- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Journal of Mathematics
Volume 2013 (2013), Article ID 439316, 9 pages
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.
- J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover Publications, New York, NY, USA, 1952.
- H. W. Engl, K. Kunisch, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic, Dordrecht, The Netherlands, 1996.
- 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.
- 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.
- A. B. Bakushinskii, “Iterative methods without saturation for solving degenerate nonlinear operator equations,” Doklady Akademii Nauk, vol. 344, pp. 7–8, 1995.
- 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.
- 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.
- 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.
- 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.
- T. Hohage, “Regularization of exponentially ill-posed problems,” Numerical Functional Analysis and Optimization, vol. 21, no. 3, pp. 439–464, 2000.
- 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.
- 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.
- 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.
- I. K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer, New York, NY, USA, 2008.
- 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.
- S. Lu and S. V. Pereverzev, “Sparsity reconstruction by the standard Tikhonov method,” RICAM-Report 2008-17, 2008.
- 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.
- 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.