Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations

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February 2011, 5(1): 1-17. doi: 10.3934/ipi.2011.5.1

Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations

Adriano De Cezaro ^1, , Johann Baumeister ^2, and Antonio Leitão ^3,

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

Received October 2009 Revised September 2010 Published February 2011

Abstract
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We investigate iterated Tikhonov methods coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. In the case of noisy data we propose a modification, the so called loping iterated Tikhonov-Kaczmarz method, where a sequence of relaxation parameters is introduced and a different stopping rule is used. Convergence analysis for this method is also provided.

Keywords: Ill-posed equations, iterated Tikhonov method, Nonlinear systems, Regularization, Kaczmarz method..

Mathematics Subject Classification: Primary: 65J20; Secondary: 47J0.

Citation: Adriano De Cezaro, Johann Baumeister, Antonio Leitão. Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations. Inverse Problems and Imaging, 2011, 5 (1) : 1-17. doi: 10.3934/ipi.2011.5.1

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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