Abstract
In this work we consider a mixed precision approach to accelerate the implementation of multi-stage methods. We show that Runge–Kutta methods can be designed so that certain costly intermediate computations can be performed as a lower-precision computation without adversely impacting the accuracy of the overall solution. In particular, a properly designed Runge–Kutta method will damp out the errors committed in the initial stages. This is of particular interest when we consider implicit Runge–Kutta methods. In such cases, the implicit computation of the stage values can be considerably faster if the solution can be of lower precision (or, equivalently, have a lower tolerance). We provide a general theoretical additive framework for designing mixed precision Runge–Kutta methods, and use this framework to derive order conditions for such methods. Next, we show how using this approach allows us to leverage low precision computation of the implicit solver while retaining high precision in the overall method. We present the behavior of some mixed-precision implicit Runge–Kutta methods through numerical studies, and demonstrate how the numerical results match with the theoretical framework. This novel mixed-precision implicit Runge–Kutta framework opens the door to the design of many such methods.
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References
Abdelfattah, A., Anzt, H., Boman, E.G., Carson, E., Cojean, T., Dongarra, J., Gates, M., Grutzmacher, T., Higham, N.J., Li, S., Lindquist, N., Liu, Y., Loe, J., Luszczek, P., Nayak, P., Pranesh, S., Rajamanickam, S., Ribizel, T., Smith, B., Swirydowicz, K., Thomas, S., Tomov, S., Tsai, Y.M., Yamazaki, I., Yang, U.M.: “A Survey of Numerical Methods Utilizing Mixed Precision Arithmetic,” (2020) arXiv:2007.06674
Alexander, R.: Diagonally implicit Runge-Kutta methods for stiff O.D.E.s. SIAM J. Numer. Anal. 14(6), 1006–1021 (1977)
Ascher, U., Ruuth, S., Wetton, B.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32, 797–823 (1995)
Burnett, B., Gottlieb, S., Grant, Z.J., Heryudono, A.: Performance evaluation of mixed-precision Runge-Kutta methods. IEEE High Perform. Extreme Comput. Conf. (HPEC) 2021, 1–6 (2021). https://doi.org/10.1109/HPEC49654.2021.9622803
Kennedy, C.A., Carpenter, M.H.: Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44(1–2), 139–181 (2003)
Field, S. E., Gottlieb, S., Grant, Z. J., Isherwood, L. F., Khanna, G.: A GPU-accelerated mixed-precision WENO method for extremal black hole and gravitational wave physics computations, arxiv.org/abs/2010.04760
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, Berlin (1991)
Higham, N.J., Pranesh, S.: Simulating low precision floating-point arithmetic. SIAM J. Sci. Comput. 41(5), C585–C602 (2019)
Higueras, I., Ketcheson, D.I., Kocsis, T.A.: Optimal monotonicity-preserving perturbations of a given Runge-Kutta method. J. Sci. Comput. 76, 1337–1369 (2018)
Ketcheson, D.I., Parsani, M., Grant, Z.J., Ahmadia, A.J., Ranocha, H.: RK-Opt: A package for the design of numerical ODE solvers. J. Open Source Softw. 5(54), 2514 (2020). https://doi.org/10.21105/joss.02514
Kouya, T.: Practical Implementation of High-Order Multiple Precision Fully Implicit Runge-Kutta Methods with Step Size Control Using Embedded Formula, arxiv.org/abs/1306.2392
Norsett, S.P., Wanner, G.: Perturbed collocation and Runge-Kutta methods. Numer. Math. 1981(38), 193–208 (1981)
Sandu, A., Gunther, M.: A generalized-structure approach to additive Runge-Kutta methods. SIAM J. Numer. Anal. 53(1), 17–42 (2015)
Funding
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, as part of their Applied Mathematics Research Program. The work was performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract No. De- AC05-00OR22725. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan http://energy.gov/downloads/doe-public-access-plan.
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Submitted to the editors on January 11, 2020.
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Grant, Z.J. Perturbed Runge–Kutta Methods for Mixed Precision Applications. J Sci Comput 92, 6 (2022). https://doi.org/10.1007/s10915-022-01801-2
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DOI: https://doi.org/10.1007/s10915-022-01801-2