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Volume 40, Issue 5
A Discretizing Levenberg-Marquardt Scheme for Solving Nonlinear Ill-Posed Integral Equations

Rong Zhang & Hongqi Yang

J. Comp. Math., 40 (2022), pp. 690-714.

Published online: 2022-05

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  • Abstract

To reduce the computational cost, we propose a regularizing modified Levenberg-Marquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems. Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved. Based on these results, we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme. By imposing certain conditions on the noise, we derive optimal convergence rates on the approximate solution under special source conditions. Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice.

  • Keywords

The regularizing Levenberg-Marquardt scheme, Multiscale Galerkin methods, Nonlinear ill-posed problems, Heuristic parameter choice rule, Optimal convergence rate.

  • AMS Subject Headings

65J15, 65J20, 65J22

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

15007082798@163.com (Rong Zhang)

mcsyhq@mail.sysu.edu.cn (Hongqi Yang)

  • BibTex
  • RIS
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@Article{JCM-40-690, author = {Rong and Zhang and 15007082798@163.com and 23352 and School of Mathematics and Computer Science, Gannan Normal University, Ganzhou, China and Rong Zhang and Hongqi and Yang and mcsyhq@mail.sysu.edu.cn and 23353 and Guangdong Province Key Laboratory of Computational Science, School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, China and Hongqi Yang}, title = {A Discretizing Levenberg-Marquardt Scheme for Solving Nonlinear Ill-Posed Integral Equations}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {5}, pages = {690--714}, abstract = {

To reduce the computational cost, we propose a regularizing modified Levenberg-Marquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems. Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved. Based on these results, we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme. By imposing certain conditions on the noise, we derive optimal convergence rates on the approximate solution under special source conditions. Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2101-m2020-0218}, url = {http://global-sci.org/intro/article_detail/jcm/20543.html} }
TY - JOUR T1 - A Discretizing Levenberg-Marquardt Scheme for Solving Nonlinear Ill-Posed Integral Equations AU - Zhang , Rong AU - Yang , Hongqi JO - Journal of Computational Mathematics VL - 5 SP - 690 EP - 714 PY - 2022 DA - 2022/05 SN - 40 DO - http://doi.org/10.4208/jcm.2101-m2020-0218 UR - https://global-sci.org/intro/article_detail/jcm/20543.html KW - The regularizing Levenberg-Marquardt scheme, Multiscale Galerkin methods, Nonlinear ill-posed problems, Heuristic parameter choice rule, Optimal convergence rate. AB -

To reduce the computational cost, we propose a regularizing modified Levenberg-Marquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems. Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved. Based on these results, we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme. By imposing certain conditions on the noise, we derive optimal convergence rates on the approximate solution under special source conditions. Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice.

Rong Zhang & Hongqi Yang. (2022). A Discretizing Levenberg-Marquardt Scheme for Solving Nonlinear Ill-Posed Integral Equations. Journal of Computational Mathematics. 40 (5). 690-714. doi:10.4208/jcm.2101-m2020-0218
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