IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v474y2024ics0096300324001607.html
   My bibliography  Save this article

Levenberg-Marquardt method with singular scaling and applications

Author

Listed:
  • Boos, Everton
  • Gonçalves, Douglas S.
  • Bazán, Fermín S.V.

Abstract

Inspired by certain regularization techniques for linear inverse problems, in this work we investigate the convergence properties of the Levenberg-Marquardt method using singular scaling matrices. Under a completeness condition, we show that the method is well-defined and establish its local quadratic convergence under an error bound assumption. We also prove that the search directions are gradient-related allowing us to show that limit points of the sequence generated by a line-search version of the method are stationary for the sum-of-squares function. The usefulness of the method is illustrated with some examples of parameter identification in heat conduction problems for which specific singular scaling matrices can be used to improve the quality of approximate solutions.

Suggested Citation

  • Boos, Everton & Gonçalves, Douglas S. & Bazán, Fermín S.V., 2024. "Levenberg-Marquardt method with singular scaling and applications," Applied Mathematics and Computation, Elsevier, vol. 474(C).
    • Handle: RePEc:eee:apmaco:v:474:y:2024:i:c:s0096300324001607
    DOI: 10.1016/j.amc.2024.128688
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300324001607
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2024.128688?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    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. S. Bellavia & B. Morini & E. Riccietti, 2016. "On an adaptive regularization for ill-posed nonlinear systems and its trust-region implementation," Computational Optimization and Applications, Springer, vol. 64(1), pages 1-30, May.
    3. Stefania Bellavia & Elisa Riccietti, 2018. "On an Elliptical Trust-Region Procedure for Ill-Posed Nonlinear Least-Squares Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 824-859, September.
    4. Roger Behling & Douglas S. Gonçalves & Sandra A. Santos, 2019. "Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 1099-1122, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Naoki Marumo & Takayuki Okuno & Akiko Takeda, 2023. "Majorization-minimization-based Levenberg–Marquardt method for constrained nonlinear least squares," Computational Optimization and Applications, Springer, vol. 84(3), pages 833-874, April.
    2. Roger Behling & Douglas S. Gonçalves & Sandra A. Santos, 2019. "Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 1099-1122, December.
    3. Won-Kwang Park, 2021. "Fast Localization of Small Inhomogeneities from Far-Field Pattern Data in the Limited-Aperture Inverse Scattering Problem," Mathematics, MDPI, vol. 9(17), pages 1-22, August.
    4. Nataša Krejić & Greta Malaspina & Lense Swaenen, 2023. "A split Levenberg-Marquardt method for large-scale sparse problems," Computational Optimization and Applications, Springer, vol. 85(1), pages 147-179, May.
    5. 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.
    6. Stefania Bellavia & Elisa Riccietti, 2018. "On an Elliptical Trust-Region Procedure for Ill-Posed Nonlinear Least-Squares Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 824-859, September.
    7. Tao Liu & Zijian Ding & Jiayuan Yu & Wenwen Zhang, 2023. "Parameter Estimation for Nonlinear Diffusion Problems by the Constrained Homotopy Method," Mathematics, MDPI, vol. 11(12), pages 1-12, June.
    8. Reza Arefidamghani & Roger Behling & Yunier Bello-Cruz & Alfredo N. Iusem & Luiz-Rafael Santos, 2021. "The circumcentered-reflection method achieves better rates than alternating projections," Computational Optimization and Applications, Springer, vol. 79(2), pages 507-530, June.
    9. E. V. Castelani & R. Lopes & W. V. I. Shirabayashi & F. N. C. Sobral, 2021. "A robust method based on LOVO functions for solving least squares problems," Journal of Global Optimization, Springer, vol. 80(2), pages 387-414, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:474:y:2024:i:c:s0096300324001607. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.