensionalize: scale your model to use some representative units
> so that the solution and residual are well-scaled
> >
> > 2. find an effective preconditioner. PCFIELDSPLIT might work well, but
> you have to understand methods to know how to drive it. Look at the
> examples and read through the literature in your field to find something
> you want to implement, then ask here if you need suggestions for how to
> implement. (It can often be done using only command line options.)
>
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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