About this Journal Submit a Manuscript Table of Contents
Journal of Mathematics
Volume 2013 (2013), Article ID 439316, 9 pages
http://dx.doi.org/10.1155/2013/439316
Research Article

Newton-Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems

Department of Mathematical and Computational Sciences, National Institute of Technology, Mangalore, Karnataka 575 025, India

Received 14 August 2012; Accepted 15 October 2012

Academic Editor: De-Xing Kong

Copyright © 2013 Santhosh George. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover Publications, New York, NY, USA, 1952. View at MathSciNet
  2. H. W. Engl, K. Kunisch, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic, Dordrecht, The Netherlands, 1996.
  3. H. W. Engl, K. Kunisch, and A. Neubauer, “Convergence rates for Tikhonov regularisation of non-linear ill-posed problems,” Inverse Problems, vol. 5, no. 4, pp. 523–540, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. A. B. Bakushinskii, “The problem of the convergence of the iteratively regularized Gauss-Newton method,” Computational Mathematics and Mathematical Physics, vol. 32, no. 9, pp. 1353–1359, 1992. View at Scopus
  5. A. B. Bakushinskii, “Iterative methods without saturation for solving degenerate nonlinear operator equations,” Doklady Akademii Nauk, vol. 344, pp. 7–8, 1995. View at MathSciNet
  6. B. Blaschke, A. Neubauer, and O. Scherzer, “On convergence rates for the iteratively regularized Gauss-Newton method,” IMA Journal of Numerical Analysis, vol. 17, no. 3, pp. 421–436, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. B. Kaltenbacher, “A note on logarithmic convergence rates for nonlinear Tikhonov regularization,” Journal of Inverse and Ill-Posed Problems, vol. 16, no. 1, pp. 79–88, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. P. A. Mahale and M. T. Nair, “A simplified generalized Gauss-Newton method for nonlinear ill-posed problems,” Mathematics of Computation, vol. 78, no. 265, pp. 171–184, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. T. Hohage, “Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem,” Inverse Problems, vol. 13, no. 5, pp. 1279–1299, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. T. Hohage, “Regularization of exponentially ill-posed problems,” Numerical Functional Analysis and Optimization, vol. 21, no. 3, pp. 439–464, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. S. Langer and T. Hohage, “Convergence analysis of an inexact iteratively regularized Gauss-Newton method under general source conditions,” Journal of Inverse and Ill-Posed Problems, vol. 15, no. 3, pp. 311–327, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. A. Bakushinsky and A. Smirnova, “On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems,” Numerical Functional Analysis and Optimization, vol. 26, no. 1, pp. 35–48, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. S. George, “On convergence of regularized modified Newton's method for nonlinear ill-posed problems,” Journal of Inverse and Ill-Posed Problems, vol. 18, no. 2, pp. 133–146, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. I. K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer, New York, NY, USA, 2008. View at Zentralblatt MATH · View at MathSciNet
  15. S. Pereverzev and E. Schock, “On the adaptive selection of the parameter in regularization of ill-posed problems,” SIAM Journal on Numerical Analysis, vol. 43, no. 5, pp. 2060–2076, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. S. Lu and S. V. Pereverzev, “Sparsity reconstruction by the standard Tikhonov method,” RICAM-Report 2008-17, 2008.
  17. E. V. Semenova, “Lavrentiev regularization and balancing principle for solving ill-posed problems with monotone operators,” Computational Methods in Applied Mathematics, vol. 4, no. 4, pp. 444–454, 2010. View at MathSciNet
  18. C. W. Groetsch, J. T. King, and D. Murio, “Asymptotic analysis of a finite element method for Fredholm equations of the first kind,” in Treatment of Integral Equations by Numerical Methods, C. T. H. Baker and G. F. Miller, Eds., pp. 1–11, Academic Press, London, UK, 1982. View at Zentralblatt MATH · View at MathSciNet