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Weighted Radial Basis Collocation Method for the Nonlinear Inverse Helmholtz Problems

Author

Listed:
  • Minghao Hu

    (School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China)

  • Lihua Wang

    (School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China)

  • Fan Yang

    (School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China)

  • Yueting Zhou

    (School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China)

Abstract

In this paper, a meshfree weighted radial basis collocation method associated with the Newton’s iteration method is introduced to solve the nonlinear inverse Helmholtz problems for identifying the parameter. All the measurement data can be included in the least-squares solution, which can avoid the iteration calculations for comparing the solutions with part of the measurement data in the Galerkin-based methods. Appropriate weights are imposed on the boundary conditions and measurement conditions to balance the errors, which leads to the high accuracy and optimal convergence for solving the inverse problems. Moreover, it is quite easy to extend the solution process of the one-dimensional inverse problem to high-dimensional inverse problem. Nonlinear numerical examples include one-, two- and three-dimensional inverse Helmholtz problems of constant and varying parameter identification in regular and irregular domains and show the high accuracy and exponential convergence of the presented method.

Suggested Citation

  • Minghao Hu & Lihua Wang & Fan Yang & Yueting Zhou, 2023. "Weighted Radial Basis Collocation Method for the Nonlinear Inverse Helmholtz Problems," Mathematics, MDPI, vol. 11(3), pages 1-29, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:662-:d:1049284
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    References listed on IDEAS

    as
    1. Zhiyong Liu & Qiuyan Xu, 2019. "A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations," Mathematics, MDPI, vol. 7(10), pages 1-15, October.
    2. Yang Yu & Xiaochuan Luo & Huaxi (Yulin) Zhang & Qingxin Zhang, 2019. "The Solution of Backward Heat Conduction Problem with Piecewise Linear Heat Transfer Coefficient," Mathematics, MDPI, vol. 7(5), pages 1-17, April.
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