189
Views
0
CrossRef citations to date
0
Altmetric
Original Articles

Quadratic convergence of Levenberg-Marquardt method for general nonlinear inverse problems with two parameters

, , , &
Pages 1949-1966 | Received 17 Jun 2019, Accepted 07 Sep 2019, Published online: 23 Sep 2019

References

  • H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhauser, Boston, 1989.
  • S. Chaabane and M. Jaoua, Identification of Robin coefficients by the means of boundary measurements, Inverse Probl. 15 (1999), pp. 1425–1438. doi: 10.1088/0266-5611/15/6/303
  • M. Choulli, An inverse problem in corrosion detection: Stability estimates, J. Inverse Ill-posed Probl.12(4) (2004), pp. 349–367. doi: 10.1515/1569394042248247
  • M. Choulli and M. Yamamoto, Uniqueness and stability in determining the heat radiative coefficient, the initial temperature and a boundary coefficient in a parabolic equation, Nonlinear Anal. 69 (2008), pp. 3983–3998. doi: 10.1016/j.na.2007.10.031
  • I. Daubechies, M. Defrise and C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math. 57(11) (2004), pp. 1413–1457. doi: 10.1002/cpa.20042
  • H.W. Engl and W. Rundell, Inverse Problems in Diffusion Processes, SIAM, Philadelphia, 1995.
  • J.Y. Fan and Y.X. Yuan, On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing 74 (2005), pp. 23–39. doi: 10.1007/s00607-004-0083-1
  • P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Publishing Program, Boston, 1985.
  • M. Hanke, A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Problems 13(1) (1997), pp. 79–95.
  • D.N. Hao and T.N. Quyen, Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations, Inverse Probl. 26 (2010), p. 125014. (20pp). doi: 10.1088/0266-5611/26/12/125014
  • D.J. Jiang and T.A. Talaat, Simultaneous identification of Robin coefficient and heat flux in an elliptic system, Intern. J. Comput. Math. 94(1) (2017), pp. 185–196. doi: 10.1080/00207160.2015.1099634
  • D.J. Jiang, H. Feng and J. Zou, Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system, Inverse Probl. 28 (2012), p. 104002. (20pp). doi: 10.1088/0266-5611/28/10/104002
  • D.J. Jiang, H. Feng and J. Zou, Quadratic convergence of Levenberg Marquardt method for elliptic and parabolic inverse Robin problems, ESAIM Math. Model. Numer. Anal. 52(3) (2018), pp. 1085–1107. doi: 10.1051/m2an/2018016
  • B.T. Jin, Conjugate gradient method for the Robin inverse problem associated with the Laplace equation, Int. J. Numer. Methods Eng. 71 (2007), pp. 433–453. doi: 10.1002/nme.1949
  • Y.L. Keung and J. Zou, Numerical identifications of parameters in parabolic systems, Inverse Probl. 14 (1998), pp. 83–100. doi: 10.1088/0266-5611/14/1/009
  • J.Z. Li and J. Zou, A multilevel model correction method for parameter identification, Inverse Probl. 23 (2007), pp. 1759–1786. doi: 10.1088/0266-5611/23/5/001
  • J. Xie and J. Zou, Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal. 43 (2005), pp. 1504–1535. doi: 10.1137/030602551
  • M. Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Probl. 17 (2001), pp. 1181–1202. doi: 10.1088/0266-5611/17/4/340
  • N. Yamashita and M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method. Topics in numerical analysis, Comput. Suppl. 15, Springer, Vienna, 2001, pp. 239–249.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.