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Mathematical notes of NEFU, 2021, Volume 28, Issue 1, Pages 78–92
DOI: https://doi.org/10.25587/SVFU.2021.81.41.007 (Mi svfu312) 		 
Mathematical modeling

Numerical methods for identifying the diffusion coefficient in a nonlinear elliptic equation

J. Huangabc, A. V. Grigorevd, D. Kh. Ivanovde

a School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China
b Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan, 411105, China
c Key Laboratory of Intelligent Computing Information Processing of Ministry of Education, Xiangtan, 411105, China
d M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677000, Russia
e Yakutsk Branch of the Regional Scientificand Educational Mathematical Center "Far Eastern Center of Mathematical Research", 48 Kulakovsky Street, Yakutsk 677000, Russia
Full-text PDF (684 kB)
DOI: https://doi.org/10.25587/SVFU.2021.81.41.007
Abstract: Two different approaches for solving a nonlinear coefficient inverse problem are investigated in this paper. As a classical approach, we use the finite element method to discretize the direct and inverse problems and solve the inverse problem by the conjugate gradient method. Meanwhile, we also apply the neural network approach to recover the coefficient of the inverse problem, which is to map measurements at some fixed points and the unknown coefficient. According to the results of applying the two approaches, our methods are shown to solve the nonlinear coefficient inverse problem efficiently, even with perturbed data.
Keywords: inverse problem, neural network, nonlinear elliptic equation, optimization, finite element method.
Funding agency Grant number
National Natural Science Foundation of China 11901497
Natural Science Foundation of Hunan Province 2019JJ50607
China Postdoctoral Science Foundation BX20180266
Russian Foundation for Basic Research 21–51–54001
Ministry of Education and Science of the Russian Federation 14.Y26.31.001
The work of J. Huang was supported by the National Natural Science Foundation of China (Grant №. 11901497), in part by the Natural Science Foundation of Hunan Province (Grant №. 2019JJ50607) and in part by the China Postdoctoral Science Foundation Funded Project (Grant №. BX20180266); The work of A. Grigorev was supported by RFBR (Grant 21-51-54001). The work of D. Ivanov was supported by the Mega-Grant of the Russian Federation Government 14.Y26.31.0013. The work of Y. Huang was supported by the National Natural Science Foundation of China (Grant №. 11971410) and in part by the Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2018WK4006).
Received: 04.04.2019
Bibliographic databases:
Document Type: Article
UDC: 004.85+519.63
Language: English
Citation: J. Huang, A. V. Grigorev, D. Kh. Ivanov, “Numerical methods for identifying the diffusion coefficient in a nonlinear elliptic equation”, Mathematical notes of NEFU, 28:1 (2021), 78–92
Citation in format AMSBIB
\Bibitem{HuaGriIva21}
\by J.~Huang, A.~V.~Grigorev, D.~Kh.~Ivanov
\paper Numerical methods for identifying the diffusion coefficient in a nonlinear elliptic equation
\jour Mathematical notes of NEFU
\yr 2021
\vol 28
\issue 1
\pages 78--92
\mathnet{http://mi.mathnet.ru/svfu312}
\crossref{https://doi.org/10.25587/SVFU.2021.81.41.007}
\elib{https://elibrary.ru/item.asp?id=45658542}
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