A simplified generalized Gauss-Newton method for nonlinear ill-posed problems

Authors:
Pallavi Mahale and M. Thamban Nair

Journal:
Math. Comp. **78** (2009), 171-184

MSC (2000):
Primary 65J20

Published electronically:
June 10, 2008

MathSciNet review:
2448702

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Iterative regularization methods for nonlinear ill-posed equations of the form , where is an operator between Hilbert spaces and , usually involve calculation of the Fréchet derivatives of at each iterate and at the unknown solution . In this paper, we suggest a modified form of the generalized Gauss-Newton method which requires the Fréchet derivative of only at an initial approximation of the solution . The error analysis for this method is done under a general source condition which also involves the Fréchet derivative only at . The conditions under which the results of this paper hold are weaker than those considered by Kaltenbacher (1998) for an analogous situation for a special case of the source condition.

**1.**A. B. Bakushinskiĭ,*On a convergence problem of the iterative-regularized Gauss-Newton method*, Zh. Vychisl. Mat. i Mat. Fiz.**32**(1992), no. 9, 1503–1509 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys.**32**(1992), no. 9, 1353–1359 (1993). MR**1185952****2.**A. B. Bakushinskiĭ,*Iterative methods without saturation for solving degenerate nonlinear operator equations*, Dokl. Akad. Nauk**344**(1995), no. 1, 7–8 (Russian). MR**1361018****3.**Barbara Kaltenbacher,*A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems*, Numer. Math.**79**(1998), no. 4, 501–528. MR**1631677**, 10.1007/s002110050349**4.**Barbara Blaschke, Andreas Neubauer, and Otmar Scherzer,*On convergence rates for the iteratively regularized Gauss-Newton method*, IMA J. Numer. Anal.**17**(1997), no. 3, 421–436. MR**1459331**, 10.1093/imanum/17.3.421**5.**Thorsten Hohage,*Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem*, Inverse Problems**13**(1997), no. 5, 1279–1299. MR**1474369**, 10.1088/0266-5611/13/5/012**6.**Hohage, T. (1999):*Iterative methods in inverse obstacle scattering: Regularization theory of linear and nonlinear exponentially ill-posed problems*, Ph.D. Thesis, Johannes Kepler University, Linz, Austria.**7.**Thorsten Hohage,*Regularization of exponentially ill-posed problems*, Numer. Funct. Anal. Optim.**21**(2000), no. 3-4, 439–464. MR**1769885**, 10.1080/01630560008816965**8.**S. Langer and T. Hohage,*Convergence analysis of an inexact iteratively regularized Gauss-Newton method under general source conditions*, J. Inverse Ill-Posed Probl.**15**(2007), no. 3, 311–327. MR**2337589**, 10.1515/jiip.2007.017**9.**Peter Mathé and Sergei V. Pereverzev,*Geometry of linear ill-posed problems in variable Hilbert scales*, Inverse Problems**19**(2003), no. 3, 789–803. MR**1984890**, 10.1088/0266-5611/19/3/319

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65J20

Retrieve articles in all journals with MSC (2000): 65J20

Additional Information

**Pallavi Mahale**

Affiliation:
Department of Mathematics, IIT Madras, Chennai 600036, India

Email:
pallavimahale@iitm.ac.in

**M. Thamban Nair**

Affiliation:
Department of Mathematics, IIT Madras, Chennai 600036, India

Email:
mtnair@iitm.ac.in

DOI:
https://doi.org/10.1090/S0025-5718-08-02149-2

Received by editor(s):
July 2, 2007

Received by editor(s) in revised form:
January 13, 2008

Published electronically:
June 10, 2008

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.