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A Synchronization-Free Algorithm for Parallel Sparse Triangular Solves

  • Weifeng LiuEmail author
  • Ang Li
  • Jonathan Hogg
  • Iain S. Duff
  • Brian Vinter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9833)

Abstract

The sparse triangular solve kernel, SpTRSV, is an important building block for a number of numerical linear algebra routines. Parallelizing SpTRSV on today’s manycore platforms, such as GPUs, is not an easy task since computing a component of the solution may depend on previously computed components, enforcing a degree of sequential processing. As a consequence, most existing work introduces a preprocessing stage to partition the components into a group of level-sets or colour-sets so that components within a set are independent and can be processed simultaneously during the subsequent solution stage. However, this class of methods requires a long preprocessing time as well as significant runtime synchronization overhead between the sets. To address this, we propose in this paper a novel approach for SpTRSV in which the ordering between components is naturally enforced within the solution stage. In this way, the cost for preprocessing can be greatly reduced, and the synchronizations between sets are completely eliminated. A comparison with the state-of-the-art library supplied by the GPU vendor, using 11 sparse matrices on the latest GPU device, show that our approach obtains an average speedup of 2.3 times in single precision and 2.14 times in double precision. The maximum speedups are 5.95 and 3.65, respectively. In addition, our method is an order of magnitude faster for the preprocessing stage than existing methods.

Keywords

Critical Section Average Speedup Atomic Operation Barrier Synchronization Scratchpad Memory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The authors would like to thank our anonymous reviewers for their invaluable feedback. We also thank Shuai Che for helpful discussion about OpenCL programming, and thank Huamin Ren for supplying access to the machine with the NVIDIA GeForce Titan X GPU. The research leading to these results has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 671633.

References

  1. 1.
    Anderson, E., Saad, Y.: Solving sparse triangular linear systems on parallel computers. Int. J. High Speed Comput. 1(1), 73–95 (1989)CrossRefzbMATHGoogle Scholar
  2. 2.
    Anzt, H., Chow, E., Dongarra, J.: Iterative sparse triangular solves for preconditioning. In: Träff, J.L., Hunold, S., Versaci, F. (eds.) Euro-Par 2015. LNCS, vol. 9233, pp. 650–661. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  3. 3.
    Björck, Å.: Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics, Philadelphia (1996)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chow, E., Patel, A.: Fine-grained parallel incomplete LU factorization. SIAM J. Sci. Comput. 37(2), C169–C193 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Davis, T.: Direct Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia (2006)CrossRefzbMATHGoogle Scholar
  6. 6.
    Davis, T.A., Hu, Y.: The University of Florida sparse matrix collection. ACM Trans. Math. Softw. 38(1), 1:1–1:25 (2011)MathSciNetGoogle Scholar
  7. 7.
    Duff, I.S., Erisman, A.M., Reid, J.K.: Direct Methods for Sparse Matrices. Oxford University Press Inc., New York (1986)zbMATHGoogle Scholar
  8. 8.
    Duff, I.S., Heroux, M.A., Pozo, R.: An overview of the sparse basic linear algebra subprograms: the new standard from the BLAS Technical forum. ACM Trans. Math. Softw. 28(2), 239–267 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hogg, J.D.: A fast dense triangular solve in CUDA. SIAM J. Sci. Comput. 35(3), C303–C322 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kabir, H., Booth, J.D., Aupy, G., Benoit, A., Robert, Y., Raghavan, P.: STS-k: a multilevel sparse triangular solution scheme for NUMA multicores. In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC 2015, pp. 55:1–55:11 (2015)Google Scholar
  11. 11.
    Li, A., van den Braak, G.J., Corporaal, H., Kumar, A.: Fine-grained synchronizations and dataflow programming on GPUs. In: Proceedings of the 29th ACM on International Conference on Supercomputing, ICS 2015, pp. 109–118 (2015)Google Scholar
  12. 12.
    Li, R., Saad, Y.: GPU-accelerated preconditioned iterative linear solvers. J. Supercomputing 63(2), 443–466 (2013)CrossRefGoogle Scholar
  13. 13.
    Liang, C.K., Prvulovic, M.: MiSAR: minimalistic synchronization accelerator with resource overflow management. In: Proceedings of the 42nd Annual International Symposium on Computer Architecture, ISCA 2015, pp. 414–426 (2015)Google Scholar
  14. 14.
    Liu, W.: Parallel and Scalable Sparse Basic Linear Algebra Subprograms. Ph.D. Thesis, University of Copenhagen (2015)Google Scholar
  15. 15.
    Liu, W., Vinter, B.: A framework for general sparse matrix-matrix multiplication on GPUs and heterogeneous processors. J. Parallel Distrib. Comput. 85, 47–61 (2015)CrossRefGoogle Scholar
  16. 16.
    Liu, W., Vinter, B.: CSR5: an efficient storage format for cross-platform sparse matrix-vector multiplication. In: Proceedings of the 29th ACM International Conference on Supercomputing, ICS 2015, pp. 339–350 (2015)Google Scholar
  17. 17.
    Liu, W., Vinter, B.: Speculative segmented sum for sparse matrix-vector multiplication on heterogeneous processors. Parallel Comput. 49, 179–193 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mayer, J.: Parallel algorithms for solving linear systems with sparse triangular matrices. Computing 86(4), 291–312 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Naumov, M.: Parallel Solution of Sparse Triangular Linear Systems in the Preconditioned Iterative Methods on the GPU. Technical report NVIDIA (2011)Google Scholar
  20. 20.
    Park, J., Smelyanskiy, M., Sundaram, N., Dubey, P.: Sparsifying synchronization for high-performance shared-memory sparse triangular solver. In: Kunkel, J.M., Ludwig, T., Meuer, H.W. (eds.) ISC 2014. LNCS, vol. 8488, pp. 124–140. Springer, Heidelberg (2014)Google Scholar
  21. 21.
    Ros, A., Kaxiras, S.: Callback: efficient synchronization without invalidation with a directory just for spin-waiting. In: Proceedings of the 42nd Annual International Symposium on Computer Architecture, ISCA 2015, pp. 427–438 (2015)Google Scholar
  22. 22.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)CrossRefzbMATHGoogle Scholar
  23. 23.
    Saltz, J.H.: Aggregation methods for solving sparse triangular systems on multiprocessors. SIAM J. Sci. Stat. Comput. 11(1), 123–144 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schreiber, R., Tang, W.P.: Vectorizing the Conjugate Gradient Method. In: Proceedings of the Symposium on CYBER 205 Applications (1982)Google Scholar
  25. 25.
    Scogland, T.R., Feng, W.C.: Design and evaluation of scalable concurrent queues for many-core architectures. In: Proceedings of the 6th ACM/SPEC International Conference on Performance Engineering, ICPE 2015, pp. 63–74 (2015)Google Scholar
  26. 26.
    Suchoski, B., Severn, C., Shantharam, M., Raghavan, P.: Adapting sparse triangular solution to GPUs. In: Proceedings of the 2012 41st International Conference on Parallel Processing Workshops, ICPPW 2012, pp. 140–148 (2012)Google Scholar
  27. 27.
    Wang, H., Liu, W., Hou, K., Feng, W.C.: Parallel Transposition of Sparse Data Structures. In: Proceedings of the 30th ACM International Conference on Supercomputing, ICS 2016 (2016)Google Scholar
  28. 28.
    Wolf, M.M., Heroux, M.A., Boman, E.G.: Factors impacting performance of multithreaded sparse triangular solve. In: Palma, J.M.L.M., Daydé, M., Marques, O., Lopes, J.C. (eds.) VECPAR 2010. LNCS, vol. 6449, pp. 32–44. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  29. 29.
    Xiao, S., Feng, W.C.: Inter-block GPU Communication via fast barrier synchronization. In: 2010 IEEE International Symposium on Parallel Distributed Processing, IPDPS 2010, pp. 1–12 (2010)Google Scholar
  30. 30.
    Yan, S., Long, G., Zhang, Y.: StreamScan: fast scan algorithms for GPUs without global barrier synchronization. In: Proceedings of the 18th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, PPopp. 2013, pp. 229–238 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Weifeng Liu
    • 1
    • 2
    Email author
  • Ang Li
    • 3
  • Jonathan Hogg
    • 2
  • Iain S. Duff
    • 2
  • Brian Vinter
    • 1
  1. 1.Niels Bohr InstituteUniversity of CopenhagenCopenhagenDenmark
  2. 2.Scientific Computing DepartmentSTFC Rutherford Appleton LaboratoryDidcotUK
  3. 3.Eindhoven University of TechnologyEindhovenNetherlands

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