[petsc-users] MatMatSolve in sequential call

[Jack Poulson]

On Tue, Aug 21, 2012 at 3:37 PM, Matthew Knepley <knepley at gmail.com> wrote:

> On Tue, Aug 21, 2012 at 11:47 AM, Jack Poulson <jack.poulson at gmail.com>wrote:
>
>> On Tue, Aug 21, 2012 at 11:42 AM, Alexander Grayver <
>> agrayver at gfz-potsdam.de> wrote:
>>
>>>  On 21.08.2012 18:32, Matthew Knepley wrote:
>>>
>>>   MUMPS takes only several minutes and 6 GB of memory to factorize it.
>>>> This factorization gives residual on the order of 10e-12 and solution
>>>> is indeed correct.
>>>>
>>>> Nevertheless, you're right, there is numerical null-space in this
>>>> matrix since it comes
>>>> from the discretization of equation that contains curl curl operator,
>>>> but practically this
>>>> case is not really the worst one.
>>>>
>>>
>>>  This makes no sense whatsoever. How can you LU factor a matrix that
>>> has a null space?
>>>
>>>
>>> Matt,
>>>
>>> I'm not sure that I correctly used term numerical null-space in my post.
>>> The equation is
>>>
>>> curl curl E + kE = -J,
>>>
>>> where k is a function of frequency and conductivity, whenever one of
>>> them becomes small this term gets vanishingly small thus we have problems
>>> since curl curl operator has nontrivial null-space by definition. So let's
>>> say solving this equation for low frequencies and for models containing air
>>> is difficult.
>>>
>>> What kind of magic is inside MUMPS I don't know, but it is able to
>>> handle such cases (e.g. SuperLU and PaStiX fail).
>>>
>>> Also, if it matters, I'm talking about LDLt factorization in MUMPS.
>>>
>>> --
>>> Regards,
>>> Alexander
>>>
>>>  You can find Vasseur's talk on this exact subject here:
>> http://graal.ens-lyon.fr/MUMPS/doc/ud_2010/Vasseur_talk.pdf
>
>
> I was wrong, this is not nonsense. However, for curl curl the null space
> grows with matrix dimension, and
> as far as I can tell from the slides, the null space determination is not
> scalable (Jack correct me if I am wrong).
> Also, they gave no timings, so I suspect null space determination is slow.
>
> I don't think any other LU we have will do this, so if you have null
> spaces you are stuck with MUMPS.
>
>     Matt
>
>

It's not something that I've studied in detail, but I believe that it isn't
that the behavior will not be that different from "difficult" nonsingular
cases (i.e., where a large number of pivots do not satisfy the thresholding
condition and must be delayed to the parent front). If the null space is
large, then I would expect this to impact performance significantly. I
would expect it to make load balancing much more difficult. In practice,
this might lead to nonscalability, as it is sophisticated algorithm.

Jack
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