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Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations
1. | Institute of Mathematics Statistics and Physics, Federal University of Rio Grande, Av. Italia km 8, 96201-900 Rio Grande, Brazil |
2. | Fachbereich Mathematik, Johann Wolfgang Goethe Universität, Robert–Mayer–Str. 6–10, 60054 Frankfurt am Main |
3. | Department of Mathematics, Federal University of St. Catarina, P.O. Box 476, 88040-900 Florianópolis |
References:
[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 Levenberg-Marquardt Kaczmarz methods for regularizing systems of nonlinear ill-posed equations, Inverse Problems and Imaging, 4 (2010), 335-350.
doi: 10.3934/ipi.2010.4.335. |
[4] |
B. Blaschke(-Kaltenbacher), "Some Newton Type Methods ror the Solution of Nonlinear Ill-Posed Problems," Ph.D. thesis, Johannes Kepler University, Linz, 2005. |
[5] |
M. Brill and E. Schock, Iterative solution of ill-posed problems: A survey, in "Model Optimization in Exploration Geophysics" (ed. A. Vogel), 13-38, Vieweg, Braunschweig, 1987. |
[6] |
C. Byrne, Block-iterative algorithms, Int. Trans. in Operational Research, 16 (2009), 01-37. |
[7] |
J. Cheng and M. Yamamoto, Identification of convection term in a parabolic equation with a single measurement, Nonlinear Analysis, 50 (2002), 163-171.
doi: 10.1016/S0362-546X(01)00742-8. |
[8] |
F. Colonius and K. Kunisch, Stability of parameter estimation in two point boundary value problems, J. Reine Angew. Math., 370 (1986), 1-29.
doi: 10.1515/crll.1986.370.1. |
[9] |
A. De Cezaro, M. Haltmeier, A. Leitão and O. Scherzer, On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations, Appl. Math. Comput., 202 (2008), 596-607.
doi: 10.1016/j.amc.2008.03.010. |
[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, Non-stationary iterated Tikhonov-Morozov method and third-order differential equations for the evaluation of unbounded operators, Math. Methods Appl. Sci., 23 (2000), 1287-1300.
doi: 10.1002/1099-1476(200010)23:15<1287::AID-MMA165>3.0.CO;2-N. |
[12] |
M. Haltmeier, A. Leitão and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations. I. Convergence analysis, Inverse Probl. Imaging, 1 (2007), 289-298. |
[13] |
M. Haltmeier, A. Leitão and E. Resmerita, On regularization methods of EM-Kaczmarz type, Inverse Problems, 25 (2009), 075008.
doi: 10.1088/0266-5611/25/7/075008. |
[14] |
M. Hanke, Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems, Numer. Funct. Anal. Optim., 18 (1997), 971-993.
doi: 10.1080/01630569708816804. |
[15] |
M. Hanke and C. W. Groetsch, Nonstationary iterated Tikhonov regularization, J. Optim. Theory Appl., 98 (1998), 37-53.
doi: 10.1023/A:1022680629327. |
[16] |
M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.
doi: 10.1007/s002110050158. |
[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), 1269-1271.
doi: 10.1080/00207179308934446. |
[19] |
B. Kaltenbacher, Some Newton-type methods for the regularization of nonlinear ill-posed problems, Inverse Problems, 13 (1997), 729-753.
doi: 10.1088/0266-5611/13/3/012. |
[20] |
B. Kaltenbacher, A. Neubauer and O. Scherzer, "Iterative Regularization Methods for Nonlinear Ill-Posed Problems," Radon Series on Computational and Applied Mathematics, vol. 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
doi: 10.1515/9783110208276. |
[21] |
S. Kindermann and A. Neubauer, On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization, Inverse Probl. Imaging, 2 (2008), 291-299. |
[22] |
R. Kowar and O. Scherzer, Convergence analysis of a Landweber-Kaczmarz method for solving nonlinear ill-posed problems, Ill posed and inverse problems (book series), 23 (2002), 69-90. |
[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), 152-157. |
[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), 1137-1150.
doi: 10.1512/iumj.1977.26.26090. |
[25] |
V. A. Morozov, "Regularization Methods for Ill-Posed 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 ill-posed problems, Numer. Math., 66 (1993), 259-279.
doi: 10.1007/BF01385697. |
[28] |
O. Scherzer, A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems, Numer. Funct. Anal. Optim., 17 (1996), 197-214.
doi: 10.1080/01630569608816691. |
[29] |
A. N. Tikhonov and V. Y. Arsenin, "Solutions of Ill-Posed Problems," John Wiley & Sons, Washington, D.C., 1977, Translation editor: Fritz John. |
show all references
References:
[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 Levenberg-Marquardt Kaczmarz methods for regularizing systems of nonlinear ill-posed equations, Inverse Problems and Imaging, 4 (2010), 335-350.
doi: 10.3934/ipi.2010.4.335. |
[4] |
B. Blaschke(-Kaltenbacher), "Some Newton Type Methods ror the Solution of Nonlinear Ill-Posed Problems," Ph.D. thesis, Johannes Kepler University, Linz, 2005. |
[5] |
M. Brill and E. Schock, Iterative solution of ill-posed problems: A survey, in "Model Optimization in Exploration Geophysics" (ed. A. Vogel), 13-38, Vieweg, Braunschweig, 1987. |
[6] |
C. Byrne, Block-iterative algorithms, Int. Trans. in Operational Research, 16 (2009), 01-37. |
[7] |
J. Cheng and M. Yamamoto, Identification of convection term in a parabolic equation with a single measurement, Nonlinear Analysis, 50 (2002), 163-171.
doi: 10.1016/S0362-546X(01)00742-8. |
[8] |
F. Colonius and K. Kunisch, Stability of parameter estimation in two point boundary value problems, J. Reine Angew. Math., 370 (1986), 1-29.
doi: 10.1515/crll.1986.370.1. |
[9] |
A. De Cezaro, M. Haltmeier, A. Leitão and O. Scherzer, On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations, Appl. Math. Comput., 202 (2008), 596-607.
doi: 10.1016/j.amc.2008.03.010. |
[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, Non-stationary iterated Tikhonov-Morozov method and third-order differential equations for the evaluation of unbounded operators, Math. Methods Appl. Sci., 23 (2000), 1287-1300.
doi: 10.1002/1099-1476(200010)23:15<1287::AID-MMA165>3.0.CO;2-N. |
[12] |
M. Haltmeier, A. Leitão and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations. I. Convergence analysis, Inverse Probl. Imaging, 1 (2007), 289-298. |
[13] |
M. Haltmeier, A. Leitão and E. Resmerita, On regularization methods of EM-Kaczmarz type, Inverse Problems, 25 (2009), 075008.
doi: 10.1088/0266-5611/25/7/075008. |
[14] |
M. Hanke, Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems, Numer. Funct. Anal. Optim., 18 (1997), 971-993.
doi: 10.1080/01630569708816804. |
[15] |
M. Hanke and C. W. Groetsch, Nonstationary iterated Tikhonov regularization, J. Optim. Theory Appl., 98 (1998), 37-53.
doi: 10.1023/A:1022680629327. |
[16] |
M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.
doi: 10.1007/s002110050158. |
[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), 1269-1271.
doi: 10.1080/00207179308934446. |
[19] |
B. Kaltenbacher, Some Newton-type methods for the regularization of nonlinear ill-posed problems, Inverse Problems, 13 (1997), 729-753.
doi: 10.1088/0266-5611/13/3/012. |
[20] |
B. Kaltenbacher, A. Neubauer and O. Scherzer, "Iterative Regularization Methods for Nonlinear Ill-Posed Problems," Radon Series on Computational and Applied Mathematics, vol. 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
doi: 10.1515/9783110208276. |
[21] |
S. Kindermann and A. Neubauer, On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization, Inverse Probl. Imaging, 2 (2008), 291-299. |
[22] |
R. Kowar and O. Scherzer, Convergence analysis of a Landweber-Kaczmarz method for solving nonlinear ill-posed problems, Ill posed and inverse problems (book series), 23 (2002), 69-90. |
[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), 152-157. |
[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), 1137-1150.
doi: 10.1512/iumj.1977.26.26090. |
[25] |
V. A. Morozov, "Regularization Methods for Ill-Posed 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 ill-posed problems, Numer. Math., 66 (1993), 259-279.
doi: 10.1007/BF01385697. |
[28] |
O. Scherzer, A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems, Numer. Funct. Anal. Optim., 17 (1996), 197-214.
doi: 10.1080/01630569608816691. |
[29] |
A. N. Tikhonov and V. Y. Arsenin, "Solutions of Ill-Posed Problems," John Wiley & Sons, Washington, D.C., 1977, Translation editor: Fritz John. |
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