A simplified generalized GaussNewton method for nonlinear illposed problems
Authors:
Pallavi Mahale and M. Thamban Nair
Journal:
Math. Comp. 78 (2009), 171184
MSC (2000):
Primary 65J20
Published electronically:
June 10, 2008
MathSciNet review:
2448702
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Iterative regularization methods for nonlinear illposed 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 GaussNewton 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. Bakushinski&ibreve;, On a convergence problem of the
iterativeregularized GaussNewton 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
(93k:65049)
 2.
A.
B. Bakushinski&ibreve;, 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 illposed
problems, Numer. Math. 79 (1998), no. 4,
501–528. MR 1631677
(99f:65091), http://dx.doi.org/10.1007/s002110050349
 4.
Barbara
Blaschke, Andreas
Neubauer, and Otmar
Scherzer, On convergence rates for the iteratively regularized
GaussNewton method, IMA J. Numer. Anal. 17 (1997),
no. 3, 421–436. MR 1459331
(98f:65066), http://dx.doi.org/10.1093/imanum/17.3.421
 5.
Thorsten
Hohage, Logarithmic convergence rates of the iteratively
regularized GaussNewton method for an inverse potential and an inverse
scattering problem, Inverse Problems 13 (1997),
no. 5, 1279–1299. MR 1474369
(98k:65031), http://dx.doi.org/10.1088/02665611/13/5/012
 6.
Hohage, T. (1999): Iterative methods in inverse obstacle scattering: Regularization theory of linear and nonlinear exponentially illposed problems, Ph.D. Thesis, Johannes Kepler University, Linz, Austria.
 7.
Thorsten
Hohage, Regularization of exponentially illposed problems,
Numer. Funct. Anal. Optim. 21 (2000), no. 34,
439–464. MR 1769885
(2001e:65095), http://dx.doi.org/10.1080/01630560008816965
 8.
S.
Langer and T.
Hohage, Convergence analysis of an inexact iteratively regularized
GaussNewton method under general source conditions, J. Inverse
IllPosed Probl. 15 (2007), no. 3, 311–327. MR 2337589
(2008e:65181), http://dx.doi.org/10.1515/jiip.2007.017
 9.
Peter
Mathé and Sergei
V. Pereverzev, Geometry of linear illposed problems in variable
Hilbert scales, Inverse Problems 19 (2003),
no. 3, 789–803. MR 1984890
(2004i:47021), http://dx.doi.org/10.1088/02665611/19/3/319
 1.
 Bakushinskii, A.B. (1992): The problem of the convergence of the iteratively regularised GaussNewton method, Comput. Math. Phys., 32, 13531359. MR 1185952 (93k:65049)
 2.
 Bakushinskii, A.B. (1995): Iterative methods without saturation for solving degenerate nonlinear operator equations, Dokl. Akad. Nauk, 344: 78. MR 1361018
 3.
 Kaltenbacher, Barbara (1998): A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear illposed problems, Numerische Mathematik, 79, 501528. MR 1631677 (99f:65091)
 4.
 Blaschke, B., Neubauer, A., Scherzer, O. (1997): On convergence rates for the iteratively regularized GaussNewton method, IMA J. Numer. Anal., 17, 421436. MR 1459331 (98f:65066)
 5.
 Hohage, T. (1997): Logarithmic convergence rates of the iteratively regularized GaussNewton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13, 12791299. MR 1474369 (98k:65031)
 6.
 Hohage, T. (1999): Iterative methods in inverse obstacle scattering: Regularization theory of linear and nonlinear exponentially illposed problems, Ph.D. Thesis, Johannes Kepler University, Linz, Austria.
 7.
 Hohage, T. (2000): Regularization of exponentially illposed problems, Numer. Funct. Anal. & Optim., 21, 439464. MR 1769885 (2001e:65095)
 8.
 Langer, S., Hohage, T. (2007): Convergence analysis of an inexact iteratively regulaized GaussNewton method under general source conditions, J. of Inverse & IllPosed Problems, 15, 1935. MR 2337589
 9.
 Mathé, P., Pereverzev, S. (2003): Geometry of illposed problems in variable Hilbert scales, Inverse Problems, 19, 789803. MR 1984890 (2004i:47021)
Similar Articles
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:
http://dx.doi.org/10.1090/S0025571808021492
PII:
S 00255718(08)021492
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.
