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Theoretical and Computational Fluid Dynamics

, Volume 29, Issue 4, pp 311–328 | Cite as

Rounding errors may be beneficial for simulations of atmospheric flow: results from the forced 1D Burgers equation

  • Peter D. DübenEmail author
  • Stamen I. Dolaptchiev
Open Access
Original Article

Abstract

Inexact hardware can reduce computational cost, due to a reduced energy demand and an increase in performance, and can therefore allow higher-resolution simulations of the atmosphere within the same budget for computation. We investigate the use of emulated inexact hardware for a model of the randomly forced 1D Burgers equation with stochastic sub-grid-scale parametrisation. Results show that numerical precision can be reduced to only 12 bits in the significand of floating-point numbers—instead of 52 bits for double precision—with no serious degradation in results for all diagnostics considered. Simulations that use inexact hardware on a grid with higher spatial resolution show results that are significantly better compared to simulations in double precision on a coarser grid at similar estimated computing cost. In the second half of the paper, we compare the forcing due to rounding errors to the stochastic forcing of the stochastic parametrisation scheme that is used to represent sub-grid-scale variability in the standard model setup. We argue that stochastic forcings of stochastic parametrisation schemes can provide a first guess for the upper limit of the magnitude of rounding errors of inexact hardware that can be tolerated by model simulations and suggest that rounding errors can be hidden in the distribution of the stochastic forcing. We present an idealised model setup that replaces the expensive stochastic forcing of the stochastic parametrisation scheme with an engineered rounding error forcing and provides results of similar quality. The engineered rounding error forcing can be used to create a forecast ensemble of similar spread compared to an ensemble based on the stochastic forcing. We conclude that rounding errors are not necessarily degrading the quality of model simulations. Instead, they can be beneficial for the representation of sub-grid-scale variability.

Keywords

Inexact hardware Stochastic parametrisation Numerical precision Turbulent closure Ensemble methods 

References

  1. 1.
    Bengtsson L., Steinheimer M., Bechtold P., Geleyn J.F.: A stochastic parametrization for deep convection using cellular automata. Q. J. R. Meteorol. Soc. 139(675), 1533–1543 (2013)CrossRefGoogle Scholar
  2. 2.
    Berner J., Shutts G.J., Leutbecher M., Palmer T.N.: A Spectral stochastic kinetic Energy backscatter scheme and its impact On flow-dependent predictability in the ECMWF ensemble prediction system. J. Atmos. Sci. 66, 603 (2009). doi: 10.1175/2008JAS2677.1 CrossRefGoogle Scholar
  3. 3.
    Buizza R., Miller M., Palmer T.N.: Stochastic representation of model uncertainties in the ECMWF ensemble prediction system. Q. J. R. Meteorol. Soc. 125, 2887–2908 (1999). doi: 10.1002/qj.49712556006 CrossRefGoogle Scholar
  4. 4.
    Crommelin D., Vanden-Eijnden E.: Subgrid-Scale parameterization with conditional Markov chains. J. Atmos. Sci. 65, 2661–2675 (2008). doi: 10.1175/2008JAS2566.1 CrossRefGoogle Scholar
  5. 5.
    Dolaptchiev S., Achatz U., Timofeyev I.: Stochastic closure for local averages in the finite-difference discretization of the forced burgers equation. Theor. Comput. Fluid Dyn. 27(3–4), 297–317 (2013)CrossRefGoogle Scholar
  6. 6.
    Dolaptchiev S., Timofeyev I., Achatz U.: Subgrid-scale closure for the inviscid burgers-hopf equation. Commun. Math. Sci. 11(3), 757–777 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Düben, P.D., Joven, J., Lingamneni, A., McNamara, H., De Micheli, G., Palem, K.V., Palmer, T.N.: On the use of inexact, pruned hardware in atmospheric modelling. Philos. Trans. R. Soc. A (2014)Google Scholar
  8. 8.
    Düben, P.D., McNamara, H., Palmer, T.: The use of imprecise processing to improve accuracy in weather & climate prediction. J. Comput. Phys. 271(0):2–18 (2014). Frontiers in Computational Physics Modeling the Earth SystemGoogle Scholar
  9. 9.
    Düben, P.D., Palmer, T.N.: Benchmark tests for numerical forecasts on inexact hardware. Mon. Weather Rev. 142, 3809–3829 (2014). doi: 10.1175/MWR-D-14-00110.1
  10. 10.
    Düben, P.D., Schlachter, J., Parishkrati, Yenugula, S., Augustine, J., Enz, C., Palem, K., Palmer, T.N.: Opportunities for energy efficient computing: a study of inexact general purpose processors for high-performance and big-data applications. In: Proceedings of the 2015 Design, Automation & Test in Europe Conference & Exhibition (DATE ’15). EDA Consortium, San Jose, CA, USA, pp. 764–769 (2015)Google Scholar
  11. 11.
    Franzke C., Majda A.J., Vanden-Eijnden E.: Low-order stochastic mode reduction for a realistic barotropic model climate. J. Atmos. Sci. 62, 1722–1745 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gan, L., Fu, H., Luk, W., Yang, C., Xue, W., Huang, X., Zhang, Y., Yang, G.: Accelerating solvers for global atmospheric equations through mixed-precision data flow engine. In: 23rd International Conference on Field Programmable Logic and Applications (FPL), pp. 1–6 (2013)Google Scholar
  13. 13.
    Hasselmann K.: Stochastic climate models part i. theory. Tellus 28(6), 473–485 (1976)CrossRefGoogle Scholar
  14. 14.
    Horenko I., Dolaptchiev S.I., Eliseev A.V., Mokhov I.I., Klein R.: Metastable decomposition of high-dimensional meteorological data with gaps. J. Atmos. Sci. 65, 3479–3496 (2008)CrossRefzbMATHGoogle Scholar
  15. 15.
    IEEE: IEEE standard for binary floating-point arithmetic. ANSI/IEEE Std, pp. 754–1985 (1985). doi: 10.1109/IEEESTD.1985.82928
  16. 16.
    Kahng, A., Kang, S., Kumar, R., Sartori, J.: Slack redistribution for graceful degradation under voltage overscaling. In: 15th Asia and South Pacific on Design Automation Conference (ASP-DAC), pp. 825–831 (2010)Google Scholar
  17. 17.
    Lingamneni, A., Enz, C., Nagel, J.L., Palem, K., Piguet, C.: Energy parsimonious circuit design through probabilistic pruning. In: Design, Automation Test in Europe Conference Exhibition (DATE), pp. 1–6 (2011)Google Scholar
  18. 18.
    Majda A., Timofeyev I., Vanden-Eijnden E.: Stochastic models for selected slow variables in large deterministic systems. Nonlinearity 19, 769–794 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Majda A.J., Timofeyev I., Vanden Eijnden E.: A mathematical framework for stochastic climate models. Commun. Pure Appl. Mathe. 54(8), 891–974 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Narayanan, S., Sartori, J., Kumar, R., Jones, D.L.: Scalable stochastic processors. In: Proceedings of Design, Automation and Test in Europe Conference, pp. 335–338 (2010)Google Scholar
  21. 21.
    Oriato, D., Tilbury, S., Marrocu, M., Pusceddu, G.: Acceleration of a meteorological limited area model with dataflow engines. In: 2012 Symposium on Application Accelerators in High Performance Computing (SAAHPC), pp. 129–132 (2012)Google Scholar
  22. 22.
    Palem K.: Energy aware computing through probabilistic switching: a study of limits. IEEE Trans. Comput. 54(9), 1123–1137 (2005)CrossRefGoogle Scholar
  23. 23.
    Palem, K.V.: Energy aware algorithm design via probabilistic computing: from algorithms and models to Moore’s law and novel (semiconductor) devices. In: Proceedings of CASES, pp. 113–116 (2003)Google Scholar
  24. 24.
    Palem, K.V.: Inexactness and a future of computing. Philos. Trans. R. Soc. A math. Phys. Eng. Sci. 372(2018), 20130281 (2014)Google Scholar
  25. 25.
    Palmer T.N.: A nonlinear dynamical perspective on model error: a proposal for non-local stochastic-dynamic parametrization in weather and climate prediction models. Q. J. R. Meteorol. Soc. 127(572), 279–304 (2001)Google Scholar
  26. 26.
    Palmer T.N.: Towards the probabilistic earth-system simulator: a vision for the future of climate and weather prediction. Q. J. R. Meteorol. Soc. 138(665), 841–861 (2012)CrossRefGoogle Scholar
  27. 27.
    Palmer, T.N., Buizza, R., Doblas-Reyes, F., Jung, T., Leutbecher, M., Shutts, G.J., Steinheimer, M., Weisheimer, M.: Stochastic parametrization and model uncertainty. Technical Memorandum, 598 ECMWF: Reading, UK (2009)Google Scholar
  28. 28.
    Plant R.S., Craig G.C.: A stochastic parameterization for deep convection based on equilibrium statistics. J. Atmos. Sci. 65, 87–105 (2008)CrossRefGoogle Scholar
  29. 29.
    Russell, F.P., Düben, P.D., Niu, X., Luk, W., Palmer, T.N.: Hardware acceleration and precision analysis of a chaotic system with application to atmospheric modelling, accepted. In: Proceedings of FCCM (2015)Google Scholar
  30. 30.
    Shutts G.: A kinetic energy backscatter algorithm for use in ensemble prediction systems. Q. J. R. Meteorol. Soc. 131, 3079–3102 (2005)CrossRefGoogle Scholar

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© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Atmospheric, Oceanic and Planetary PhysicsUniversity of OxfordOxfordUK
  2. 2.Institut für Atmosphäre und UmweltJohann Wolfgang Goethe-Universität Frankfurt/MainFrankfurtGermany

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