# American Institute of Mathematical Sciences

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

 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

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.
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.
 [1] Markus Haltmeier, Richard Kowar, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations II: Applications. Inverse Problems and Imaging, 2007, 1 (3) : 507-523. doi: 10.3934/ipi.2007.1.507 [2] Johann Baumeister, Barbara Kaltenbacher, Antonio Leitão. On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations. Inverse Problems and Imaging, 2010, 4 (3) : 335-350. doi: 10.3934/ipi.2010.4.335 [3] Guozhi Dong, Bert Jüttler, Otmar Scherzer, Thomas Takacs. Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces. Inverse Problems and Imaging, 2017, 11 (2) : 221-246. doi: 10.3934/ipi.2017011 [4] Markus Haltmeier, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis. Inverse Problems and Imaging, 2007, 1 (2) : 289-298. doi: 10.3934/ipi.2007.1.289 [5] Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems and Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033 [6] Paola Favati, Grazia Lotti, Ornella Menchi, Francesco Romani. An inner-outer regularizing method for ill-posed problems. Inverse Problems and Imaging, 2014, 8 (2) : 409-420. doi: 10.3934/ipi.2014.8.409 [7] Matthew A. Fury. Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space. Conference Publications, 2013, 2013 (special) : 259-272. doi: 10.3934/proc.2013.2013.259 [8] Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609 [9] Zonghao Li, Caibin Zeng. Center manifolds for ill-posed stochastic evolution equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2483-2499. doi: 10.3934/dcdsb.2021142 [10] Felix Lucka, Katharina Proksch, Christoph Brune, Nicolai Bissantz, Martin Burger, Holger Dette, Frank Wübbeling. Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations. Inverse Problems and Imaging, 2018, 12 (5) : 1121-1155. doi: 10.3934/ipi.2018047 [11] Ye Zhang, Bernd Hofmann. Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems. Inverse Problems and Imaging, 2021, 15 (2) : 229-256. doi: 10.3934/ipi.2020062 [12] Stefan Kindermann, Andreas Neubauer. On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Problems and Imaging, 2008, 2 (2) : 291-299. doi: 10.3934/ipi.2008.2.291 [13] Lianwang Deng. Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator. Communications on Pure and Applied Analysis, 2020, 19 (1) : 145-174. doi: 10.3934/cpaa.2020009 [14] Sergiy Zhuk. Inverse problems for linear ill-posed differential-algebraic equations with uncertain parameters. Conference Publications, 2011, 2011 (Special) : 1467-1476. doi: 10.3934/proc.2011.2011.1467 [15] Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016 [16] Matthew A. Fury. Estimates for solutions of nonautonomous semilinear ill-posed problems. Conference Publications, 2015, 2015 (special) : 479-488. doi: 10.3934/proc.2015.0479 [17] Eliane Bécache, Laurent Bourgeois, Lucas Franceschini, Jérémi Dardé. Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case. Inverse Problems and Imaging, 2015, 9 (4) : 971-1002. doi: 10.3934/ipi.2015.9.971 [18] Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial and Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345 [19] Youri V. Egorov, Evariste Sanchez-Palencia. Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1293-1305. doi: 10.3934/dcds.2011.31.1293 [20] Olha P. Kupenko, Rosanna Manzo. On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1363-1393. doi: 10.3934/dcdsb.2018155

2021 Impact Factor: 1.483