DEVELOPMENT OF MOMENT-SEMIINVARIANT SOFTWARE FOR STOCHASTIC ANALYSIS
Abstract
The article describes the methodological foundations and the results of comparative testing of scientific software for stochastic analysis of multidimensional nonlinear stochastic systems (StS) described by stochastic differential equations, developed on the basis of the modified moment-semi-invariant method (M2SM) and orthogonal factorization of the unknown one-dimensional density of the system state vector by the bi-orthogonal system of polynomials. We have obtained exact differential equations for the components of the expectation vector, the covariance matrix, and the higher initial moments of the state vector of the nonlinear StS, described by the Ito stochastic differential equation containing non-polynomial coefficients (continuous or having first-kind discontinuous points) and a vector random process with independent increments. Software “StS-Analysis-M2SM” contains the modules developed by the author, which allow you to partially automate actions to compile a closed-loop system of differential equations for parameters of the state vector of a nonlinear StS. Software “StS-Analysis-M2SM” is developed in the Python language and is compatible with new domestic processors and high-performance platforms. Verification of the performance of the software “StS-Analysis-M2SM“ was carried out using examples of non-linear StS that have exact solutions. The results were compared with the normal approximation method (NAM) and the ellipsoidal approximation method (EAM). M2SM gives a high accuracy (less than 2%) of determining the expectation function and the covariance matrix of the state vector of the StS, complying with the system of ordinary differential equations for the expectation function components, the covariance matrix and probabilistic higher initial moments included in the right-hand parts of the exact open equations for the expectation and covariance matrix. Software “StS-Analysis-M2SM” can be applied in scientific research; for training in higher education in the courses “Random Processes”, “Theory of Stochastic Differential Systems”, etc.; in the management of technical, organizational and economical, environmental and other systems.
References
[2] Sinitsyn I.N., Shalamov A.S. Lektsii po teorii sistem integrirovannoy logisticheskoy podderzhki [Lectures on the theory of systems of integrated logistics support]. Moscow: Torus Press, 2012. 624 p. (In Russ.)
[3] Sinitsyn I.N., Sinitsyn V.I. Lektsii po teorii normal’noi i ellipsoidal’noi approksimatsii raspredelenii v stokhasticheskikh sistemakh [Lectures on the Theory of Normal and Ellipsoidal Approximation of Distributions in the Stochastic Systems]. Moscow: Torus Press, 2013. 488 p. (In Russ.)
[4] Guo L., Cao S. Anti-disturbance control theory for Systems with multiple disturbances: a survey. ISA Transactions. 2014; 53(4):846-849. (In Eng.) DOI: 10.1016/j.isatra.2013.10.005
[5] Moon W., Wettlaufer J.S. A stochastic perturbation theory for non-autonomous systems. Journal of Mathematical Physics. 2013; 54(12):123303. (In Eng.) DOI: 10.1063/1.4848776
[6] Silvestrov S., Malyarenko A., Rancic M. Stochastic Processes and Applications. SPAS2017, Västerås and Stockholm, Sweden, October 4-6, 2017. Springer Proceedings in Mathematics & Statistics. Vol. 271. Springer International Publishing, 2018. 475 p. (In Eng.) DOI: 10.1007/978-3-030-02825-1
[7] Kozachenko Yu., Pogorilyak O., Rozora I., Tegza A. 3 – Simulation of Gaussian Stochastic Processes with Respect to Output Processes of the System. Simulation of Stochastic Processes with Given Accuracy and Reliability. 2016:105-168. (In Eng.) DOI: 10.1016/B978-1-78548-217-5.50003-9
[8] Liu Z., Wang W. Favard separation method for almost periodic stochastic differential equations. Journal of Differential Equations. 2016; 260(11):8109-8136. (In Eng.) DOI: 10.1016/j.jde.2016.02.019
[9] Riedel S., Scheutzow M. Rough differential equations with unbounded drift term. Journal of Differential Equations. 2017; 262(1):283-312. (In Eng.) DOI: 10.1016/j.jde.2016.02.021
[10] Shena J., Zhaob J., Lu K., Wang B. The Wong–Zakai approximations of invariant manifolds and foliations for stochastic evolution equations. Journal of Differential Equations. 2019; 266(8):4568-4623. (In Eng.) DOI: 10.1016/j.jde.2018.10.008
[11] Anton C. Error Expansion for a Symplectic Scheme for Stochastic Hamiltonian Systems. In: Kilgour D., Kunze H., Makarov R., Melnik R., Wang X. (eds) Recent Advances in Mathematical and Statistical Methods. AMMCS 2017. Springer Proceedings in Mathematics & Statistics. Springer, Cham, 2018; 259:567-577. (In Eng.) DOI: 10.1007/978-3-319-99719-3_51
[12] Averina T.A., Rybakov K.A. Two Methods for Analysis of Stochastic Systems with Poisson Component. Differential Equations and Control Processes. 2013; 3:85-116. Available at: https://elibrary.ru/item.asp?id=24896759 (accessed 21.09.2018). (In Russ.)
[13] Rybakov K.A. Statistical Methods of Analysis and Filtering for Continuous Stochastic Systems. Differential Equations and Control Processes. 2017; 4:1001-1005. Available at: https://elibrary.ru/item.asp?id=30777674 (accessed 21.09.2018). (In Russ.)
[14] Konashenkova T. D., Shin V.I. Approximate method of determination of moments of phase variables of multivariable stochastic systems. Autom. Remote Control. 1990; 51(1):35-42. (In Eng.)
[15] Andreeva E.V., Konashenkova T.D., Maisheva E.Yu., Ogneva OS, Petrova M.V., Shin V.I. Modified quasi-momentary and moment-semi-invariant methods for analyzing multidimensional stochastic systems and their software implementation. Systems and Means of Informatics. 1992; 2:160-171. (In Russ.)
[16] Sinitsyn I.N., Sinitsyn V.I., Korepanov E.R., Belousov V.V., Konashenkova T.D., Semendyaev N.N., Basilashvili D.A. Software tools for analysis and synthesis of stochastic systems of with high availability (I). Science Intensive Technologies. 2009; 10(10):4-52. Available at: https://elibrary.ru/item.asp?id=13070122 (accessed 21.09.2018). (In Russ.)
[17] Sinitsyn I.N., Sinitsyn V.I., Korepanov E.R., Belousov V.V., Konashenkova T.D., Semendyaev N.N., Basilashvili D.A. Software tools for analysis and synthesis of stochastic systems with high availability (II). Science Intensive Technologies. 2010; 11(5):4-45. Available at: https://elibrary.ru/item.asp?id=19135495 (accessed 21.09.2018). (In Russ.)
[18] Sinitsyn I.N., Sinitsyn V.I., Korepanov E.R., Belousov V.V., Konashenkova T.D., Semendyaev N.N., Basilashvili D.A. Software tools for analysis and synthesis of stochastic systems with high availability (III). Highly available systems. 2010; 6(4):23-47. Available at: https://elibrary.ru/item.asp?id=22831889 (accessed 21.09.2018). (In Russ.)
[19] Sinitsyn I.N., Sergeev I.V., Korepanov E.R., Konashenkova T.D. Software tools for analysis and synthesis of stochastic systems with high abailability (IV). Highly available systems. 2017; 13(3):55-69. Available at: https://elibrary.ru/item.asp?id=30554622 (accessed 21.09.2018). (In Russ.)
[20] Sinitsyn I.N., Sergeev I.V., Korepanov E.R., Konashenkova T.D. Stochastic canonical wavelet expansions in problems of simulation of vibro-shock reliability of computer equipment. Computer mathematics systems and their applications. 2017; 18:123-124. Available at: https://elibrary.ru/item.asp?id=30469426 (accessed 21.09.2018). (In Russ.)
[21] Sinitsyn I.N., Sergeev I.V., Korepanov E.R., Konashenkova T.D. Software tools for analysis and synthesis of stochastic systems with high abailability (V). Highly available systems. 2018; 14(1):59-70. Available at: https://elibrary.ru/item.asp?id=32795845 (accessed 21.09.2018). (In Russ.)
[22] Sinitsyn I.N., Sergeev I.V., Korepanov E.R., Konashenkova T.D. Software tools for analysis and synthesis of stochastic systems with high availability (VI). Highly available systems. 2018; 14(2):40-56. Available at: https://elibrary.ru/item.asp?id=35256128 (accessed 21.09.2018). (In Russ.)
[23] Sinitsyn I.N., Sergeev I.V., Korepanov E.R., Konashenkova T.D. express modeling of stochastic highly available systems based on wavelet canonical expansions. Computer mathematics systems and their applications. 2018; 19:213-220. Available at: https://elibrary.ru/item.asp?id=35177124 (accessed 21.09.2018). (In Russ.)
[24] Sinitsyn I.N. Method of interpolational analytical modeling of processes in stochastic systems. Informatics and Applications. 2018; 12(1):55-61. Available at: https://elibrary.ru/item.asp?id=32686788 (accessed 21.09.2018). (In Russ.)
[25] Belousov V.V. Experience in tools developing for organization-economical systems modeling based on the open source software. Highly available systems. 2018; 14(5):3-11. Available at: https://elibrary.ru/item.asp?id=36833182 (accessed 21.09.2018). (In Russ.)

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