[petsc-users] LU factorization and solution of independent matrices does not scale, why?

[Thomas Witkowski]

So, I run the benchmark for 16, 64, 256 and 1024 MPI tasks. I replaced  
a local UMFPACK LU factorization of interior matrices and the 192  
solves with them by MUMPS. It all three cases, the size, the structure  
and the values of the matrices are all the same. As expected, with  
UMFPACK both times for factorization and solve are the same for  
different number of cores. The MUMPS, this is not the case:

16 cores: factorization: 3.87 sec   solves: 46 sec
64 cores: factorization: 4.29 sec   solves: 70 sec
256 cores: factorization: 6.11 sec   solves: 254 sec
1024 cores: factorization: 25.64 sec   solves: forever :)


This is really baaad! There is no communication (PETSC_COMM_SELF,  
MatAIJSeq) and all matrices are of the same size. What's going on  
here? May be its possible to reproduce this scenario with one of the  
PETSc examples?

Thomas

Zitat von Thomas Witkowski <Thomas.Witkowski at tu-dresden.de>:

> Jack, I also considered this problem. The 4 MPI tasks of each coarse
> space matrix should run all on one node (each node contains 4 dual core
> CPUs). I'm not 100% sure, but I discussed this with the administrators
> of the system. The system should schedule always the first 8 ranks to
> the first node, and so on. And the coarse space matrices are build on
> ranks 0-3, 4-7 ...
>
> I'm running at the moment some benchmarks, where I replaced the local
> LU factorization from using UMFPACK to MUMPS. Each matrix and the
> corresponding ksp object are defined on PETSC_COMM_SELF and the problem
> is perfectly balanced (the grid is a unit square uniformly refined).
> Lets see...
>
> Thomas
>
> Zitat von Jack Poulson <jack.poulson at gmail.com>:
>
>> Hi Thomas,
>>
>> Network topology is important. Since most machines are not fully connected,
>> random subsets of four processes will become more scattered about the
>> cluster as you increase your total number of processes.
>>
>> Jack
>> On Dec 20, 2012 12:39 PM, "Thomas Witkowski"   
>> <Thomas.Witkowski at tu-dresden.de>
>> wrote:
>>
>>> I cannot use the information from log_summary, as I have three different
>>> LU factorizations and solve (local matrices and two hierarchies of coarse
>>> grids). Therefore, I use the following work around to get the timing of the
>>> solve I'm intrested in:
>>>
>>>    MPI::COMM_WORLD.Barrier();
>>>    wtime = MPI::Wtime();
>>>    KSPSolve(*(data->ksp_schur_**primal_local), tmp_primal, tmp_primal);
>>>    FetiTimings::fetiSolve03 += (MPI::Wtime() - wtime);
>>>
>>> The factorization is done explicitly before with "KSPSetUp", so I can
>>> measure the time for LU factorization. It also does not scale! For 64
>>> cores, I takes 0.05 seconds, for 1024 cores 1.2 seconds. In all
>>> calculations, the local coarse space matrices defined on four cores have
>>> exactly the same number of rows and exactly the same number of non zero
>>> entries. So, from my point of view, the time should be absolutely constant.
>>>
>>> Thomas
>>>
>>> Zitat von Barry Smith <bsmith at mcs.anl.gov>:
>>>
>>>
>>>>  Are you timing ONLY the time to factor and solve the subproblems?  Or
>>>> also the time to get the data to the collection of 4 cores at a  time?
>>>>
>>>>   If you are only using LU for these problems and not elsewhere in  the
>>>> code you can get the factorization and time from MatLUFactor()  and
>>>> MatSolve() or you can use stages to put this calculation in its  own stage
>>>> and use the MatLUFactor() and MatSolve() time from that  stage.
>>>> Also look at the load balancing column for the factorization and  solve
>>>> stage, it is well balanced?
>>>>
>>>>   Barry
>>>>
>>>> On Dec 20, 2012, at 2:16 PM, Thomas Witkowski  <
>>>> thomas.witkowski at tu-dresden.**de <thomas.witkowski at tu-dresden.de>> wrote:
>>>>
>>>> In my multilevel FETI-DP code, I have localized course matrices,  which
>>>>> are defined on only a subset of all MPI tasks, typically    
>>>>> between 4 and 64
>>>>> tasks. The MatAIJ and the KSP objects are both  defined on a MPI
>>>>> communicator, which is a subset of  MPI::COMM_WORLD. The LU factorization
>>>>> of the matrices is computed  with either MUMPS or superlu_dist, but both
>>>>> show some scaling  property I really wonder of: When the overall problem
>>>>> size is  increased, the solve with the LU factorization of the local
>>>>> matrices does not scale! But why not? I just increase the number  
>>>>>   of  local
>>>>> matrices, but all of them are independent of each other. Some  example: I
>>>>> use 64 cores, each coarse matrix is spanned by 4 cores  so there  
>>>>>   are 16 MPI
>>>>> communicators with 16 coarse space matrices.  The problem need   
>>>>> to  solve 192
>>>>> times with the coarse space systems,  and this takes together    
>>>>> 0.09 seconds.
>>>>> Now I increase the number of  cores to 256, but let the local   
>>>>> coarse space
>>>>> be defined again on  only 4 cores. Again, 192 solutions with these coarse
>>>>> spaces are  required, but now this takes 0.24 seconds. The same for 1024
>>>>> cores,  and we are at 1.7 seconds for the local coarse space solver!
>>>>>
>>>>> For me, this is a total mystery! Any idea how to explain, debug and
>>>>> eventually how to resolve this problem?
>>>>>
>>>>> Thomas
>>>>>
>>>>
>>>>
>>>>
>>>
>>>
>>