Modified iterated Tikhonov methods for solving systems of nonlinear
illposed equations
Pages: 1  17,
Volume 5,
Issue 1,
February
2011
doi:10.3934/ipi.2011.5.1 Abstract
References
Full text (438.9K)
Related Articles
Adriano De Cezaro  Institute of Mathematics Statistics and Physics, Federal University of Rio Grande, Av. Italia km 8, 96201900 Rio Grande, Brazil (email)
Johann Baumeister  Fachbereich Mathematik, Johann Wolfgang Goethe Universität, Robert–Mayer–Str. 6–10, 60054 Frankfurt am Main, Germany (email)
Antonio Leitão  Department of Mathematics, Federal University of St. Catarina, P.O. Box 476, 88040900 Florianópolis, Brazil (email)
1 
A. B. Bakushinsky and M. Y. Kokurin, "Iterative Methods for Approximate Solution of Inverse Problems," Mathematics and Its Applications, vol. 577, Springer, Dordrecht, 2004. 

2 
H. T. Banks and K. Kunisch, "Estimation Techniques for Distributed Parameter Systems," Birkhäuser, 1989. 

3 
J. Baumeister, B. Kaltenbacher and A. Leitão, On LevenbergMarquardt Kaczmarz methods for regularizing systems of nonlinear illposed equations, Inverse Problems and Imaging, 4 (2010), 335350. 

4 
B. Blaschke(Kaltenbacher), "Some Newton Type Methods ror the Solution of Nonlinear IllPosed Problems," Ph.D. thesis, Johannes Kepler University, Linz, 2005. 

5 
M. Brill and E. Schock, Iterative solution of illposed problems: A survey, in "Model Optimization in Exploration Geophysics" (ed. A. Vogel), 1338, Vieweg, Braunschweig, 1987. 

6 
C. Byrne, Blockiterative algorithms, Int. Trans. in Operational Research, 16 (2009), 0137. 

7 
J. Cheng and M. Yamamoto, Identification of convection term in a parabolic equation with a single measurement, Nonlinear Analysis, 50 (2002), 163171. 

8 
F. Colonius and K. Kunisch, Stability of parameter estimation in two point boundary value problems, J. Reine Angew. Math., 370 (1986), 129. 

9 
A. De Cezaro, M. Haltmeier, A. Leitão and O. Scherzer, On steepestdescentKaczmarz methods for regularizing systems of nonlinear illposed equations, Appl. Math. Comput., 202 (2008), 596607. 

10 
H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Kluwer Academic Publishers, Dordrecht, 1996. 

11 
C. W. Groetsch and O. Scherzer, Nonstationary iterated TikhonovMorozov method and thirdorder differential equations for the evaluation of unbounded operators, Math. Methods Appl. Sci., 23 (2000), 12871300. 

12 
M. Haltmeier, A. Leitão and O. Scherzer, Kaczmarz methods for regularizing nonlinear illposed equations. I. Convergence analysis, Inverse Probl. Imaging, 1 (2007), 289298. 

13 
M. Haltmeier, A. Leitão and E. Resmerita, On regularization methods of EMKaczmarz type, Inverse Problems, 25 (2009), 075008. 

14 
M. Hanke, Regularizing properties of a truncated NewtonCG algorithm for nonlinear inverse problems, Numer. Funct. Anal. Optim., 18 (1997), 971993. 

15 
M. Hanke and C. W. Groetsch, Nonstationary iterated Tikhonov regularization, J. Optim. Theory Appl., 98 (1998), 3753. 

16 
M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of Landweber iteration for nonlinear illposed problems, Numer. Math., 72 (1995), 2137. 

17 
V. Isakov, "Inverse Problems for Partial Differential Equations," Second ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006. 

18 
S. Kaczmarz, Approximate solution of systems of linear equations, Internat. J. Control, 57 (1993), 12691271. 

19 
B. Kaltenbacher, Some Newtontype methods for the regularization of nonlinear illposed problems, Inverse Problems, 13 (1997), 729753. 

20 
B. Kaltenbacher, A. Neubauer and O. Scherzer, "Iterative Regularization Methods for Nonlinear IllPosed Problems," Radon Series on Computational and Applied Mathematics, vol. 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. 

21 
S. Kindermann and A. Neubauer, On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization, Inverse Probl. Imaging, 2 (2008), 291299. 

22 
R. Kowar and O. Scherzer, Convergence analysis of a LandweberKaczmarz method for solving nonlinear illposed problems, Ill posed and inverse problems (book series), 23 (2002), 6990. 

23 
L. J. Lardy, A series representation for the generalized inverse of a closed linear operator, Atti della Accademia Nazionale dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche, e Naturali, Serie VIII, 58 (1975), 152157. 

24 
S. McCormick, The methods of Kaczmarz and row orthogonalization for solving linear equations and least squares problems in Hilbert space, Indiana Univ. Math. J., 26 (1977), 11371150. 

25 
V. A. Morozov, "Regularization Methods for IllPosed Problems," CRC Press, Boca Raton, 1993. 

26 
F. Natterer, Algorithms in tomography, in "The State of the Art in Numerical Analysis," vol. 63, Oxford University Press, New York, 1997. 

27 
O. Scherzer, Convergence rates of iterated Tikhonov regularized solutions of nonlinear illposed problems, Numer. Math., 66 (1993), 259279. 

28 
O. Scherzer, A convergence analysis of a method of steepest descent and a twostep algorithm for nonlinear illposed problems, Numer. Funct. Anal. Optim., 17 (1996), 197214. 

29 
A. N. Tikhonov and V. Y. Arsenin, "Solutions of IllPosed Problems," John Wiley & Sons, Washington, D.C., 1977, Translation editor: Fritz John. 

Go to top
