I.N. Sinitsyn – Dr.Sc.(Eng.), Professor, Main Research Scientist, 

FRC «Computer Science and Control» of RAS (Moscow)

E-mail: sinitsin@dol.ru

D.V. Zhukov – Head of Department, 

FRC «Computer Science and Control» of RAS (Moscow)

E-mail: dzhukov@ipiran.ru

E.R. Korepanov – Ph.D.(Eng.), Leading Research Scientist, 

FRC «Computer Science and Control» of RAS (Moscow)

E-mail: ekorepanov@ipiran.ru

T.D. Konashenkova – Leading Programmer, 

FRC «Computer Science and Control» of RAS (Moscow) E-mail: tkonzshenkova@ipiran.ru

Survey concerning methods and software tools for direct numerical interpolation methods for analysis and analytical modeling in linear, linear with parametric noises and nonlinear stochastic systems with high availability (StSHA) is given. The article proceeds the thematic cycle dedicated to analytical modeling of linear stationary and nonstationary StSHA based on wavelet and wavelet canonical expansions. Integral of Haar wavelets are introduced. The Haar wavelet interpolational methods of characteristic function and probability density in scalar nonlinear StSHA from stochastic systems and Haar wavelet theory are developed. Elements of linear, linear with parametric noises and nonlinear StSHA based on the one dimensional characteristic function and density are given. In Section 1 main definitions are presented. Basic algorithms of interpolational analytical modeling of one-dimensional characteristic function and probability density are given in Section 2. Results of computer experiments for 3 typical test examples are compared with exact solutions in Section 3. Computer experiments for comparison with interpolation methods based on Kotelnikov V.A. confirm the effectiveness of direct numerical wavelet methods and wavelet software tools. Some generalizations concerning adduction of high dimensional StSHA to the one dimensional StSHA are presented. Special attention is paid to analytical modeling problems for integrated support systems: mass organizational-technical-economical systems (MOTES). So 2 types of interpolational analytical modeling for StSHA Gaussian and nongaussian methods are developed: (1) numerical methods based on direct numerical solution of Pugachev equation for one-dimensional characteristic function an probability density; (2) numerical wavelet direct interpolation methods for analytical modeling based wavelet canonical expansions. Special complex of computer experiments was realized for the following industrial applications: industrial controllers with discontinuous elements and vibroproof and damping devices. The developed methods was implemented for the stochastic modeling and parametric optimal design of MOTES on user and supplier side. Wide and narrow band stochastic elements were considered. Various terminal and nonterminal criteria were used for estimation of critical level of resources in MOTES in presence of internal and external noises and stochastic factors. Problems of express modeling in MOTERS at nonlinear nongaussian transitions are discussed.

1. Pugachev V.S., Sinitsyn I.N. Teoriya stokhasticheskikh sistem. M.: Logos. 2000; 2004. 1000 s. (In Russian).
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3. Kazakov I.E., Mal’chikov S.V. Analiz stokhasticheskikh sistem v prostranstve sostoyaniy. M.: Nauka. 1983. 348 s. (In Russian).
4. Kotel’nikov V.A. Teoriya potentsial’noy pomekhoustoychivosti. M.: Gosenergoizdat. 1956. 412 s. (In Russian).
5. Sinitsyn I.N. Metod interpolyatsionnogo analiticheskogo modelirovaniya odnomernykh raspredeleniy v stokhasticheskikh sistemakh. Informatika i ee primeneniya. 2018. T. 12. № 1. S. 55−61. (In Russian).
6. Sinitsyn I.N. Interpolyatsionnoe analiticheskoe modelirovanie raspredeleniy v slozhnykh stokhasticheskikh sistemakh. Informatika i ee primenenie. 2019. T. 13. № 1. S. 2−8. (In Russian).
7. Pugachev V.S. and Sinitsyn I.N. Lectures on Functional Analysis and Applications. 2nd edition. Preprint. M.: TORUS PRESS. 2017.
8. Sinitsyn I.N., Sergeev I.V., Korepanov E.R., Konashenkova T.D. Instrumental’noe programmnoe obespechenie analiza i sinteza stokhasticheskikh sistem vysokoy dostupnosti (IV). Sistemy vysokoy dostupnosti. 2017. T. 13. № 3. S. 55−59. (In Russian).
9. Sinitsyn I.N., Sinitsyn V.I., Korepanov E.R., Belousov V.V., Konashenkova T.D., Sergeev I.V., Semendyaev N.N., Basilashvili D.A. Instrumental’noe programmnoe obespechenie analiza i sinteza sistem vysokoy dostupnosti (V). Sistemy vysokoy dostupnosti. 2018. T. 14. № 1. S. 59−70. (In Russian).
10. Sinitsyn I.N., Sergeev I.V., Korepanov E.R., Konashenkova T.D. Instrumental’noe programmnoe obespechenie analiza i sinteza sistem vysokoy dostupnosti (VI). Sistemy vysokoy dostupnosti. 2018. T. 14. № 2. S. 40−56. (In Russian).
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12. Gagnon L. and Lina J.M. Symmetric Daubechies' wavelets and numerical solutions of NLS2 equations. J. Phys. A: Math. Gen. 1994. № 27. P. 8207−8230.
13. Sinitsyn I.N., Korepanov E.R., Konashenkova T.D. Instrumental’noe programmnoe obespechenie analiza i sinteza sistem vysokoy dostupnosti (VII). Sistemy vysokoy dostupnosti. 2019. T. 15. № 1 (v pechati). (In Russian).
14. Sinitsyn I.N., Zhukov D.V., Korepanov E.R., Konashenkova T.D. Razvitie pryamykh metodov interpolyatsionnogo analiticheskogo modelirovaniya raspredeleniy v stokhasticheskikh sistemakh. Materialy XX Mezhdunar. nauchnoy konf. «Sistemy komp’yuternoy matematiki i ikh prilozheniya» (SKMP-2019). 17−19 maya 2019. Smolensk: izd-vo SmolGU. 2019 (v pechati). (In Russian).
15. Lepik U. Numeral solution of evolution equations by the Haar wavelet methods. Appl. Math. Comput. 2007. № 185. P. 695−704.
16. Lepik U. Application of the Haar wavelet transform to solving integral and differential equations. Proc. Estonian Acad. Sci. Phys. Math. 2007. № 56. P. 28−46.
17. Sinitsyn I.N., Sergeev I.V., Korepanov E.R., Konashenkova T.D. Stokhasticheskie kanonicheskie veyvlet razlozheniya v zadachakh modelirovaniya vibroudaronadezhnosti komp’yuternogo oborudovaniya. XVIII Mezhdunar. nauchnaya konf. «Sistemy komp’yuternoy matematiki i ikh prilozheniya» (SKMP – 2017). 19−21 maya 2017. Smolensk: izd-vo SmolGu. S. 123−124. (In Russian).
18. Sinitsyn I.N., Shalamov A.S. Lektsii po teorii sistem integrirovannoy logisticheskoy podderzhki. M.: TORUC PRESS. 2012 (1-e izd.). 2019 (2-e izd.). (In Russian).