[petsc-users] MatMatSolve in sequential call
Jack Poulson
jack.poulson at gmail.com
Tue Aug 21 18:24:32 CDT 2012
On Tue, Aug 21, 2012 at 6:22 PM, Jack Poulson <jack.poulson at gmail.com>wrote:
> 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
>
Please ignore the atrocity that was my attempt at a first sentence in the
previous email: the point is that the delayed pivot mechanism is also used
within the standard threshold pivoted LU factorization.
Jack
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