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Hybrid GPU–CPU Efficient Implementation of a Parallel Numerical Algorithm for Solving the Cauchy Problem for a Nonlinear Differential Riccati Equation of Fractional Variable Order

Author

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  • Dmitrii Tverdyi

    (Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Mirnaya Street 7, 684034 Kamchatka, Russia
    International Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatsky, Pogranichnaya Street 4, 683032 Kamchatka, Russia)

  • Roman Parovik

    (Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Mirnaya Street 7, 684034 Kamchatka, Russia
    International Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatsky, Pogranichnaya Street 4, 683032 Kamchatka, Russia)

Abstract

The numerical solution for fractional dynamics problems can create a high computational load, which makes it necessary to implement efficient algorithms for their solution. The main contribution to the computational load of such computations is created by heredity (memory), which is determined by the dependence of the current value of the solution function on previous values in the time interval. In terms of mathematics, the heredity here is described using a fractional differentiation operator in the Gerasimov–Caputo sense of variable order. As an example, we consider the Cauchy problem for the non-linear fractional Riccati equation with non-constant coefficients. An efficient parallel implementation algorithm has been proposed for the known sequential non-local explicit finite-difference numerical solution scheme. This implementation of the algorithm is a hybrid one, since it uses both GPU and CPU computational nodes. The program code of the parallel implementation of the algorithm is described in C and CUDA C languages, and is developed using OpenMP and CUDA hardware, as well as software architectures. This paper presents a study on the computational efficiency of the proposed parallel algorithm based on data from a series of computational experiments that were obtained using a computing server NVIDIA DGX STATION. The average computation time is analyzed in terms of the following: running time, acceleration, efficiency, and the cost of the algorithm. As a result, it is shown on test examples that the hybrid version of the numerical algorithm can give a significant performance increase of 3–5 times in comparison with both the sequential version of the algorithm and OpenMP implementation.

Suggested Citation

  • Dmitrii Tverdyi & Roman Parovik, 2023. "Hybrid GPU–CPU Efficient Implementation of a Parallel Numerical Algorithm for Solving the Cauchy Problem for a Nonlinear Differential Riccati Equation of Fractional Variable Order," Mathematics, MDPI, vol. 11(15), pages 1-21, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:15:p:3358-:d:1207670
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    References listed on IDEAS

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    1. Vladimir Okrepilov & Valeriy Makarov & Albert Bakhtizin & Svetlana Kuzmina, 2015. "Application of Supercomputer Technologies for Simulation Of Socio-Economic Systems," Economy of region, Centre for Economic Security, Institute of Economics of Ural Branch of Russian Academy of Sciences, vol. 1(2), pages 301-313.
    2. Dmitrii Tverdyi & Evgeny Makarov & Roman Parovik, 2023. "Hereditary Mathematical Model of the Dynamics of Radon Accumulation in the Accumulation Chamber," Mathematics, MDPI, vol. 11(4), pages 1-20, February.
    3. Vasily E. Tarasov, 2020. "Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 8(5), pages 1-3, April.
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