One dynamical input reconstruction problem: tuning of solving algorithm via numerical experiments

Lidiya Melnikova; Valeriy Rozenberg; Lidiya Melnikova; Valeriy Rozenberg

doi:10.3934/math.2019.3.699

AIMS Mathematics

2019, Volume 4, Issue 3: 699-713. doi: 10.3934/math.2019.3.699

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Research article Topical Sections

One dynamical input reconstruction problem: tuning of solving algorithm via numerical experiments

Lidiya Melnikova ¹,
Valeriy Rozenberg ^{1,2
,
,}

1.
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S. Kovalevskoi str. 16, Yekaterinburg, 620990, Russia
2.
Ural Federal University, Mira str. 19, Yekaterinburg, 620002, Russia

Received: 31 January 2019 Accepted: 31 May 2019 Published: 19 June 2019
MSC : 49K15, 60H10, 93E12

The input reconstruction problem for a stochastic differential equation is investigated by means of the approach of the theory of dynamic inversion. The statement when the simultaneous reconstruction of disturbances in both the deterministic and stochastic terms of the equation is performed from the discrete information on several realizations of the stochastic process is considered. A finite-step software-oriented solving algorithm based on the method of auxiliary feedback controlled models is designed; an estimate for its convergence rate with respect to the number of measurable realizations is obtained. An empirical procedure for the automatic tuning of algorithm's parameters in order to get best approximation results for a specific dynamical system is proposed. To optimize this time-taking process, the parallelization of calculations is applied. A model example illustrating the method proposed is given.
- stochastic differential equation,
- dynamical input reconstruction,
- controlled model,
- tuning procedure,
- parallelization of calculations
Citation: Lidiya Melnikova, Valeriy Rozenberg. One dynamical input reconstruction problem: tuning of solving algorithm via numerical experiments[J]. AIMS Mathematics, 2019, 4(3): 699-713. doi: 10.3934/math.2019.3.699

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Abstract

The input reconstruction problem for a stochastic differential equation is investigated by means of the approach of the theory of dynamic inversion. The statement when the simultaneous reconstruction of disturbances in both the deterministic and stochastic terms of the equation is performed from the discrete information on several realizations of the stochastic process is considered. A finite-step software-oriented solving algorithm based on the method of auxiliary feedback controlled models is designed; an estimate for its convergence rate with respect to the number of measurable realizations is obtained. An empirical procedure for the automatic tuning of algorithm's parameters in order to get best approximation results for a specific dynamical system is proposed. To optimize this time-taking process, the parallelization of calculations is applied. A model example illustrating the method proposed is given.

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