Advertisement

A Modular Precision Format for Decoupling Arithmetic Format and Storage Format

  • Thomas Grützmacher
  • Hartwig AnztEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11339)

Abstract

In this work, we propose to decouple the arithmetic format from the storage format in numerical algorithms. We complement this idea with a modular precision storage layout that allows runtime precision adaptation such that a value can be accessed faster if lower accuracy is acceptable. Combined with precision-aware numerical algorithms that use full precision in all arithmetic computations, this strategy can result in runtime savings without impacting the memory footprint or the accuracy of the final result. In an experimental analysis using the adaptive precision Jacobi method we assess the benefits of the modular precision format on a recent high-end GPU architecture.

Keywords

Mixed precision numerics Modular precision ecosystem Customized precision GPUs Adaptive precision Jacobi 

Notes

Acknowledgements

This work was supported by the “Impuls und Vernetzungsfond” of the Helmholtz Association under grant VH-NG-1241. The authors want to acknowledge the access to the PizDaint supercomputer at the Swiss National Supercomputing Centre granted under the project #d65. The authors would like to thank Goran Flegar and Enrique Quintana-Ortí for commenting on an earlier version of the paper.

References

  1. 1.
    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).  https://doi.org/10.1007/978-3-662-48096-0_50CrossRefGoogle Scholar
  2. 2.
    Anzt, H., Dongarra, J., Flegar, G., Higham, N.J., QuintanaOrtí, E.S.: Adaptive precision in block-Jacobi preconditioning for iterative sparse linear system solvers. Concurr. Comput. Pract. Experience 0(0), e4460.  https://doi.org/10.1002/cpe.4460. https://onlinelibrary.wiley.com/doi/abs/10.1002/cpe.4460, e4460 cpe.4460
  3. 3.
    Anzt, H., Dongarra, J., Quintana-Ortí, E.S.: Adaptive precision solvers for sparse linear systems. In: Proceedings of the 3rd International Workshop on Energy Efficient Supercomputing, pp. 2:1–2:10. ACM, New York (2015).  https://doi.org/10.1145/2834800.2834802. http://doi.acm.org/10.1145/2834800.2834802
  4. 4.
    Anzt, H., Huckle, T.K., Bräckle, J., Dongarra, J.: Incomplete sparse approximate inverses for parallel preconditioning. Parallel Comput. 71(Supplement C), 1–22 (2018).  https://doi.org/10.1016/j.parco.2017.10.003. http://www.sciencedirect.com/science/article/pii/S016781911730176XMathSciNetCrossRefGoogle Scholar
  5. 5.
    Buttari, A., Dongarra, J.J., Langou, J., Langou, J., Luszczek, P., Kurzak, J.: Mixed precision iterative refinement techniques for the solution of dense linear systems. Int. J. High Perf. Comput. Appl. 21(4), 457–486 (2007)CrossRefGoogle Scholar
  6. 6.
    Carson, E., Higham, N.J.: A new analysis of iterative refinement and its application to accurate solution of ill-conditioned sparse linear systems. SIAM J. Sci. Comput. 39(6), A2834–A2856 (2017).  https://doi.org/10.1137/17M1122918MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carson, E., Higham, N.J.: Accelerating the solution of linear systems by iterative refinement in three precisions. SIAM J. Sci. Comput. 40(2), A817–A847 (2018).  https://doi.org/10.1137/17M1140819MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Göddeke, D., Strzodka, R., Turek, S.: Performance and accuracy of hardware-oriented native-, emulated- and mixed-precision solvers in FEM simulations. Int. J. Parallel Emergent Distrib. Syst. 22(4), 221–256 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)CrossRefGoogle Scholar
  10. 10.
    IEEE Computer Society: 754–2008 - IEEE Standard for Floating-Point ArithmeticGoogle Scholar
  11. 11.
    NVIDIA Corp.: CUDA C Programming Guide, 9.0 edn. http://docs.nvidia.com/cuda/cuda-c-programming-guide/index.html
  12. 12.
    Prikopa, K.E., Gansterer, W.N.: On mixed precision iterative refinement for eigenvalue problems. Proc. Comput. Sci. 18, 2647–2650 (2013).  https://doi.org/10.1016/j.procs.2013.06.002. http://www.sciencedirect.com/science/article/pii/S1877050913006108. 2013 International Conference on Computational ScienceCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.University of TennesseeKnoxvilleUSA

Personalised recommendations