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First Experiences on Parallelizing Peer Methods for Numerical Solution of a Vegetation Model

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Computational Science and Its Applications – ICCSA 2022 (ICCSA 2022)

Abstract

The purpose of this paper is to provide a parallel acceleration of peer methods for the numerical solution of systems of Ordinary Differential Equations (ODEs) arising from the space discretization of Partial Differential Equations (PDEs) modeling the growth of vegetation in semi-arid climatic zones. The parallel algorithm is implemented by using the CUDA environment for Graphics Processing Units (GPUs) architectures. Numerical experiments, showing the performance gain of the proposed strategy, are provided.

The authors are members of the GNCS group. This work is supported by GNCS-INDAM project and by the Italian Ministry of University and Research (MUR), through the PRIN 2020 project (No. 2020JLWP23) “Integrated Mathematical Approaches to Socio-Epidemiological Dynamics” (CUP: E15F21005420006) and the PRIN 2017 project (No. 2017JYCLSF) “Structure preserving approximation of evolutionary problems”.

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References

  1. Butcher, J.C.: Implicit Runge-Kutta processes. Math. Comp. 18, 50–64 (1964)

    Article  MathSciNet  Google Scholar 

  2. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)

    Book  Google Scholar 

  3. Butcher, J.C.: General linear methods. Acta Numer. 15, 157–256 (2006)

    Article  MathSciNet  Google Scholar 

  4. Calvo, M.P., Gerisch, A.: Linearly implicit Runge-Kutta methods and approximate matrix factorization. Appl. Math. 53(2–4), 183–200 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Conte, D., D’Ambrosio, R., Paternoster, B.: GPU-acceleration of waveform relaxation methods for large differential systems. Numer. Algorithms 71(2), 293–310 (2015). https://doi.org/10.1007/s11075-015-9993-6

    Article  MathSciNet  MATH  Google Scholar 

  6. Conte, D., Paternoster, B.: Parallel methods for weakly singular Volterra integral equations on GPUs. Appl. Numer. Math. 114, 30–37 (2016)

    Article  MathSciNet  Google Scholar 

  7. Cuomo, S., De Michele, P., Galletti, A., Marcellino, L.: A GPU-parallel algorithm for ECG signal denoising based on the NLM method. In: 2016 30th International Conference on Advanced Information Networking and Applications Workshops (WAINA), pp. 35–39, March 2016

    Google Scholar 

  8. Conte, D., D’Ambrosio, R., Pagano, G., Paternoster, B.: Jacobian-dependent vs. Jacobian-free discretizations for nonlinear differential problems. Comput. Appl. Math. 39(3), 1–12 (2020). https://doi.org/10.1007/s40314-020-01200-z

    Article  MathSciNet  MATH  Google Scholar 

  9. Conte, D., Mohammadi, F., Moradi, L., Paternoster, B.: Exponentially fitted two-step peer methods for oscillatory problems. Comput. Appl. Math. 39(3), 1–19 (2020). https://doi.org/10.1007/s40314-020-01202-x

    Article  MathSciNet  MATH  Google Scholar 

  10. Conte, D., Pagano, G., Paternoster, B.: Jacobian-dependent two-stage peer method for ordinary differential equations. In: Gervasi, O., et al. (eds.) ICCSA 2021, Part I. LNCS, vol. 12949, pp. 309–324. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-86653-2_23

    Chapter  Google Scholar 

  11. Conte, D., Pagano, G., Paternoster, B.: Two-step peer methods with equation-dependent coefficients. Comput. Appl. Math. 41(4), 140 (2022)

    Article  MathSciNet  Google Scholar 

  12. Eigentler, L., Sherratt, J.A.: Metastability as a coexistence mechanism in a model for dryland vegetation patterns. Bull. Math. Biol. 81, 2290–2322 (2019). https://doi.org/10.1007/s11538-019-00606-z

    Article  MathSciNet  MATH  Google Scholar 

  13. Schmitt, B.A., Weiner, R.: Parallel start for explicit parallel two-step peer methods. Numer. Algorithms 53(2), 363–381 (2010). https://doi.org/10.1007/s11075-009-9267-2

    Article  MathSciNet  MATH  Google Scholar 

  14. Schmitt, B.A., Weiner, R., Jebens, S.: Parameter optimization for explicit parallel peer two-step methods. Appl. Numer. Math. 59(3–4), 769–782 (2009)

    Article  MathSciNet  Google Scholar 

  15. Schmitt, B.A., Weiner, R., Podhaisky, H.: Multi-implicit peer two-step W-methods for parallel time integration. BIT Numer. Math. 45(1), 197–217 (2005). https://doi.org/10.1007/s10543-005-2635-y

    Article  MathSciNet  MATH  Google Scholar 

  16. Schmitt, B.A., Weiner, R., Erdmann, K.: Implicit parallel peer methods for stiff initial value problems. Appl. Numer. Math. 53(2–4), 457–470 (2005)

    Article  MathSciNet  Google Scholar 

  17. Weiner, R., Schmitt, B.A., Podhaisky, H.: Parallel “Peer” two-step W-methods and their application to MOL-systems. Appl. Numer. Math., 48(3–4), 425–439 (2004)

    Google Scholar 

  18. Schmitt, B.A., Weiner, R.: Parallel two-step W-methods with peer variables. SIAM J. Numer. Anal. 42(1), 265–282 (2004)

    Article  MathSciNet  Google Scholar 

  19. https://developer.nvidia.com/cuda-zone

  20. De Luca, P., Galletti, A., Marcellino, L.: A Gaussian recursive filter parallel implementation with overlapping. In: 2019 15th International Conference on Signal-Image Technology and Internet-Based systems (SITIS), pp. 641–648 (2019)

    Google Scholar 

  21. De Luca, P., Galletti, A., Giunta, G., Marcellino, L.: Accelerated Gaussian convolution in a data assimilation scenario. In: Krzhizhanovskaya, V.V., et al. (eds.) ICCS 2020, Part VI. LNCS, vol. 12142, pp. 199–211. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-50433-5_16

    Chapter  Google Scholar 

  22. De Luca, P., Galletti, A., Ghehsareh, H.R., Marcellino, L., Raei, M.: A GPU-CUDA framework for solving a two-dimensional inverse anomalous diffusion problem. In: Foster, I., Joubert, G.R., Kučera, L., Nagel, W.E., Peters, F. (eds.) Parallel Computing: Technology Trends, Advances in Parallel Computing, vol. 36, pp. 311–320 (2020)

    Google Scholar 

  23. De Luca, P., Galletti, A., Giunta, G., Marcellino, L.: Recursive filter based GPU algorithms in a data assimilation scenario. J. Comput. Sci. 53, 101339 (2021)

    Article  MathSciNet  Google Scholar 

  24. Jones, S.: Introduction to dynamic parallelism. In: GPU Technology Conference Presentation, vol. 338 (2012)

    Google Scholar 

  25. Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer, Berlin (1993). https://doi.org/10.1007/978-3-540-78862-1

    Book  MATH  Google Scholar 

  26. Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, Berlin (2003). https://doi.org/10.1007/978-3-662-09017-6

    Book  MATH  Google Scholar 

  27. Ixaru, L.G.: Runge-Kutta methods with equation dependent coefficients. Comput. Phys. Commun. 183(1), 63–69 (2012)

    Article  MathSciNet  Google Scholar 

  28. Jebens, S., Weiner, R., Podhaisky, H., Schmitt, B.: Explicit multi-step peer methods for special second-order differential equations. Appl. Math. Comput. 202(2), 803–813 (2008)

    MathSciNet  MATH  Google Scholar 

  29. Klinge, M., Weiner, R., Podhaisky, H.: Optimally zero stable explicit peer methods with variable nodes. BIT Numer. Math. 58(2), 331–345 (2017). https://doi.org/10.1007/s10543-017-0691-8

    Article  MathSciNet  MATH  Google Scholar 

  30. Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5(4), 329–330 (1963)

    Article  MathSciNet  Google Scholar 

  31. Sanz-Serna, J.M., Verwer, J.G., Hundsdorfer, W.H.: Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations. Numer. Math. 50, 405–418 (1986). https://doi.org/10.1007/BF01396661

    Article  MathSciNet  MATH  Google Scholar 

  32. Weiner, R., Biermann, K., Schmitt, B., Podhaisky, H.: Explicit two-step peer methods. Comput. Math. Appl. 55(4), 609–619 (2008)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are members of the GNCS group. This work is supported by GNCS-INDAM project and by the Italian Ministry of University and Research (MUR), through the PRIN 2020 project (No. 2020JLWP23) “Integrated Mathematical Approaches to Socio-Epidemiological Dynamics” (CUP: E15F21005420006) and the PRIN 2017 project (No. 2017JYCLSF) “Structure preserving approximation of evolutionary problems”.

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Conte, D. et al. (2022). First Experiences on Parallelizing Peer Methods for Numerical Solution of a Vegetation Model. In: Gervasi, O., Murgante, B., Hendrix, E.M.T., Taniar, D., Apduhan, B.O. (eds) Computational Science and Its Applications – ICCSA 2022. ICCSA 2022. Lecture Notes in Computer Science, vol 13376. Springer, Cham. https://doi.org/10.1007/978-3-031-10450-3_33

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  • DOI: https://doi.org/10.1007/978-3-031-10450-3_33

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