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

  • Huang Jian, huangjian213@xtu.edu.cn 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
  • Grigorev Aleksandr V., re5itsme@gmail.com M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677000, Russia
  • Ivanov Dulus Kh., i.am.djoos@gmail.com 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
Keywords: inverse problem, neural network, nonlinear elliptic equation, optimization, finite element method

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.

References


[1]
V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York (2006).

[2]
A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, de Gruyter (2008) (Inverse Ill-Posed Probl. Ser.; vol. 52).

[3]
L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York (2012).

[4]
M. Benning and M. Burger, “Modern regularization methods for inverse problems,” Acta Numerica, 27, 1–111 (2018).

[5]
H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Acad. Publ., Dordrecht; Boston; London (1996).

[6]
M. Hanke, Conjugate Gradient Type Methods for Ill-posed Problems, CRC Press (2019).

[7]
C. H. Huang and C. W. Chen, “A boundary element-based inverse problem in estimating transient boundary conditions with conjugate gradient method,” Int. J. Numer. Methods Eng., 42, No. 5, 943–965 (1998).

[8]
B. Jin, “Conjugate gradient method for the Robin inverse problem associated with the Laplace equation,” Int. J. Numer. Methods Eng., 71, No. 4, 433–453 (2007).

[9]
I. Elshafiey, L. Udpa, and S. Udpa, “Solution of inverse problems in electromagnetics using Hopfield neural networks,” IEEE Trans. Magnetics, 31, No. 1, 852–861 (1995).

[10]
Y. Fan and L. Ying, “Solving inverse wave scattering with deep learning,” arXiv:1911.13202 (2019).

[11]
S. Hoole, “Artificial neural networks in the solution of inverse electromagnetic field problems,” IEEE Trans. Magnetics, 29, No. 2, 1931–1934 (1993).

[12]
J. Schwab, S. Antholzer, and M. Haltmeier, “Deep null space learning for inverse problems: convergence analysis and rates,” Inverse Probl., 35, No. 2, 025008 (2019).

[13]
J. Adler and O. Öktem, “Solving ill-posed inverse problems using iterative deep neural networks,” Inverse Probl., 33, No. 12, 124007 (2017).

[14]
L. Beilina and M. V. Klibanov, “A globally convergent numerical method for a coefficient inverse problem,” SIAM J. Sci. Comput., 31, No. 1, 478–509 (2008).

[15]
A. Bonito, A. Cohen, R. DeVore, G. Petrova, and G. Welper, “Diffusion coefficients estimation for elliptic partial differential equations,” SIAM J. Math. Anal., 49, No. 2, 1570–1592 (2017).

[16]
V. I. Gorbachenko, T. V. Lazovskaya, D. A. Tarkhov, A. N. Vasilyev, and M. V. Zhukov, “Neural network technique in some inverse problems of mathematical physics,” in: Int. Symp. Neural Networks, pp. 310–316, Springer, Cham (2016).

[17]
D. N. H`ao and T. N. T. Quyen, “Finite element methods for coefficient identification in an elliptic equation,” Appl. Anal., 93, No. 7, 1533–1566 (2014).

[18]
I. Knowles, “Parameter identification for elliptic problems,” J. Comput. Appl. Math., 131, No. 1–2, 175–194 (2001).

[19]
T. N. T. Quyen, “Finite element analysis for identifying the reaction coefficient in PDE from boundary observations,” Appl. Numer. Math., 145, 297–314 (2019).

[20]
L. Wang and J. Zou, “Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems,” Discrete Contin. Dyn. Syst., B, 14, No. 4, 1641 (2010).

[21]
J. Zou, “Numerical methods for elliptic inverse problems,” Int. J. Computer Math., 70, No. 2, 211–232 (1998).

[22]
H. Li, J. Schwab, S. Antholzer, and M. Haltmeier, “NETT: Solving inverse problems with deep neural networks,” Inverse Probl., (2020).

[23]
A. Logg, K. A. Mardal, and G. Wells, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Springer-Verl., Berlin; Heidelberg (2012) (Lect. Notes Comput. Sci. Eng.; vol. 84).

[24]
S. S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice Hall PTR, New York (1994).

[25]
S. S. Haykin, Neural Networks and Learning Machines, Prentice Hall, New York (2009).

[26]
K. H. Jin, M. T. McCann, E. Froustey, and M. Unser, “Deep convolutional neural network for inverse problems in imaging,” IEEE Trans. Image Process., 26, No. 9, 4509–4522 (2017).

[27]
A. Lucas, M. Iliadis, R. Molina, and A. K. Katsaggelos, “Using deep neural networks for inverse problems in imaging: beyond analytical methods,” IEEE Signal Process. Mag., 35, No. 1, 20–36 (2018).

[28]
V. M. Krasnopolsky and H. Schiller, “Some neural network applications in environmental sciences. Part I: forward and inverse problems in geophysical remote measurements,” Neural Networks, 16, No. 3–4, 321–334 (2003).
How to Cite
Huang, J., Grigorev, A. and Ivanov, D. (2021) “Numerical methods for identifying the diffusion coefficient in a nonlinear elliptic equation”, Mathematical notes of NEFU, 28(1), pp. 78-92. doi: https://doi.org/10.25587/SVFU.2021.81.41.007.
Section
Mathematical Modeling