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A Nonlinear Multigrid Method for the Parameter Identification Problem of Partial Differential Equations with Constraints

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Listed:
  • Tao Liu

    (School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066000, China)

  • Jiayuan Yu

    (School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066000, China)

  • Yuanjin Zheng

    (School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore)

  • Chao Liu

    (School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066000, China)

  • Yanxiong Yang

    (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)

Abstract

In this paper, we consider the parameter identification problem of partial differential equations with constraints. A nonlinear multigrid method is introduced to the process of parameter inversion. By keeping the objective functions on coarse grids consistent with those on fine grids, the proposed method reduces the dimensions of objective functions enormously and mitigates the risk of trapping in local minima effectively. Furthermore, constraints significantly improve the convergence ability of the method. We performed the numerical simulation based on the porosity identification of elastic wave equations in the fluid-saturated porous media, which suggests that the nonlinear multigrid method with constraints decreases the computational expenditure, suppresses the noise, and improves the inversion results.

Suggested Citation

  • Tao Liu & Jiayuan Yu & Yuanjin Zheng & Chao Liu & Yanxiong Yang & Yunfei Qi, 2022. "A Nonlinear Multigrid Method for the Parameter Identification Problem of Partial Differential Equations with Constraints," Mathematics, MDPI, vol. 10(16), pages 1-12, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:16:p:2938-:d:888821
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    References listed on IDEAS

    as
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    4. Tomasz Rymarczyk & Konrad Niderla & Edward Kozłowski & Krzysztof Król & Joanna Maria Wyrwisz & Sylwia Skrzypek-Ahmed & Piotr Gołąbek, 2021. "Logistic Regression with Wave Preprocessing to Solve Inverse Problem in Industrial Tomography for Technological Process Control," Energies, MDPI, vol. 14(23), pages 1-21, December.
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