Abstract
Results obtained by the authors in solving inverse coefficient problems are overviewed. The inverse problem under consideration is to determine a temperature-dependent thermal conductivity coefficient from experimental observations of the temperature field in the studied substance and (or) the heat flux on the surface of the object. The study is based on the Dirichlet boundary value problem for the nonstationary heat equation stated in the general \(n\)-dimensional formulation. For this general case, an analytical expression for the cost functional gradient is obtained. The features of solving the inverse problem and the difficulties encountered in the solution process are discussed.
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REFERENCES
Yu. G. Evtushenko, Optimization and Fast Automatic Differentiation (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 2013) [in Russian].
Yu. G. Evtushenko and V. I. Zubov, “Generalized fast automatic differentiation technique,” Comput. Math. Math. Phys. 56 (11), 1819–1833 (2016).
A. F. Albu, V. I. Zubov, and V. A. Inyakin, “Optimal control of the process of melting and solidification of a substance,” Comput. Math. Math. Phys. 44 (8), 1291–1305 (2004).
A. V. Albu, A. F. Albu, and V. I. Zubov, “Functional gradient evaluation in the optimal control of a complex dynamical system,” Comput. Math. Math. Phys. 51 (5), 762–780 (2011).
A. V. Albu, A. F. Albu, and V. I. Zubov, “Control of substance solidification in a complex-geometry mold,” Comput. Math. Math. Phys. 52 (12), 1612–1623 (2012).
A. F. Albu and V. I. Zubov, “On the influence of setup parameters on the control of solidification in metal casting,” Comput. Math. Math. Phys. 53 (2), 170–179 (2013).
A. F. Albu and V. I. Zubov, “Investigation of the optimal control of metal solidification for a complex-geometry object in a new formulation,” Comput. Math. Math. Phys. 54 (12), 1804–1816 (2014).
V. I. Zubov, “Application of fast automatic differentiation for solving the inverse coefficient problem for the heat equation,” Comput. Math. Math. Phys. 56 (10), 1743–1757 (2016).
A. F. Albu and V. I. Zubov, “Identification of the thermal conductivity coefficient using a given surface heat flux,” Comput. Math. Math. Phys. 58 (12), 2031–2042 (2018).
A. F. Albu, Y. G. Evtushenko, and V. I. Zubov, “Identification of Discontinuous Thermal Conductivity Coefficient Using Fast Automatic Differentiation,” in Learning and Intelligent Optimization LION 2017, Ed. by R. Battiti, D. Kvasov, and Y. Sergeyev, Lecture Notes in Computer Science (Springer, Cham, 2017), Vol. 10556, pp. 295–300.
V. I. Zubov and A. F. Albu, “The FAD-methodology and recovery of the thermal conductivity coefficient in two dimension case,” Proceedings of the 8th International Conference on Optimization Methods and Applications ‘Optimization and Applications’ (2017), pp. 39–44.
A. F. Albu and V. I. Zubov, “Identification of thermal conductivity coefficient using a given temperature field,” Comput. Math. Math. Phys. 58 (10), 1585–1599 (2018).
A. Albu and V. Zubov, “Identification of the thermal conductivity coefficient in two dimension case,” Optim. Lett. 13 (8), 1727–1743 (2019).
A. Albu and V. Zubov, “On the stability of the algorithm of identification of the thermal conductivity coefficient,” in Optimization and Applications: OPTIMA 2018, Ed. by Y. Evtushenko, M. Jaćimović, M. Khachay, Y. Kochetov, V. Malkova, and M. Posypkin, Communications in Computer and Information Science (Springer, Cham, 2019), Vol. 974.
A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1977; Marcel Dekker, New York, 2001).
J. Douglas and H. H. Rachford, “On the numerical solution of heat conduction problems in two and three space variables,” Trans. Am. Math. Soc. 82, 421–439 (1956).
A. A. Samarskii and P. N. Vabishchevich, Computational Heat Transfer (Wiley, New York, 1996; Editorial URSS, Moscow, 2003).
Ch. Gao and Y. Wang, “A general formulation of Peaceman and Rachford ADI method for the N-dimensional heat diffusion equation,” Int. Commun. Heat Mass Transfer 23 (6), 845–854 (1996).
A. V. Albu and V. I. Zubov, “Choosing a cost functional and a difference scheme in the optimal control of metal solidification,” Comput. Math. Math. Phys. 51 (1), 24–38 (2011).
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This work was supported in part by the Russian Science Foundation, project no. 20-11-20081.
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Translated by I. Ruzanova
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Albu, A.F., Evtushenko, Y.G. & Zubov, V.I. Application of the Fast Automatic Differentiation Technique for Solving Inverse Coefficient Problems. Comput. Math. and Math. Phys. 60, 15–25 (2020). https://doi.org/10.1134/S0965542520010042
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DOI: https://doi.org/10.1134/S0965542520010042