A suggestion: take your second ordering and now interlace the second half of the rows with the first half of the rows (keeping the some column ordering) That is, order the rows 0, n/2, 1, n/2+1, 2, n/2+2 etc this will take the two separate "diagonal" bands and form a single "diagonal band". This will increase the "diagonal block weight" to be pretty high and the only scatters will need to be for the final rows of the input vector that all processes need to do their part of the multiply. Generate the image to make sure what I suggest make sense and then run this ordering with 1, 2, and 3 processes. Send the logs.
Barry > On Aug 23, 2015, at 10:12 AM, Nelson Filipe Lopes da Silva > <[email protected]> wrote: > > Thank you for the fast response! > > Yes. The last rows of the matrix are indeed more dense, compared with the > remaining ones. > For this example, concerning load balance between machines, the last process > had 46% of the matrix nonzero entries. A few weeks ago I suspected of this > problem and wrote a little function that could permute the matrix rows based > on their number of nonzeros. However, the matrix would become less pleasant > regarding "diagonal block weight", and I stop using it as i thought I was > becoming worse. > > Also, due to this problem, I thought I could have a complete vector copy in > each processor, instead of a distributed vector. I tried to implement this > idea, but had no luck with the results. However, even if this solution would > work, the communication for vector update was inevitable once each iteration > of my algorithm. > Since this is a rectangular matrix, I cannot apply RCM or such permutations, > however I can permute rows and columns though. > > More specifically, the problem I'm trying to solve is one of balance the best > guess and uncertainty estimates of a set of Input-Output subject to linear > constraints and ancillary informations. The matrix is called an aggregation > matrix, and each entry can be 1, 0 or -1. I don't know the cause of its > nonzero structure. I'm addressing this problem using a weighted least-squares > algorithm. > > I ran the code with a different, more friendly problem topology, logging the > load of nonzero entries and the "diagonal load" per processor. > I'm sending images of both matrices nonzero structure. The last email example > used matrix1, the example in this email uses matrix2. > Matrix1 (last email example) is 1.098.939 rows x 2.039.681 columns and > 5.171.901 nnz. > The matrix2 (this email example) is 800.000 rows x 8.800.000 columns and > 16.800.000 nnz. > > > With 1,2,3 machines, I have these distributions of nonzeros (using matrix2). > I'm sending the logs in this email. > 1 machine > [0] Matrix diagonal_nnz:16800000 (100.00 %) > [0] Matrix local nnz: 16800000 (100.00 %), local rows: 800000 (100.00 %) > ExecTime: 4min47sec > > 2 machines > [0] Matrix diagonal_nnz:4400000 (52.38 %) > [1] Matrix diagonal_nnz:4000000 (47.62 %) > > [0] Matrix local nnz: 8400000 (50.00 %), local rows: 400000 (50.00 %) > [1] Matrix local nnz: 8400000 (50.00 %), local rows: 400000 (50.00 %) > ExecTime: 13min23sec > > 3 machines > [0] Matrix diagonal_nnz:2933334 (52.38 %) > [1] Matrix diagonal_nnz:533327 (9.52 %) > [2] Matrix diagonal_nnz:2399999 (42.86 %) > > [0] Matrix local nnz: 5600007 (33.33 %), local rows: 266667 (33.33 %) > [1] Matrix local nnz: 5600007 (33.33 %), local rows: 266667 (33.33 %) > [2] Matrix local nnz: 5599986 (33.33 %), local rows: 266666 (33.33 %) > ExecTime: 20min26sec > > As for the network, I ran the make streams NPMAX=3 again. I'm also sending it > in this email. > > I too think that these bad results are caused by a combination of bad matrix > structure, especially the "diagonal weight", and maybe network. > > I really should find a way to permute these matrices to a more friendly > structure. > > Thank you very much for the help. > Nelson > > Em 2015-08-22 22:49, Barry Smith escreveu: >>> On Aug 22, 2015, at 4:17 PM, Nelson Filipe Lopes da Silva >>> <[email protected]> wrote: >>> >>> Hi. >>> >>> >>> I managed to finish the re-implementation. I ran the program with >>> 1,2,3,4,5,6 machines and saved the summary. I send each of them in this >>> email. >>> In these executions, the program performs Matrix-Vector (MatMult, >>> MatMultAdd) products and Vector-Vector operations. From what I understand >>> while reading the logs, the program takes most of the time in >>> "VecScatterEnd". >>> In this example, the matrix taking part on the Matrix-Vector products is >>> not "much diagonal heavy". >>> The following numbers are the percentages of nnz values on the matrix >>> diagonal block for each machine, and each execution time. >>> NMachines %NNZ ExecTime >>> 1 machine0 100%; 16min08sec >>> >>> 2 machine0 91.1%; 24min58sec >>> machine1 69.2%; >>> >>> 3 machine0 90.9% 25min42sec >>> machine1 82.8% >>> machine2 51.6% >>> >>> 4 machine0 91.9% 26min27sec >>> machine1 82.4% >>> machine2 73.1% >>> machine3 39.9% >>> >>> 5 machine0 93.2% 39min23sec >>> machine1 82.8% >>> machine2 74.4% >>> machine3 64.6% >>> machine4 31.6% >>> >>> 6 machine0 94.2% 54min54sec >>> machine1 82.6% >>> machine2 73.1% >>> machine3 65.2% >>> machine4 55.9% >>> machine5 25.4% >> >> Based on this I am guessing the last rows of the matrix have a lot >> of nonzeros away from the diagonal? >> >> There is a big load imbalance in something: for example with 2 >> processes you have >> >> VecMax 10509 1.0 2.0602e+02 4.2 0.00e+00 0.0 0.0e+00 >> 0.0e+00 1.1e+04 9 0 0 0 72 9 0 0 0 72 0 >> VecScatterEnd 18128 1.0 8.9404e+02 1.3 0.00e+00 0.0 0.0e+00 >> 0.0e+00 0.0e+00 53 0 0 0 0 53 0 0 0 0 0 >> MatMult 10505 1.0 6.5591e+02 1.4 3.16e+10 1.4 2.1e+04 >> 1.2e+06 0.0e+00 37 33 58 38 0 37 33 58 38 0 83 >> MatMultAdd 7624 1.0 7.0028e+02 2.3 3.26e+10 2.1 1.5e+04 >> 2.8e+06 0.0e+00 34 29 42 62 0 34 29 42 62 0 69 >> >> the 5th column has the imbalance between slowest and fastest >> process. It is 4.2 for max, 1.4 for multi and 2.3 for matmultadd, to >> get good speed ups these need to be much closer to 1. >> >> How many nonzeros in the matrix are there per process? Is it very >> different for difference processes? You really need to have each >> process have similar number of matrix nonzeros. Do you have a >> picture of the nonzero structure of the matrix? Where does the matrix >> come from, why does it have this structure? >> >> Also likely there are just to many vector entries that need to be >> scattered to the last process for the matmults. >>> >>> In this implementation I'm using MatCreate and VecCreate. I'm also leaving >>> the partition sizes in PETSC_DECIDE. >>> >>> Finally, to run the application, I'm using mpirun.hydra from mpich, >>> downloaded by PETSc configure script. >>> I'm checking the process assignment as suggested on the last email. >>> >>> Am I missing anything? >> >> Your network is very poor; likely ethernet. It is had to get much >> speedup with such slow reductions and sends and receives. >> >> Average time to get PetscTime(): 1.19209e-07 >> Average time for MPI_Barrier(): 0.000215769 >> Average time for zero size MPI_Send(): 5.94854e-05 >> >> I think you are seeing such bad results due to an unkind matrix >> nonzero structure giving per load balance and too much communication >> and a very poor computer network that just makes all the needed >> communication totally dominate. >> >> >>> >>> Regards, >>> Nelson >>> >>> Em 2015-08-20 16:17, Matthew Knepley escreveu: >>> >>>> On Thu, Aug 20, 2015 at 6:30 AM, Nelson Filipe Lopes da Silva >>>> <[email protected]> wrote: >>>> Hello. >>>> >>>> I am sorry for the long time without response. I decided to rewrite my >>>> application in a different way and will send the log_summary output when >>>> done reimplementing. >>>> >>>> As for the machine, I am using mpirun to run jobs in a 8 node cluster. I >>>> modified the makefile on the steams folder so it would run using my >>>> hostfile. >>>> The output is attached to this email. It seems reasonable for a cluster >>>> with 8 machines. From "lscpu", each machine cpu has 4 cores and 1 socket. >>>> 1) You launcher is placing processes haphazardly. I would figure out how >>>> to assign them to certain nodes >>>> 2) Each node has enough bandwidth for 1 core, so it does not make much >>>> sense to use more than 1. >>>> Thanks, >>>> Matt >>>> >>>> Cheers, >>>> Nelson >>>> >>>> >>>> Em 2015-07-24 16:50, Barry Smith escreveu: >>>> It would be very helpful if you ran the code on say 1, 2, 4, 8, 16 >>>> ... processes with the option -log_summary and send (as attachments) >>>> the log summary information. >>>> >>>> Also on the same machine run the streams benchmark; with recent >>>> releases of PETSc you only need to do >>>> >>>> cd $PETSC_DIR >>>> make streams NPMAX=16 (or whatever your largest process count is) >>>> >>>> and send the output. >>>> >>>> I suspect that you are doing everything fine and it is more an issue >>>> with the configuration of your machine. Also read the information at >>>> http://www.mcs.anl.gov/petsc/documentation/faq.html#computers on >>>> "binding" >>>> >>>> Barry >>>> >>>> On Jul 24, 2015, at 10:41 AM, Nelson Filipe Lopes da Silva >>>> <[email protected]> wrote: >>>> >>>> Hello, >>>> >>>> I have been using PETSc for a few months now, and it truly is fantastic >>>> piece of software. >>>> >>>> In my particular example I am working with a large, sparse distributed >>>> (MPI AIJ) matrix we can refer as 'G'. >>>> G is a horizontal - retangular matrix (for example, 1,1 Million rows per >>>> 2,1 Million columns). This matrix is commonly very sparse and not diagonal >>>> 'heavy' (for example 5,2 Million nnz in which ~50% are on the diagonal >>>> block of MPI AIJ representation). >>>> To work with this matrix, I also have a few parallel vectors (created >>>> using MatCreate Vec), we can refer as 'm' and 'k'. >>>> I am trying to parallelize an iterative algorithm in which the most >>>> computational heavy operations are: >>>> >>>> ->Matrix-Vector Multiplication, more precisely G * m + k = b (MatMultAdd). >>>> From what I have been reading, to achive a good speedup in this operation, >>>> G should be as much diagonal as possible, due to overlapping communication >>>> and computation. But even when using a G matrix in which the diagonal >>>> block has ~95% of the nnz, I cannot get a decent speedup. Most of the >>>> times, the performance even gets worse. >>>> >>>> ->Matrix-Matrix Multiplication, in this case I need to perform G * G' = A, >>>> where A is later used on the linear solver and G' is transpose of G. The >>>> speedup in this operation is not worse, although is not very good. >>>> >>>> ->Linear problem solving. Lastly, In this operation I compute "Ax=b" from >>>> the last two operations. I tried to apply a RCM permutation to A to make >>>> it more diagonal, for better performance. However, the problem I faced was >>>> that, the permutation is performed locally in each processor and thus, the >>>> final result is different with different number of processors. I assume >>>> this was intended to reduce communication. The solution I found was >>>> 1-calculate A >>>> 2-calculate, localy to 1 machine, the RCM permutation IS using A >>>> 3-apply this permutation to the lines of G. >>>> This works well, and A is generated as if RCM permuted. It is fine to do >>>> this operation in one machine because it is only done once while reading >>>> the input. The nnz of G become more spread and less diagonal, causing >>>> problems when calculating G * m + k = b. >>>> >>>> These 3 operations (except the permutation) are performed in each >>>> iteration of my algorithm. >>>> >>>> So, my questions are. >>>> -What are the characteristics of G that lead to a good speedup in the >>>> operations I described? Am I missing something and too much obsessed with >>>> the diagonal block? >>>> >>>> -Is there a better way to permute A without permute G and still get the >>>> same result using 1 or N machines? >>>> >>>> >>>> I have been avoiding asking for help for a while. I'm very sorry for the >>>> long email. >>>> Thank you very much for your time. >>>> Best Regards, >>>> Nelson >>>> >>>> >>>> -- >>>> 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 >>> >>> <Log01P.txt><Log02P.txt><Log03P.txt><Log04P.txt><Log05P.txt><Log06P.txt> > <Log01P.txt><Log02P.txt><Log03P.txt><matrix1.png><matrix2.png><streams.out>
