Monte-Carlo for solving large linear systems of ordinary differential equations

Authors

  • Sergey M. Ermakov St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
  • Maxim G. Smilovitskiy St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2021.104

Abstract

Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

Keywords:

Monte-Carlo, ODE system, integral equation, queuing theory, optimal density, unbiased estimate, statistical modelling

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References

Литература

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References

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Published

2021-05-29

How to Cite

Ermakov, S. M., & Smilovitskiy, M. G. (2021). Monte-Carlo for solving large linear systems of ordinary differential equations. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(1), 37–48. https://doi.org/10.21638/spbu01.2021.104

Issue

Vol. 8 No. 1 (2021)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.