2021
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Browsing 2021 by Author "Ivashchenko, Olena Valerievna"
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Item Efficient Algorithms for Parallelizing Tridiagonal Systems of Equations(НМетАУ, Дніпро, 2021) Shvachych, Gennady Grygorovych; Vozna, Nataliіa; Ivashchenko, Olena Valerievna; Bilyi, Oleksandr; Moroz, DmytroENG: The article is devoted to the development of the maximal parallel forms of mathematical models with a tridiagonal structure. The example of solving the Dirichlet and Neumann problems by the method of straight lines and the sweep method for the heat equation illustrates the direct fundamental features of constructing parallel algorithms. It is noted that the study of the heat and mass transfer processes is run through their numerical modeling based on modern computer technology. It is shown that with the multiprocessor computing systems’ development, there disappear the problems of increasing their peak performance. On the other hand, building such systems, as a rule, requires standard network technologies, mass-produced processors, and free software. The noted circumstances aim at solving the so-called big problems. It should be borne in mind that the classical approach to solving the tridiagonal structure models based on multiprocessor computing systems is far more time-consuming compared to single-processor computing facilities. That is explained by the recurrence relations that make the basis of classical methods. Therefore, the proposed studies are relevant and aim at the distributed algorithms development for solving applied problems. The proposed research aims to construct the maximal parallel forms of mathematical models with a tridiagonal structure. The paper proposes the schemes to implement parallelization algorithms for applied problems and their mapping to parallel computing systems. Parallelization of tridiagonal mathematical models by the method of straight lines and the sweeping method allows designing absolutely stable algorithms with the maximum parallel form and, therefore, the minimum possible time for their implementation on parallel computing devices. It is noteworthy that in the proposed algorithms, the computational errors of the input data are separated from the round-off errors inherent in a PC. The proposed approach can be used in various branches of metallurgical, thermal physics, economics, and ecology problems in the metallurgical industry.Item Research of Computing Efficiency in Modular Multiprocessor Systems(НМетАУ, Дніпро, 2021) Ivashchenko, Valery Petrovych; Shvachych, Gennady Grygorovych; Ivashchenko, Olena ValerievnaENG: The article is devoted to the research of efficiency of a multiprocessor computing system in solving problems aimed at expanding the computing area. The purpose of the research is to further develop the approach associated with definition of methodology for effectiveness evaluation of the multiprocessor modular computing system and the impact on this indicator of the calculations deceleration. At the same time, the main attention is paid to the impact peculiarities on this indicator of network interface of multiprocessor system. We should note that the research of these problems is important and relevant. This is not only due to the fundamental limitation of the maximum possible performance of ordinary serial computers, but also to the almost constant existence of computational problems which solution requires additional capabilities of existing computer facilities. The working methods of analyzing the multiprocessor systems efficiency do not allow to determine the optimal number of its nodes for solving a certain class of problems taking into account the calculations deceleration. At the same time, to solution of such problems is short of proper research development on the analysis of the network interface impact on the efficiency of modular multiprocessor computer systems. In addition, to estimate the efficiency of a computing multiprocessor system, the basic analytical relationships are usually not provided through the parameters of the system being studied. The basic regularities concerning the time of solving the problem are revealed, depending on the change in the multiprocessor system calculations area. The research is aimed at determining the deceleration factor associated with the increase of the computing area of a multiprocessor system when compared with the computer version with an unlimited computing area. The analytical ratios are derived for determining the calculations deceleration coefficient. A stage of simulation for calculations of the deceleration factor was carried out to determine the regularities of its change, depending on the application of a particular computing platform. The revealed tendencies of such a change point to the need to reconcile the components of the network interface and computing capabilities of the chosen computing platform. The derived analytical relations were aimed at determining the optimal number of nodes of a multiprocessor system which allow the minimum delay of calculations.