DOI: https://doi.org/10.15587/1729-4061.2019.171632

### Modeling the parallelism of empirical models of optimal complexity using a Petri net

Mikhail Gorbiychuk, Olga Bila, Taras Humeniuk

#### Abstract

Many physical processes and phenomena in view of their complexity cannot be described analytically. In these cases, empirical modeling is applied. In this research, the method based on the genetic approach is used to construct empirical models of optimal complexity that have the form of a polynomial of assigned power. Implementation of the developed method requires a multiple solution of the system of linear algebraic equations. Solution of the system of linear algebraic equations is found by reducing the corresponding matrix to the upper diagonal form with unities on the main diagonal. Analysis of the algorithm showed that the procedure of reducing the matrix to the upper diagonal form has internal parallelism. Based on the created model of the computational process in the form of a Petri net, the strategy of construction of the parallel algorithm for solving the system of linear algebraic equations was developed. The essence of the strategy is that computations are performed on some parallel processors. One of them was assigned coordinating functions, and it was named master. Other processors – slaves are subordinated to the master. Division of computation volume is such that the number of rows of the matrix, which master operates is at least by unity more than the corresponding number of rows allocated to the servant. The effectiveness of the parallel algorithm for the proposed strategy was evaluated based of the criterion of the total number of arithmetic operations. The proposed strategy is an integral part of the process of synthesis of the empirical model of optimal complexity based on the genetic algorithms. Distribution of computational load between processors working in parallel (master and slaves) ensures the acceleration of the computational process by five times or more.

#### Keywords

empirical model; genetic algorithm; parallelism; Petri net; the number of operations

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#### References

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Gorbiychuk, M. I., Medvedchuk, V. M., Lazoriv, A. N. (2016). Analysis of Parallel Algorithm of Empirical Models Synthesis on Principles of Genetic Algorithms. Journal of Automation and Information Sciences, 48 (2), 54–73. doi: https://doi.org/10.1615/jautomatinfscien.v48.i2.60

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#### GOST Style Citations

Gorbiychuk M. I., Schupak I. V., Oskolip T. Metod sinteza empiricheskih modeley s uchetom pogreshnostey izmereniy // Metody i pribory kontrolya kachestva. 2011. Issue 2 (27). P. 67–76.

Gorbiychuk M. I., Shufnarovych M. A. The Method of Constructing Mathematical Models of Complex Processes on the Basis of Genetic Algorithms // Iskusstvenniy intellekt. 2010. Issue 4. P. 50–57.

Metod syntezu empirychnykh modelei na zasadakh henetychnykh alhorytmiv / Gorbiychuk M. I., Kohutiak M. I., Vasylenko O. B., Shchupak I. V. // Rozvidka ta rozrobka naftovykh i hazovykh rodovyshch. 2009. Issue 4. P. 72–79.

Stepashko V. S., Bulgakova A. S. Obobschennyy iteratsionnyy algoritm metoda gruppovogo ucheta argumentov // Upravlyayuschie sistemy i mashiny. 2013. Issue 2. P. 5–17.

Gupta S., Bhardwaj S., Bhatia P. K. A reminiscent study of nature inspired computation // International Journal of Advances in Engineering & Technology. 2011. Vol. 1, Issue 2. P. 117–125.

Voevodin V., Antonov A., Popova N. Studying the Structure of Parallel Algorithms as a Key Element of High-Performance Computing Education // Lecture Notes in Computer Science. 2019. P. 199–210. doi: https://doi.org/10.1007/978-3-030-10549-5_16

Ortega J. M. Introduction to parallel and vector solution of linear systems. Springer, 1988. doi: https://doi.org/10.1007/978-1-4899-2112-3

Dvukhglavov D. E., Kulynych V. E. Development of software solution for building route of a orders group delivery in presence of time constraints // Bulletin of National Technical University "KhPI". Series: System Analysis, Control and Information Technologies. 2017. Issue 55. P. 64–71. doi: https://doi.org/10.20998/2079-0023.2017.55.11

Parallel'nye algoritmy resheniya zadach vychislitel'noy matematiki / Himich A. N., Molchanov I. N., Popov A. V., Chistyakova T. V., Yakovlev M. F. Kyiv: Naukova dumka, 2008. 248 p.

Khimich A. N., Popov A. V., Polyankoa V. V. Algorithms of parallel computations for linear algebra problems with irregularly structured matrices // Cybernetics and Systems Analysis. 2011. Vol. 47, Issue 6. P. 973–985. doi: https://doi.org/10.1007/s10559-011-9377-4

Rutkovskaya D., Pilin'skiy M., Rutkovskiy L. Neyronnye seti, geneticheskie algoritmy i nechetkie sistemy. Moscow: Goryachaya liniya-Telekom, 2004. 452 p.

Bogatyrev M. Yu. Invarianty i simmetrii v geneticheskih algoritmah. URL: http://www.raai.org/conference/cai-08/files/cai-08_paper_287.pdf

Gladkov L. A., Kureychik V. V., Kureychik V. M. Geneticheskie algoritmy. 2-e izd., ispr i dop. Moscow: FIZMATLIT, 2006. 320 p.

Gorbiychuk M. I., Medvedchuk V. M., Pashkovskyi B. V. The parallelism in the algorithm of the synthesis of models of optimal complexity based on the genetic algorithms // Eastern-European Journal of Enterprise Technologies. 2014. Vol. 4, Issue 2 (70). P. 42–48. doi: https://doi.org/10.15587/1729-4061.2014.26305

Gorbiychuk M. I., Shufnarovych M. A. Metod syntezu matematychnykh modelei kolyvnykh protsesiv z nekratnymy chastotamy // Naukovyi visnyk Ivano-Frankivskoho natsionalnoho tekhnichnoho universytetu nafty i hazu. 2010. Issue 1. P. 105–112.

Luszczek P. Parallel Programming in MATLAB // The International Journal of High Performance Computing Applications. 2009. Vol. 23, Issue 3. P. 277–283. doi: https://doi.org/10.1177/1094342009106194

Verzhbitskiy V. M. Osnovy chislennyh metodov: ucheb. Moscow: Vysshaya shkola, 2002. 840 p.

Paterson Dzh. Teoriya setey Petri i modelirovanie sistem. Moscow: Mir, 2000. 263 p.

Marahovskiy V. B., Rozenblyum L. Ya., Yakovlev A. V. Modelirovanie parallel'nyh protsessov. Seti Petri: kurs dlya sistemnyh arhitektorov, programmistov, sistemnyh analitikov, proektirovschikov slozhnyh sistem upravleniya. Sankt-Peterburg: Professional'naya literatura, 2014. 398 p.

Gorbiychuk M. I., Medvedchuk V. M., Lazoriv A. N. Analysis of Parallel Algorithm of Empirical Models Synthesis on Principles of Genetic Algorithms // Journal of Automation and Information Sciences. 2016. Vol. 48, Issue 2. P. 54–73. doi: https://doi.org/10.1615/jautomatinfscien.v48.i2.60

Korn G., Korn T. Spravochnik po matematike dlya nauchnyh rabotnikov i inzhenerov. Moscow: Nauka, 1978. 832 p.

Volkov E. A. Chislennye metody: ucheb. pos. 2-e izd., isp. Moscow: Nauka, 1987. 248 p.

Copyright (c) 2019 Mikhail Gorbiychuk, Olga Bila, Taras Humeniuk