[petsc-users] Petsc mesh scalability issue with iterative solver and direct solver

Fri Aug 5 17:58:51 CDT 2016

Hi,

Thanks for your answers.

I just figured out the issues which are mainly due to the ill-conditioning
of my matrix. I found the conditional number blows up when the beam is
discretized into large number of elements.

Now, I am using the 1D bar model to solve the same problem. The good news
is the solution is always accurate and stable even I discretized into 10
million elements.

When I run the model with both iterative solver(CG+BJACOBI/ASM) and direct
solver(SUPER_LU) in parallelization, I got the following results:

Mesh size: 1 million unknowns
Processes 1 2 4 6 8 10 12 16 20
CG+BJ 0.36 0.22 0.15 0.12 0.11 0.1 0.096 0.097 0.099
CG+ASM 0.47 0.46 0.267 0.2 0.17 0.15 0.145 0.16 0.15
SUPER_LU_DIST 4.73 5.4 4.69 4.58 4.38 4.2 4.27 4.28 4.38

It seems the CG+BJ works correctly, i.e. time decreases fast with a few
more processes and reach stable with many more cores.

However, I have some concerns about CG+ASM and SUPER_LU_DIST. The time of
both two methods goes up when I use two processes compared with uniprocess.
The tendency is more obvious when I use larger mesh size.
I especially doubt the results of SUPER_LU_DIST in parallelism since the
overall expedition is very small which is not expected.
The runtime option I use for ASM pc and SUPER_LU_DIST solver is shown as
below:
ASM preconditioner:  -pc_type asm -pc_asm_type basic
SUPER_LU_DIST solver:   -ksp_type preonly -pc_type lu
-pc_factor_mat_solver_package superlu_dist

I use same mpiexec -n np ./xxxx for all solvers.

Am I using them correctly? If so, is there anyway to speed up the
computation further, especially for SUPER_LU_DIST?

Thank you very much!

Bests,
Jinlei

On Mon, Aug 1, 2016 at 2:10 PM, Matthew Knepley <knepley at gmail.com> wrote:

> On Mon, Aug 1, 2016 at 12:52 PM, Jinlei Shen <jshen25 at jhu.edu> wrote:
>
>> Hi Barry,
>>
>> Thanks for your reply.
>>
>> Firstly, as you suggested, I checked my program under valgrind. The
>> results for both sequential and parallel cases showed there are no memory
>> errors detected.
>>
>> Second, I coded a sequential program without using PETSC to generate the
>> global matrix of small mesh for the same problem. I then checked the matrix
>> both from petsc(sequential and parallel) and serial code, and they are same.
>> The way I assembled the global matrix in parallel is first distributing
>> the nodes and elements into processes, then I loop with  elements on the
>> calling process to put the element stiffness into the global. Since the
>> nodes and elements in cantilever beam are numbered successively, the
>> connectivity is simple. I didn't use any partition tools to optimize mesh.
>> It's also easy to determine the preallocation d_nnz and o_nnz since each
>> node only connects the left and right nodes except for beginning and end,
>> the maximum nonzeros in each row is 6. The MatSetValue process is shown as
>> follows:
>>     do iEL = idElStart, idElEnd
>>         g_EL = (/2*iEL-1-1,2*iEL-1,2*iEL+1-1,2*iEL+2-1/)
>>         call MatSetValues(SG,4,g_EL,4,g_El,SE,ADD_VALUES,ierr)
>>     end do
>> where idElStart and idElEnd are the global number of first element and
>> end element that the process owns, g_EL is the global index for DOF in
>> element iEL, SE is the element stiffness which is same for all elements.
>> From above assembling, most of the elements are assembled within own
>> process while there are few elements crossing two processes.
>>
>> The BC for my problem(cantilever under end point load) is to fix the
>> first two DOF, so I called the MatZeroRowsColumns to set the first two
>> rows and columns into zero with diagonal equal to one, without changing the
>> RHS.
>>
>> Now some new issues show up :
>>
>> I run with -ksp_monitor_true_residual and -ksp_converged_reason, the
>> monitor showed two different residues, one is the residue I can
>> set(preconditioned, unpreconditioned, natural), the other is called true
>> residue.
>> 
>> I initially thought the true residue is same as unpreconditioned based on
>> definition. But it seems not true.  Is it the norm of the residue (b-Ax)
>> between computed RHS and true RHS?    But, how to understand
>> unprecondition residue since its definition is b-Ax as well?
>>
>
> It is the unpreconditioned residual. You must be misinterpreting. And we
> could determine exactly if you sent the output with the suggested options.
>
>
>> Can I set the true residue as my converging criteria?
>>
>
> Use right preconditioning.
>
>
>> I found the accuracy of large mesh in my problem didn't necessary depend
>> on the tolerance I set, either preconditioned or unpreconditioned,
>> sometimes, it showed converged while the solution is not correct. But the
>> true residue looks reflecting the true convergence very well, if the true
>> residue is diverging, no matter what the first residue says, the results
>> are bad!
>>
>
> Yes, your preconditioner looks singular. Note that BJACOBI has an inner
> solver, and by default the is GMRES/ILU(0). I think
> ILU(0) is really ill-conditioned for your problem.
>
>
>> For the preconditioner concerns, actually, I used BJACOBI before I sent
>> the first email, since the JACOBI or PBJACOBI didn't even converge when the
>> size was large.
>> But BJACOBI also didn't perform well in the paralleliztion for large mesh
>> as posed in my last email, while it's fine for small size (below 10k
>> elements)
>>
>> Yesterday, I tried the ASM  with CG using the runtime option: -pc_type
>> asm -pc_asm_type basic -sub_pc_type lu (default is ilu).
>> For 15k elements mesh, I am now able to get the correct answer with 1-3,
>> 6 and more processes, using either -sub_pc_type lu or ilu.
>>
>
> Yes, LU works for your subdomain solver.
>
>
>> Based on all the results I have got, it shows the results varies a lot
>> with different PC and seems ASM is better for large problem.
>>
>
> Its not ASM so much as an LU subsolver that is better.
>
>
>> But what is the major factor to produce such difference between different
>> PCs, since it's not just the issue of computational efficiency, but also
>> the accuracy.
>> Also, I noticed for large mesh, the solution is unstable with small
>> number of processes, for the 15k case, the solution is not correct with 4
>> and 5 processes, however, the solution becomes always correct with more
>> than 6 processes. For the 50k mesh case, more processes are required to
>> show the stability.
>>
>
> Yes, partitioning is very important here. Since you do not have a good
> partition, you can get these wild variations.
>
>   Thanks,
>
>      Matt
>
>
>> What do you think about this? Anything wrong?
>> Since the iterative solver in parallel is first computed locally(if this
>> is correct), can it be possible that there are 'good' and 'bad' locals when
>> dividing the global matrix, and the result from 'bad' local will
>> contaminate the global results. But with more processes, such risk is
>> reduced.
>>
>> It is highly appreciated if you could give me some instruction for above
>> questions.
>>
>> Thank you very much.
>>
>> Bests,
>> Jinlei
>>
>>
>> On Fri, Jul 29, 2016 at 2:09 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
>>
>>>
>>>   First run  under valgrind all the cases to make sure there is not some
>>> use of uninitialized data or overwriting of data. Go to
>>> http://www.mcs.anl.gov/petsc follow the link to FAQ and search for
>>> valgrind (the web server seems to be broken at the moment).
>>>
>>>   Second it is possible that your code the assembles the matrices and
>>> vectors is not correctly assembling it for either the sequential or
>>> parallel case. Hence a different number of processes could be generating a
>>> different linear system hence inconsistent results. How are you handling
>>> the parallelism? How do you know the matrix generated in parallel is
>>> identically to that sequentially?
>>>
>>> Simple preconditioners such as pbjacobi will converge slower and slower
>>> with more elements.
>>>
>>> Note that you should run with -ksp_monitor_true_residual and
>>> -ksp_converged_reason to make sure that the iterative solver is even
>>> converging. By default PETSc KSP solvers do not stop with a big error
>>> message if they do not converge so you need make sure they are always
>>> converging.
>>>
>>>    Barry
>>>
>>>
>>>
>>> > On Jul 29, 2016, at 11:46 AM, Jinlei Shen <jshen25 at jhu.edu> wrote:
>>> >
>>> > Dear PETSC developers,
>>> >
>>> > Thank you for developing such a powerful tool for scientific
>>> computations.
>>> >
>>> > I'm currently trying to run a simple cantilever beam FEM to test the
>>> scalability of PETSC on multi-processors. I also want to verify whether
>>> iterative solver or direct solver is more efficient for parallel large FEM
>>> problem.
>>> >
>>> > Problem description, An Euler elementary cantilever beam with point
>>> load at the end along -y direction. Each node has 2 DOF (deflection and
>>> rotation)). MPIBAIJ is used with bs = 2, dnnz and onnz are determined based
>>> on the connectivity. Loop with elements in each processor to assemble the
>>> global matrix with same element stiffness matrix. The boundary condition is
>>> set using call MatZeroRowsColumns(SG,2,g_BC,one,PETSC_NULL_OBJECT,PETSC_
>>> NULL_OBJECT,ierr);
>>> >
>>> > Based on what I have done, I find the computations work well, i.e the
>>> results are correct compared with theoretical solution, for small mesh size
>>> (small than 5000 elements) using both solvers with different numbers of
>>> processes.
>>> >
>>> > However, there are several confusing issues when I increase the mesh
>>> size to 10000 and more elements with iterative solve(CG + PCBJACOBI)
>>> >
>>> > 1. For 10k elements, I can get accurate solution using iterative
>>> solver with uni-processor(i.e. only one process). However, when I use 2-8
>>> processes, it tells the linear solver converged with different iterations,
>>> but, the results are all different for different processes and erroneous.
>>> The wired thing is when I use >9 processes, the results are correct again.
>>> I am really confused by this. Could you explain me why?  If my
>>> parallelization is not correct, why it works for small cases? And I check
>>> the global matrix and RHS vector and didn't see any mallocs during the
>>> process.
>>> >
>>> > 2. For 30k elements, if I use one process, it says: Linear solve did
>>> not converge due to DIVERGED_INDEFINITE_PC. Does this commonly happen for
>>> large sparse matrix? If so, is there any stable solver or pc for large
>>> problem?
>>> >
>>> >
>>> > For parallel computing using direct solver(SUPERLU_DIST + PCLU), I can
>>> only get accuracy when the number of elements are below 5000. There must be
>>> something wrong. The way I use the superlu_dist solver is first convert
>>> MatType to AIJ, then call PCFactorSetMatSolverPackage, and change the PC to
>>> PCLU. Do I miss anything else to run SUPER_LU correctly?
>>> >
>>> >
>>> > I also use SUPER_LU and iterative solver(CG+PCBJACOBI) to solve the
>>> sequential version of the same problem. The results shows that iterative
>>> solver works well for <50k elements, while SUPER_LU only gets right
>>> solution below 5k elements. Can I say iterative solver is better than
>>> SUPER_LU for large problem? How can I improve the solver to copy with very
>>> large problem, such as million by million? Another thing is it's still
>>> doubtable of performance of SUPER_LU.
>>> >
>>> > For the inaccuracy issue, do you think it may be due to the memory?
>>> However, there is no memory error showing during the execution.
>>> >
>>> > I really appreciate someone could resolve those puzzles above for me.
>>> My goal is to replace the current SUPER_LU  solver in my parallel CPFEM
>>> main program with the iterative solver using PETSC.
>>> >
>>> >
>>> > Please let me if you would like to see my code in detail.
>>> >
>>> > Thank you very much.
>>> >
>>> > Bests,
>>> > Jinlei
>>> >
>>> >
>>> >
>>> >
>>> >
>>> >
>>> >
>>>
>>>
>>
>
>
> --
> What most experimenters take for granted before they begin their
> experiments is infinitely more interesting than any results to which their
> experiments lead.
> -- Norbert Wiener
>
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