[petsc-users] Petsc mesh scalability issue with iterative solver and direct solver
Hong
hzhang at mcs.anl.gov
Tue Aug 9 10:42:31 CDT 2016
Jinlei:
See http://www.mcs.anl.gov/petsc/documentation/faq.html#computers
Hong
Hi Barry,
>
> Thanks for your answer.
>
> But logically for large problem, we are always expecting to see paralleled
> program perform better with regard to both speed and memory since each of
> the multi-processes independently deal with its own submatrix, especially
> for iterative solver, which is revealed by CG+BJ.
> I just don't understand, in the computing with CG+ASM and SUPER_LU, why
> the two-process is most inefficient among these cases. If this is due to
> the communication cost compared with uni-process, why the speed goes down
> for triple and more processes. I'm new to parallelism, could you speculate
> any possible reason for such situation?
>
> Great thanks
>
>
>
> On Fri, Aug 5, 2016 at 10:09 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
>
>>
>> > On Aug 5, 2016, at 5:58 PM, Jinlei Shen <jshen25 at jhu.edu> wrote:
>> >
>> > 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.
>>
>> This is actually not surprising at all but since the mantra is
>> "parallelism will always make things faster" it can confuse people. When
>> run with one process the ASM and SuperLU_DIST utilize essentially
>> sequential algorithms, when run with two processes they "switch" to
>> parallel algorithms which simply are not as good as the essentially
>> sequential algorithm that is obtained with one process hence they run
>> slower. This is just life, there really isn't something one can do about it
>> except to perhaps use a poorer quality algorithm on one process so that two
>> processes look better but the goal of PETSc is not to make parallelism to
>> look good but to provide efficient solvers (as best we can) for one and
>> multiple processes.
>>
>> Barry
>>
>>
>>
>>
>> > 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,o
>> ne,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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