Skip to main content
SpringerLink
Log in
Book cover

International Conference on High Performance Computing and Applications

HPCA 2015: High Performance Computing and Applications pp 55–72Cite as

Development of Krylov and AMG Linear Solvers for Large-Scale Sparse Matrices on GPUs

  • Conference paper
  • First Online:

  • 735 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9576))

Abstract

This paper introduces our work on developing Krylov subspace and AMG solvers on NVIDIA GPUs. As SpMV is a crucial part for these iterative methods, SpMV algorithms for a single GPU and multiple GPUs are implemented. A HEC matrix format and a communication mechanism are established. Also, a set of specific algorithms for solving preconditioned systems in parallel environments are designed, including ILU(k), RAS and parallel triangular solvers. Based on these work, several Krylov solvers and AMG solvers are developed. According to numerical experiments, favorable acceleration performance is obtained from our Krylov solver and AMG solver under various parameter conditions.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Chen, Z.: Reservoir Simulation: Mathematical Techniques in Oil Recovery. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 77. SIAM, Philadelphia (2007)

    Book  MATH  Google Scholar 

  2. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  3. Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., Vorst, H.V.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edn. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

  4. Hu, X., Liu, W., Qin, G., Xu, J., Yan, Y., Zhang, C.: Development of a fast auxiliary subspace pre-conditioner for numerical reservoir simulators. In: SPE Reservoir Characterisation and Simulation Conference and Exhibition, Abu Dhabi, UAE, SPE-148388-MS, pp. 9–11, October 2011

    Google Scholar 

  5. Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media. Computational Science and Engineering Series, vol. 2. SIAM, Philadelphia (2006)

    Book  MATH  Google Scholar 

  6. Liu, H., Wang, K., Chen, Z., Jordan, K.E.: Efficient multi-stage preconditioners for highly heterogeneous reservoir simulations on parallel distributed systems. In: SPE Reservoir Simulation Symposium, Houston, Texas, USA, SPE-173208-MS, pp. 23–25 (2015)

    Google Scholar 

  7. Stüben, K.: A review of algebraic multigrid. J. Comput. Appl. Math. 128(1–2), 281–309 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ruge, J.W., Stüben, K.: Algebraic multigrid (AMG). In: McCormick, S.F. (ed.) Multigrid Methods, Frontiers in Applied Mathematics, vol. 5. SIAM, Philadelphia (1986)

    Google Scholar 

  9. Brandt, A., McCormick, S.F., Ruge, J.: Algebraic multigrid (AMG) for sparse matrix equations. In: Evans, D.J. (ed.) Sparsity and its Applications, pp. 257–284. Cambridge University Press, Cambridge (1984)

    Google Scholar 

  10. Wagner, C.: Introduction to algebraic multigrid, course notes of an algebraic multigrid course at the University of Heidelberg in the Wintersemester (1999)

    Google Scholar 

  11. Vassilevski, P.S.: Lecture Notes on Multigrid Methods. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory (2010)

    Google Scholar 

  12. Falgout, R., Cleary, A., Jones, J., Chow, E., Henson, V., Baldwin, C., Brown, P., Vassilevski, P., Yang, U.M.: Hypre home page (2011). http://acts.nersc.gov/hypre

  13. Cleary, A.J., Falgout, R.D., Henson, V.E., Jones, J.E., Manteuffel, T.A., McCormick, S.F., Miranda, G.N., Ruge, J.W.: Robustness and scalability of algebraic multigrid. SIAM J. Sci. Comput. 21, 1886–1908 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gao, Y., Iqbal, S., Zhang, P., Qiu, M.: Performance and power analysis of high-density multi-GPGPU architectures.: a preliminary case study. In: 2015 IEEE 17th International Conference on High Performance Computing and Communications (HPCC-ICESS-CSS 2015), New York, USA, 24–26 August (2015)

    Google Scholar 

  15. NVIDIA official website. http://www.nvidia.com/object/tesla-servers.html

  16. Fujitsu official website. http://techcommunity.ts.fujitsu.com/en/client-computing-devices-2/d/uid-5911b36b-324b-fc23-45fa-2438e4c546f3.html

  17. Karypis, G., Kumar, V.: A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1999)

    Article  MATH  Google Scholar 

  18. Cai, X.-C., Sarkis, M.: A restricted additive schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21, 792–797 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, H., Yu, S., Chen, Z., Hsieh, B., Shao, L.: Sparse matrix-vector multiplication on NVIDIA GPU. Int. J. Numer. Anal. Model. Ser. B 3(2), 185–191 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Liu, H., Yu, S., Chen, Z., Hsieh, B., Shao, L.: Parallel preconditioners for reservoir simulation on GPU. In: SPE Latin American and Caribbean Petroleum Engineering Conference, Mexico City, Mexico, SPE 152811-PP, pp. 16–18 (2012)

    Google Scholar 

  21. Li, R., Saad, Y.: GPU-accelerated preconditioned iterative linear solvers. Technical Report Umsi-2010-112. University of Minnesota, Minneapolis, MN, Minnesota Supercomputer Institute (2010)

    Google Scholar 

  22. Grimes, R., Kincaid, D., Young, D.: ITPACK 2.0 user’s guide. Technical Report CNA-150, Center for Numerical Analysis, University of Texas, August 1979

    Google Scholar 

  23. Bell, N., Garland, M.: Implementing sparse matrix-vector multiplication on throughput-oriented processors. In: Proceedings of Supercomputing, pp. 1–11, November 2009

    Google Scholar 

  24. Davis, T.A.: University of Florida Sparse Matrix Collection, NA digest (1994). https://www.cise.ufl.edu/research/sparse/matrices/

Download references

Acknowledgments

The support of Department of Chemical and Petroleum Engineering, University of Calgary and Reservoir Simulation Research Group is gratefully acknowledged. The research is partly supported by NSERC/AIEES/Foundation CMG, AITF iCore, IBM Thomas J. Watson Research Center, and the Frank and Sarah Meyer FCMG Collaboration Centre for Visualization and Simulation. The research is also enabled in part by support provided by WestGrid (www.westgrid.ca) and Compute Canada Calcul Canada (www.computecanada.ca).

Author information

Authors and Affiliations

  1. Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Canada

    Bo Yang, Hui Liu & Zhangxin Chen

Authors
  1. Bo Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hui Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhangxin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhangxin Chen .

Editor information

Editors and Affiliations

  1. School of Computer Engineering, Shanghai University, Shanghai, China

    Jiang Xie

  2. Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada

    Zhangxin Chen

  3. Mathematics Department, University of Wyoming, Laramie, Wyoming, USA

    Craig C. Douglas

  4. School of Computer Engineering, Shanghai University, Shanghai, China

    Wu Zhang

  5. Mathematics and Informatics, South China Agricultural University, Guangzhou, China

    Yan Chen

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Yang, B., Liu, H., Chen, Z. (2016). Development of Krylov and AMG Linear Solvers for Large-Scale Sparse Matrices on GPUs. In: Xie, J., Chen, Z., Douglas, C., Zhang, W., Chen, Y. (eds) High Performance Computing and Applications. HPCA 2015. Lecture Notes in Computer Science(), vol 9576. Springer, Cham. https://doi.org/10.1007/978-3-319-32557-6_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-32557-6_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-32556-9

  • Online ISBN: 978-3-319-32557-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation