Algorithmic stability and complexity of implicit adaptation of nonstationary thermal conductivity mesh model to heated substance
- Authors: Zhukov P.I.1, Fomin A.V.1, Glushchenko A.I.2
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Affiliations:
- STI NUST “MISIS”
- V.A. Trapeznikov Institute of Control Sciences of RAS
- Issue: No 101 (2023)
- Pages: 39-63
- Section: Control systems analysis and design
- URL: https://journals.rcsi.science/1819-2440/article/view/360589
- DOI: https://doi.org/10.25728/ubs.2023.101.3
- ID: 360589
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Abstract
This paper deals with the process of adaptation of a numerical model of nonstationary thermal conductivity implemented with the help of finite difference methods. The algorithmic stability has already been proved for the classical representation of these models in most applications and problems, but in this case we consider a problem related to the parametric adaptation of the equation of nonstationary heat conduction to the heated substance implemented by solving of the related variational problem. The basis of this approach implies replacement of thermophysical parameters of the equation in question by freely adjustable parameters and their adaptation ("model training") by a stochastic gradient method. Optimization of algorithmic equations that do not have an analytical form is associated with unstable initial conditions and "training" trajectories. To avoid falling into these regions we need to impose restrictions on the adjustable parameters. In this paper, such constraints are derived on the basis of proven stability conditions for the classical finite-difference model of non-stationary thermal conductivity. As a result of the numerical experiments, it is shown that the proposed constraints allow one to increase, on average, the number of stable initial conditions by 13%, as well as the number of experiments when stable trajectories are achieved - by 61%. In addition to this result, an analytical comparison of the growth orders of algorithmic complexity of the classical model and the modified one is also made. As a result of the calculations, it is found that both models have a growth order of O(n4), which is confirmed by numerical experiments.
About the authors
Petr Igorevich Zhukov
STI NUST “MISIS”
Author for correspondence.
Email: Zhukov.petr86@yandex.ru
Stary Oskol
Andrey Vyacheslavovich Fomin
STI NUST “MISIS”
Email: verner444@yandex.ru
Stary Oskol
Anton Igorevich Glushchenko
V.A. Trapeznikov Institute of Control Sciences of RAS
Email: aiglush@ipu.ru
Moscow
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