An Analysis of the Computational Complexity and Efficiency of Various Algorithms for Solving a Nonlinear Model of Radon Volumetric Activity with a Fractional Derivative of a Variable Order

The article presents a study of the computational complexity and efficiency of various parallel algorithms that implement the numerical solution of the equation in the hereditary α(t)-model of radon volumetric activity (RVA) in a storage chamber. As a test example, a problem based on such a model is...

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Vydáno v:	Computation Ročník 13; číslo 11; s. 252
Hlavní autor:	Tverdyi, Dmitrii
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Basel MDPI AG 01.11.2025
Témata:	algorithm efficiency Algorithms Analysis asymptotic complexity estimates Cauchy problems Complexity Derivatives Differential equations Earthquake prediction Earthquakes Effectiveness Efficiency Estimates finite difference schemes Fractals Fractional calculus fractional derivatives of variable order Geological processes Geology Geophysics Germany Inverse problems Iterative methods Mathematical analysis Modelling New York nonlinear parallel computing Radon Russia Seismology United States New York United States Germany Russia
ISSN:	2079-3197, 2079-3197
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Abstract	The article presents a study of the computational complexity and efficiency of various parallel algorithms that implement the numerical solution of the equation in the hereditary α(t)-model of radon volumetric activity (RVA) in a storage chamber. As a test example, a problem based on such a model is solved, which is a Cauchy problem for a nonlinear fractional differential equation with a Gerasimov–Caputo derivative of a variable order and variable coefficients. Such equations arise in problems of modeling anomalous RVA variations. Anomalous RVA can be considered one of the short-term precursors to earthquakes as an indicator of geological processes. However, the mechanisms of such anomalies are still poorly understood, and direct observations are impossible. This determines the importance of such mathematical modeling tasks and, therefore, of effective algorithms for their solution. This subsequently allows us to move on to inverse problems based on RVA data, where it is important to choose the most suitable algorithm for solving the direct problem in terms of computational resource costs. An analysis and an evaluation of various algorithms are based on data on the average time taken to solve a test problem in a series of computational experiments. To analyze effectiveness, the acceleration, efficiency, and cost of algorithms are determined, and the efficiency of CPU thread loading is evaluated. The results show that parallel algorithms demonstrate a significant increase in calculation speed compared to sequential analogs; hybrid parallel CPU–GPU algorithms provide a significant performance advantage when solving computationally complex problems, and it is possible to determine the optimal number of CPU threads for calculations. For sequential and parallel algorithms implementing numerical solutions, asymptotic complexity estimates are given, showing that, for most of the proposed algorithm implementations, the complexity tends to be n2 in terms of both computation time and memory consumption.
AbstractList	The article presents a study of the computational complexity and efficiency of various parallel algorithms that implement the numerical solution of the equation in the hereditary α(t)-model of radon volumetric activity (RVA) in a storage chamber. As a test example, a problem based on such a model is solved, which is a Cauchy problem for a nonlinear fractional differential equation with a Gerasimov–Caputo derivative of a variable order and variable coefficients. Such equations arise in problems of modeling anomalous RVA variations. Anomalous RVA can be considered one of the short-term precursors to earthquakes as an indicator of geological processes. However, the mechanisms of such anomalies are still poorly understood, and direct observations are impossible. This determines the importance of such mathematical modeling tasks and, therefore, of effective algorithms for their solution. This subsequently allows us to move on to inverse problems based on RVA data, where it is important to choose the most suitable algorithm for solving the direct problem in terms of computational resource costs. An analysis and an evaluation of various algorithms are based on data on the average time taken to solve a test problem in a series of computational experiments. To analyze effectiveness, the acceleration, efficiency, and cost of algorithms are determined, and the efficiency of CPU thread loading is evaluated. The results show that parallel algorithms demonstrate a significant increase in calculation speed compared to sequential analogs; hybrid parallel CPU–GPU algorithms provide a significant performance advantage when solving computationally complex problems, and it is possible to determine the optimal number of CPU threads for calculations. For sequential and parallel algorithms implementing numerical solutions, asymptotic complexity estimates are given, showing that, for most of the proposed algorithm implementations, the complexity tends to be n2 in terms of both computation time and memory consumption. The article presents a study of the computational complexity and efficiency of various parallel algorithms that implement the numerical solution of the equation in the hereditary α(t)-model of radon volumetric activity (RVA) in a storage chamber. As a test example, a problem based on such a model is solved, which is a Cauchy problem for a nonlinear fractional differential equation with a Gerasimov–Caputo derivative of a variable order and variable coefficients. Such equations arise in problems of modeling anomalous RVA variations. Anomalous RVA can be considered one of the short-term precursors to earthquakes as an indicator of geological processes. However, the mechanisms of such anomalies are still poorly understood, and direct observations are impossible. This determines the importance of such mathematical modeling tasks and, therefore, of effective algorithms for their solution. This subsequently allows us to move on to inverse problems based on RVA data, where it is important to choose the most suitable algorithm for solving the direct problem in terms of computational resource costs. An analysis and an evaluation of various algorithms are based on data on the average time taken to solve a test problem in a series of computational experiments. To analyze effectiveness, the acceleration, efficiency, and cost of algorithms are determined, and the efficiency of CPU thread loading is evaluated. The results show that parallel algorithms demonstrate a significant increase in calculation speed compared to sequential analogs; hybrid parallel CPU–GPU algorithms provide a significant performance advantage when solving computationally complex problems, and it is possible to determine the optimal number of CPU threads for calculations. For sequential and parallel algorithms implementing numerical solutions, asymptotic complexity estimates are given, showing that, for most of the proposed algorithm implementations, the complexity tends to be n[sup.2] in terms of both computation time and memory consumption.
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Cites_doi	10.1137/1.9781611971200 10.1007/s40314-020-01191-x 10.1137/1.9781611972344 10.35330/1991-6639-2023-6-116-83-94 10.1098/rspa.2019.0498 10.20944/preprints202406.1939.v1 10.1007/BFb0067700 10.1007/s11012-017-0652-y 10.1007/978-3-031-90546-9_11 10.1134/S2070048210020055 10.1007/978-3-642-18101-6 10.1016/j.jenvrad.2013.04.014 10.1615/JAutomatInfScien.v48.i10.10 10.1190/1.1441293 10.1097/00004032-199705000-00012 10.1145/321812.321815 10.1016/j.jenvrad.2023.107310 10.1515/fca-2017-0040 10.3390/fractalfract6010023 10.3390/math6020016 10.3390/math11040850 10.3390/w10091276 10.1007/b97553 10.1007/978-3-642-33911-0 10.3390/geosciences15010030 10.1007/978-3-031-18150-4 10.3390/math11153358 10.1007/s40314-019-0878-5
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SubjectTerms	algorithm efficiency Algorithms Analysis asymptotic complexity estimates Cauchy problems Complexity Derivatives Differential equations Earthquake prediction Earthquakes Effectiveness Efficiency Estimates finite difference schemes Fractals Fractional calculus fractional derivatives of variable order Geological processes Geology Geophysics Germany Inverse problems Iterative methods Mathematical analysis Modelling New York nonlinear parallel computing Radon Russia Seismology United States
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Title	An Analysis of the Computational Complexity and Efficiency of Various Algorithms for Solving a Nonlinear Model of Radon Volumetric Activity with a Fractional Derivative of a Variable Order
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