matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/SVFU.2021.81.41.007matmess-188Research ArticleМАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕNUMERICAL METHODS FOR IDENTIFYING THE DIFFUSION COEFFICIENT IN A NONLINEAR ELLIPTIC EQUATIONNUMERICAL METHODS FOR IDENTIFYING THE DIFFUSION COEFFICIENT IN A NONLINEAR ELLIPTIC EQUATIONHuangJ.HuangJ.

Jian Huang

School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China;

Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan, 411105, China;

Key Laboratory of Intelligent Computing Information Processing of Ministry of Education, Xiangtan, 411105, China

Jian Huang,

School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China;

Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan, 411105, China;

Key Laboratory of Intelligent Computing Information Processing of Ministry of Education, Xiangtan, 411105, China

huangjian213@xtu.edu.cnGrigorevA. V.GrigorevA. V.

Aleksandr V. Grigorev, M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677000, Russia 

Aleksandr V. Grigorev, M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677000, Russia 

re5itsme@gmail.comIvanovD. Kh.IvanovD. Kh.

Dulus Kh. Ivanov, M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677000, Russia;

Yakutsk Branch of the Regional Scientific and Educational Mathematical Center "Far Eastern Center of Mathematical Research", 48 Kulakovsky Street, Yakutsk 677000, Russia 

Dulus Kh. Ivanov, M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677000, Russia;

Yakutsk Branch of the Regional Scientific and Educational Mathematical Center "Far Eastern Center of Mathematical Research", 48 Kulakovsky Street, Yakutsk 677000, Russia 

i.am.djoos@gmail.comSchool of Mathematics and Computational Science, Xiangtan University; Hunan Key Laboratory for Computation and Simulation in Science and Engineering; Key Laboratory of Intelligent Computing Information Processing of Ministry of EducationКитайSchool of Mathematics and Computational Science; Hunan Key Laboratory for Computation and Simulation in Science and Engineering; Key Laboratory of Intelligent Computing Information Processing of Ministry of EducationChinaM. K. Ammosov North-Eastern Federal University, Institute of Mathematics and InformaticsРоссияM. K. Ammosov North-Eastern Federal University, Institute of Mathematics and InformaticsRussian FederationM. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics; Yakutsk Branch of the Regional Scientific and Educational Mathematical Center "Far Eastern Center of Mathematical Research"РоссияM. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics; Yakutsk Branch of the Regional Scientific and Educational Mathematical Center "Far Eastern Center of Mathematical Research"Russian Federation2021260320262817892Copyright © Huang J., Grigorev A.V., Ivanov D.K., 20262026Huang J., Grigorev A.V., Ivanov D.K.Huang J., Grigorev A.V., Ivanov D.K.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/188

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.

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.

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The authors declare that there are no conflicts of interest present.