Skip to main content

Parallel simulation of drift–diffusion–recombination by cellular automata and global random walk algorithm

Abstract

We suggest in this paper a parallel implementation of cellular automation and global random walk algorithms for solving drift–diffusion–recombination problems which in contrast to the classical random walk on spheres (RWS) methods calculate the solution in any desired family of m prescribed points. The method uses only N trajectories in contrast to mN trajectories in the conventional RWS algorithm. The idea is based on the adjoint symmetry property of the Green function and a double randomization approach. The synchronous multi-particle cellular automaton model of drift–diffusion–recombination is based on known cellular automata. The global RWS and cellular automaton models are tested against the exact solutions of the equation. The accuracy and computer time of both algorithms is analyzed. Parallel codes for the Monte Carlo and cellular automaton algorithms are implemented. The domain decomposition method is employed for the cellular automaton parallel implementation. The global random walk algorithm is parallelized by the Monte Carlo trajectories distribution among the cluster cores. The efficiency of the parallel codes is studied.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    Oksendal B (1998) Stochastic differential equations. Springer, Berlin

    Book  Google Scholar 

  2. 2.

    Sabelfeld KK (1991) Monte Carlo methods in boundary value problems. Springer, Berlin

    Google Scholar 

  3. 3.

    Sabelfeld KK, Simonov NA (2016) Stochastic methods for boundary value problems. In: Numerics for high-dimensional PDEs and applications. Walter de Gruyter, Berlin. https://www.degruyter.com/view/title/521247

  4. 4.

    Sabelfeld KK (2016) Random walk on spheres method for solving drift–diffusion problems. Monte Carlo Methods Appl. 22(4):265–275. https://doi.org/10.1515/mcma-2016-0118

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Sabelfeld KK (2019) A global random walk on spheres algorithm for transient heat equation and some extensions. Monte Carlo Methods Appl. 25(1):85–96. https://doi.org/10.1515/mcma-2019-2032

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Sabelfeld KK, Kireeva A (2020) A new global random walk algorithm for calculation of the solution and its derivatives of elliptic equations with constant coefficients in an arbitrary set of points. Appl. Math. Lett. 107:106466(1–9). https://doi.org/10.1016/j.aml.2020.106466

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Bandman O (2013) Implementation of large-scale cellular automata models on multi-core computers and cluster. In: High Performance Computing and Simulation (HPCS), 2013 International Conference. IEEE Conference Publications. Helsinki, pp 304 – 310. https://doi.org/10.1109/HPCSim.2013.6641431

  8. 8.

    Rosenthal JS (2000) Parallel computing and Monte Carlo algorithms. Far East J Theor Stat 4:207–236

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Esselink K, Loyens LDJC, Smit B (1995) Parallel Monte Carlo simulations. Phys Rev E 51(2):1560–1568. https://doi.org/10.1103/physreve.51.1560

    Article  Google Scholar 

  10. 10.

    Hoekstra AG, Kroc J, Sloot PMA (2010) Simulating complex systems by cellular automata. Understanding complex systems. Springer, Berlin

    MATH  Google Scholar 

  11. 11.

    Medvedev Yu (2010) Multi-particle cellular-automata models for diffusion simulation. In: Hsu CH, Malyshkin V (eds) MTPP 2010, LNCS 6083, vol 6083. Springer, Berlin, pp 204–211. https://doi.org/10.1007/978-3-642-14822-4_23

    Chapter  Google Scholar 

  12. 12.

    Karapiperis T, Blankleider B (1994) Cellular automation model of reaction-transport processes. Physica D 78(1–2):30–64. https://doi.org/10.1016/0167-2789(94)00093-X

    Article  MATH  Google Scholar 

  13. 13.

    Worsch T (1999) Simulation of cellular automata. Future Gener Comput Syst 16(2–3):157–170. https://doi.org/10.1016/S0167-739X(99)00044-8

    Article  MATH  Google Scholar 

  14. 14.

    Kireeva A, Sabelfeld K, Kireev S (2019) Synchronous multi-particle cellular automaton model of diffusion with self-annihilation. PaCT-2019 Proceedings. LNCS, vol 11657. Springer, Berlin, pp 345–359. https://doi.org/10.1007/978-3-030-25636-4_27

    Chapter  Google Scholar 

  15. 15.

    Medvedev Yu (2010) Automata noise in diffusion cellular-automata models. Bull Novosib Comput Center Comput Sci 30:43–52

    MATH  Google Scholar 

  16. 16.

    Polyanin AD (2002) Handbook of linear partial differential equations for engineers and scientists. Chapman and Hall/CRC Press, Boca Raton

    MATH  Google Scholar 

  17. 17.

    MVS-10P cluster, JSCC RAS. http://www.jscc.ru. Accessed 17 Nov 2020

Download references

Acknowledgements

This work has been carried out as a follow-up of our study [6] where the idea and mathematics of the GRW algorithm for drift–diffusion equation have been suggested. This study and a part of the present work are supported by the Mathematical Center in Akademgorodok. The part of the work of the present paper on the development of CA and the overall implementation including the parallel versions of GRW and CA algorithms is supported by the Russian Science Foundation. Supported by the Russian Science Foundation under Grant 19-11-00019 and the Mathematical Center in Akademgorodok, the agreement with the Ministry of Science and High Education of the Russian Federation Number 075-15-2019-1675.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Anastasiya Kireeva.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kireeva, A., Sabelfeld, K.K. & Kireev, S. Parallel simulation of drift–diffusion–recombination by cellular automata and global random walk algorithm. J Supercomput 77, 6889–6903 (2021). https://doi.org/10.1007/s11227-020-03529-y

Download citation

Keywords

  • Multi-particle cellular automaton
  • Monte Carlo
  • Global random walk
  • Drift–diffusion–recombination
  • Parallel computing