[petsc-dev] Bordered systems and low-rank corrections

[Jed Brown]

On Fri, Nov 4, 2011 at 09:13, Matthew Knepley <knepley at gmail.com> wrote:

> So that means the construction layout of C should mirror B. I wonder if
> that is strange.
>

The ownership rows of B are normally the same as the ownership rows of A.
Similarly for columns of C.


> Also, does D have a few rows on each proc, meaning B would be spread out
> too?
>

It doesn't matter where D (usually dense) is stored because it will
typically be used redundantly. (These algorithms only make sense when D is
pretty small. In practice, it is usually less than 10x10.)


> We can easily implement MatSetValues_Transpose() to facilitate convenient
>> assembly of bordered systems using MatSetValuesLocal(). Does anyone have a
>> better idea for constructing these things?
>>
>
> Constructing B and C together I think might be the easiest, and having an
> option for C = B^T.
>

You construct them together, but you want C to have a column partition,
otherwise some process will be overloaded.


>
>
>> The Woodbury formula stuff can probably be a new PC that operates on a
>> MATSCHURCOMPLEMENT by doing direct solves with the eliminated matrix
>> (typically redundantly in this case, because the dimension should be small
>> for this to make sense). Other API suggestions?
>>
>
> I think we need an example.
>

You just apply the formula. Setup involves solving (perhaps approximately)
with a few vectors, then the convergence is rate is normally the same as
for A, with a few extra vector operations per iteration.

http://en.wikipedia.org/wiki/Woodbury_matrix_identity
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.mcs.anl.gov/pipermail/petsc-dev/attachments/20111104/55069111/attachment.html>