Asymptotic Complexity Estimates of Hybrid Algorithms for the Numerical Solution of a Model Equation of Radon Volume Activity with a Variable-Order Fractional Derivative
DOI:
https://doi.org/10.71310/pcam.2_72.2026.11Keywords:
fractional derivatives, time nonlocality, variable nonlocality, finite difference schemes, parallel computing, GPU, asymptotic estimates, algorithms efficiencyAbstract
The article considers hybrid CPU-GPU parallel implementations of numerical solution algorithms for the hereditary model equation of radon volume activity. The test example is a direct Cauchy problem for a nonlinear fractional differential equation with a Gerasimov-Caputo operator of variable order and variable coefficients. The importance of developing efficient algorithms for solving direct problems of the radon volume activity model is due to their use in solving corresponding inverse problems based on radon monitoring data in order to solve practical problems of identifying certain parameters of the geological environment. Based on data on the average execution time of the test problem, asymptotic estimates of the complexity of sequential and proposed parallel algorithms are given. It is shown that the use of hybrid parallel CPU-GPU algorithms provides a performance gain of up to 17 times and can give a significant advantage in solving problems with large amounts of experimental data, due to the use of a GPU node. It is also shown that asymptotically exact complexity estimates are: for memory, for all hybrid algorithms of order Θ(????2); for the hybrid implementation of a non-local explicit scheme of order Θ(????); for the hybrid implementation of a non-local implicit scheme of order Θ(????2).
References
Nikolopoulos D., Cantzos D., Alam A., Dimopoulos S., Petraki E. Electromagnetic and Radon Earthquake Precursors // Geosciences – 2024. – Vol. 14. – No. 10. – 271. – doi: http://dx.doi.org/10.3390/geosciences14100271
Barberio M.D., Gori F., Barbieri M., Billi A., Devoti R., Doglioni C., Petitta M., Riguzzi F., Rusi S. Diurnal and Semidiurnal Cyclicity of Radon (222Rn) in Groundwater, Giardino Spring, Central Apennines, Italy // Water – 2018. – Vol. 10. – No. 9. – 1276. – doi: http://dx.doi.org/10.3390/w10091276
Tverdyi D.A., Makarov E.O., Parovik R.I. Hereditary Mathematical Model of the Dynamics of Radon Accumulation in the Accumulation Chamber // Mathematics – 2023. – Vol. 11. – No. 4. – 850. – doi: http://dx.doi.org/10.3390/math11040850
Фирстов П.П., Макаров Е.О. Динамика подпочвенного радона на Камчатке и сильные землетрясения: монография. – Петропавловск-Камчатский: КамГУ им. Витуса Беринга, 2018. – 148 с.
Huang P., Lv W., Huang R., Luo Q., Yang Y. Earthquake precursors: A review of key factors influencing radon concentration // Journal of Environmental Radioactivity – 2024. – Vol. 271. – 107310. – doi: http://dx.doi.org/10.1016/j.jenvrad.2023.107310
Uchaikin V.V. Fractional Derivatives for Physicists and Engineers. – Berlin, Heidelberg: Springer, 2013. – Vol. 1. – 373 p.
Parovik R.I., Shevtsov B.M. Radon transfer processes in fractional structure medium // Mathematical Models and Computer Simulations – 2010. – Vol. 2. – P. 180-185. – doi: http://dx.doi.org/10.1134/S2070048210020055
Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. – 1st ed. – Boston: Elsevier Science, 2006. – 540 p.
Cai M., Li C. Theory and Numerical Approximations of Fractional Integrals and Derivatives. – Philadelphia: SIAM, 2020. – 317 p.
Нахушев А.М. Дробное исчисление и его применение. – М.: Физматлит, 2003. – 272 с.
Caputo M., Fabrizio M. On the notion of fractional derivative and applications to the hysteresis phenomena // Meccanica – 2017. – Vol. 52. – P. 3043-3052. – doi: http://dx.doi.org/10.1007/s11012-017-0652-y
Novozhenova O.G. Life And Science of Alexey Gerasimov, One of the Pioneers of Fractional Calculus in Soviet Union // Fractional Calculus and Applied Analysis – 2017. – Vol. 20. – P. 790-809. – doi: http://dx.doi.org/10.1515/fca-2017-0040
Patnaik S., Hollkamp J.P., Semperlotti F. Applications of variable-order fractional operators: A review // Proceedings of the Royal Society A – 2020. – Vol. 476. – No. 2234. – 20190498. – doi: http://dx.doi.org/10.1098/rspa.2019.0498
Chicco D., Warrens M.J., Jurman G. The coefficient of determination R-squared is more informative than SMAPE, MAE, MAPE, MSE and RMSE in regression analysis evaluation // Peerj Comp. Sci. – 2021. – Vol. 7. – 623. – doi: http://dx.doi.org/10.7717/peerj-cs.623
Mueller J.L., Siltanen S. Linear and Nonlinear Inverse Problems with Practical Applications. – Philadelphia: Society for Industrial and Applied Mathematics, 2012. – 351 p.
Твёрдый Д.А. Восстановление порядка дробной производной в задаче математического моделирования накопления радона в избыточном объеме накопительной камеры по данным Петропавловск-Камчатского геодинамического полигона // Известия Кабардино-Балкарского научного центра РАН – 2023. – Том 116. – Выпуск 6. – С. 83-94. – doi: http://dx.doi.org/10.35330/1991-6639-2023-6-116-83-94
Tverdyi D.A., Parovik R.I., Makarov E.O. Estimation of radon flux density changes in temporal vicinity of the Shipunskoe earthquake with MW = 7.0, 17 August 2024 with the use of the hereditary mathematical model // Geosciences – 2025. – Vol. 15. – No. 1. – 30. – doi: http://dx.doi.org/10.3390/geosciences15010030
Твёрдый Д.А., Паровик Р.И. О задаче оптимизации для определения вида функциональной зависимости переменного порядка дробной производной типа Герасимова-Капуто // Вестник КРАУНЦ. Физико-математические науки – 2024. – Том 47. – Выпуск 2. – С. 35-57. – doi: http://dx.doi.org/10.26117/2079-6641-2024-47-2-35-57
Bogaenko V.A., Bulavatskiy V.M., Kryvonos I.G. On Mathematical modeling of Fractional-Differential Dynamics of Flushing Process for Saline Soils with Parallel Algorithms Usage // Journal of Automation and Information Sciences – 2016. – Vol. 40. – No. 10. – P. 1-12. – doi: http://dx.doi.org/10.1615/JAutomatInfScien.v48.i10.10
Bohaienko V.O. Parallel finite-difference algorithms for three-dimensional space-fractional diffusion equation with phi–Caputo derivatives // Computational and Applied Mathematics – 2020. – Vol. 39. – No. 163. – doi: http://dx.doi.org/10.1007/s40314-020-01191-x
Tverdyi D.A., Parovik R.I. 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 – 2023. – Vol. 11. – No. 15. – 3358. – doi: http://dx.doi.org/10.3390/math11153358
Kenneth R. Pointers on C. – 1st ed. – London: Pearson, 1997. – 640 p.
Corman T.H., Leiserson C.E., Rivet R.L., Stein C. Introduction to Algorithms. – 3rd ed. – Cambridge: The MIT Press, 2009. – 1292 p.
Sanders J., Kandrot E. CUDA by Example: An Introduction to General-Purpose GPU Programming. – London: Addison-Wesley Professional, 2010. – 311 p.
Cheng J., Grossman M., McKercher T. Professional Cuda C Programming. – 1st ed. – New-York: Wrox Pr Inc, 2014. – 497 p.
Supinski B., Klemm M. OpenMP Application Programming Interface Specification Version 5.2. – North Charleston: Independently published, 2021. – 669 p.
Storer J.A. An Introduction to Data Structures and Algorithms. – Boston: Birkh¨auser Boston, 2012. – 599 p.
Курносов М.Г. Введение в структуры и алгоритмы обработки данных: учебное пособие. – Новосибирск: Автограф, 2015. – 178 с.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 D.A. Tverdyi
This work is licensed under a Creative Commons Attribution 4.0 International License.
