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

[w_ang_temp]

    Thanks.
    I agree with you. Maybe I need to get the representative units. Also, maybe it is not reasonable to evaluate the
performance in my project just under the condition number.




At 2012-08-20 21:10:08,"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 bet>ter 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 on>e sentence like a kindergartener can do it.
>
>    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 cou>>ld 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 exa>>mples 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 imple>>ment. (It can often be done using only command line options.)
>
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