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Combination of Multigrid with Constraint Data for Inverse Problem of Nonlinear Diffusion Equation

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  • Tao Liu

    (School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
    School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore)

  • Di Ouyang

    (School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China)

  • Lianjun Guo

    (Eighth Geological Brigade of Hebei Bureau of Geology and Mineral Resources Exploration, Qinhuangdao 066000, China)

  • Ruofeng Qiu

    (Eighth Geological Brigade of Hebei Bureau of Geology and Mineral Resources Exploration, Qinhuangdao 066000, China)

  • Yunfei Qi

    (Eighth Geological Brigade of Hebei Bureau of Geology and Mineral Resources Exploration, Qinhuangdao 066000, China)

  • Wu Xie

    (Eighth Geological Brigade of Hebei Bureau of Geology and Mineral Resources Exploration, Qinhuangdao 066000, China)

  • Qiang Ma

    (Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China)

  • Chao Liu

    (School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China)

Abstract

This paper delves into a rapid and accurate numerical solution for the inverse problem of the nonlinear diffusion equation in the context of multiphase porous media flow. For the realization of this, the combination of the multigrid method with constraint data is utilized and investigated. Additionally, to address the ill-posedness of the inverse problem, the Tikhonov regularization is incorporated. Numerical results demonstrate the computational performance of this method. The proposed combination strategy displays remarkable capabilities in reducing noise, avoiding local minima, and accelerating convergence. Moreover, this combination method performs better than any one method used alone.

Suggested Citation

  • Tao Liu & Di Ouyang & Lianjun Guo & Ruofeng Qiu & Yunfei Qi & Wu Xie & Qiang Ma & Chao Liu, 2023. "Combination of Multigrid with Constraint Data for Inverse Problem of Nonlinear Diffusion Equation," Mathematics, MDPI, vol. 11(13), pages 1-15, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:13:p:2887-:d:1180878
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    References listed on IDEAS

    as
    1. El Houcine Bergou & Youssef Diouane & Vyacheslav Kungurtsev, 2020. "Convergence and Complexity Analysis of a Levenberg–Marquardt Algorithm for Inverse Problems," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 927-944, June.
    2. Hassan K. Ibrahim Al-Mahdawi & Mostafa Abotaleb & Hussein Alkattan & Al-Mahdawi Zena Tareq & Amr Badr & Ammar Kadi, 2022. "Multigrid Method for Solving Inverse Problems for Heat Equation," Mathematics, MDPI, vol. 10(15), pages 1-15, August.
    3. Zhang, Zhi-Yong & Zhang, Hui & Liu, Ye & Li, Jie-Ying & Liu, Cheng-Bao, 2023. "Generalized conditional symmetry enhanced physics-informed neural network and application to the forward and inverse problems of nonlinear diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    4. Hassan K. Ibrahim Al-Mahdawi & Hussein Alkattan & Mostafa Abotaleb & Ammar Kadi & El-Sayed M. El-kenawy, 2022. "Updating the Landweber Iteration Method for Solving Inverse Problems," Mathematics, MDPI, vol. 10(15), pages 1-13, August.
    5. Liu, Tao, 2018. "A nonlinear multigrid method for inverse problem in the multiphase porous media flow," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 271-281.
    6. Rodrigues, F.A & Orlande, H.R.B & Dulikravich, G.S, 2004. "Simultaneous estimation of spatially dependent diffusion coefficient and source term in a nonlinear 1D diffusion problem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 66(4), pages 409-424.
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