# American Institute of Mathematical Sciences

February  2014, 8(1): 149-172. doi: 10.3934/ipi.2014.8.149

## Convergence rates for Kaczmarz-type regularization methods

 1 Industrial Mathematics Institute, Johannes Kepler University Linz, A-4040 Linz 2 Department of Mathematics, Federal University of St. Catarina, P.O. Box 476, 88040-900 Florianópolis

Received  August 2012 Revised  November 2013 Published  March 2014

This article is devoted to the convergence analysis of a special family of iterative regularization methods for solving systems of ill--posed operator equations in Hilbert spaces, namely Kaczmarz-type methods. The analysis is focused on the Landweber--Kaczmarz (LK) explicit iteration and the iterated Tikhonov--Kaczmarz (iTK) implicit iteration. The corresponding symmetric versions of these iterative methods are also investigated (sLK and siTK). We prove convergence rates for the four methods above, extending and complementing the convergence analysis established originally in [22,13,12,8].
Citation: Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems and Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149
##### References:
 [1] P. K. Anh and C. V. Chung, Parallel regularized Newton method for nonlinear ill-posed equations, Numer. Algorithms, 58 (2011), 379-398. doi: 10.1007/s11075-011-9460-y. [2] A. B. Bakushinsky, M. Y. Kokurin and A. Smirnova, Iterative Methods for Ill-Posed Problems, Walter de Gruyter GmbH & Co. KG, Berlin, 2011. [3] J. Baumeister, B. Kaltenbacher and A. Leitão, On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations, Inverse Probl. Imaging, 4 (2010), 335-350. doi: 10.3934/ipi.2010.4.335. [4] H. Bui and Q. Nguyen, Thermomechanical Couplings in Solids, North-Holland Publishing Co., Amsterdan, 1987. [5] M. Burger and B. Kaltenbacher, Regularizing Newton-Kaczmarz methods for nonlinear ill-posed problems, SIAM J. Numer. Anal., 44 (2006), 153-182. doi: 10.1137/040613779. [6] M. Crouzeix, Numerical range and functional calculus in Hilbert space, J. Funct. Anal., 244 (2007), 668-690. doi: 10.1016/j.jfa.2006.10.013. [7] 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. [8] A. De Cezaro, J. Baumeister and A. Leitão, Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations, Inverse Probl. Imaging, 5 (2011), 1-17. doi: 10.3934/ipi.2011.5.1. [9] T. Elfving and T. Nikazad, Properties of a class of block-iterative methods, Inverse Problems, 25 (2009), 115011, 13. doi: 10.1088/0266-5611/25/11/115011. [10] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996. [11] M. Haltmeier, A. Leitão and E. Resmerita, On regularization methods of EM-Kaczmarz type, Inverse Problems, 25 (2009), 075008, 17pp. doi: 10.1088/0266-5611/25/7/075008. [12] M. Haltmeier, Convergence analysis of a block iterative version of the loping Landweber-Kaczmarz iteration, Nonlinear Anal., 71 (2009), e2912-e2919. doi: 10.1016/j.na.2009.07.016. [13] M. Haltmeier, R. Kowar, A. Leitão and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations. II. Applications, Inverse Probl. Imaging, 1 (2007), 507-523. doi: 10.3934/ipi.2007.1.507. [14] 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. doi: 10.3934/ipi.2007.1.289. [15] M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37. doi: 10.1007/s002110050158. [16] M. Hanke and W. Niethammer, On the acceleration of Kaczmarz's method for inconsistent linear systems, Linear Algebra Appl., 130 (1990), 83-98, Linear algebra in image reconstruction from projections. doi: 10.1016/0024-3795(90)90207-S. [17] S. Kaczmarz, Angenäherte Auflösung von Systemen linearer Gleichungen, Bull. Internat. Acad. Polon. Sci. A, 1937 (1937), 355-357. [18] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276. [19] N. Kalton, S. Montgomery-Smith, K. Oleszkiewicz and Y. Tomilov, Power-bounded operators and related norm estimates, J. London Math. Soc. (2), 70 (2004), 463-478. doi: 10.1112/S0024610704005514. [20] T. Kato, A generalization of the Heinz inequality, Proc. Japan Acad., 37 (1961), 305-308. doi: 10.3792/pja/1195523678. [21] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition. [22] R. Kowar and O. Scherzer, Convergence analysis of a Landweber-Kaczmarz method for solving nonlinear ill-posed problems, in Ill-posed and inverse problems, VSP, Zeist, (2002), 253-270. [23] Y. Lyubich, Spectral localization, power boundedness and invariant subspaces under Ritt's type condition, Studia Math., 134 (1999), 153-167. [24] S. McCormick, An iterative procedure for the solution of constrained nonlinear equations with application to optimization problems, Numer. Math., 23 (1975), 371-385. doi: 10.1007/BF01437037. [25] 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. [26] K.-H. Meyn, Solution of underdetermined nonlinear equations by stationary iteration methods, Numer. Math., 42 (1983), 161-172. doi: 10.1007/BF01395309. [27] B. Nagy and J. Zemánek, A resolvent condition implying power boundedness, Studia Math., 134 (1999), 143-151. [28] M. Z. Nashed, Continuous and semicontinuous analogues of iterative methods of Cimmino and Kaczmarz with applications to the inverse Radon transform, in Mathematical aspects of computerized tomography (Oberwolfach, 1980), vol. 8 of Lecture Notes in Med. Inform., Springer, Berlin, (1981), 160-178. doi: 10.1007/978-3-642-93157-4_14. [29] F. Natterer, The Mathematics of Computerized Tomography, SIAM, Philadelphia, PA, 2001. doi: 10.1137/1.9780898719284. [30] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. doi: 10.1137/1.9780898718324. [31] O. Nevanlinna, Convergence of Iterations for Linear Equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8547-8. [32] R. Plato and U. Hämarik, On pseudo-optimal parameter choices and stopping rules for regularization methods in Banach spaces, Numer. Funct. Anal. Optim., 17 (1996), 181-195. doi: 10.1080/01630569608816690. [33] R. Plato, Iterative and Other Methods for Linear Ill-Posed Equations, Habilitation thesis, Department of Mathematics, TU Berlin, 1995. [34] A. Rieder, On the regularization of nonlinear ill-posed problems via inexact Newton iterations, Inverse Problems, 15 (1999), 309-327. doi: 10.1088/0266-5611/15/1/028. [35] O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl., 194 (1995), 911-933. doi: 10.1006/jmaa.1995.1335.

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##### References:
 [1] P. K. Anh and C. V. Chung, Parallel regularized Newton method for nonlinear ill-posed equations, Numer. Algorithms, 58 (2011), 379-398. doi: 10.1007/s11075-011-9460-y. [2] A. B. Bakushinsky, M. Y. Kokurin and A. Smirnova, Iterative Methods for Ill-Posed Problems, Walter de Gruyter GmbH & Co. KG, Berlin, 2011. [3] J. Baumeister, B. Kaltenbacher and A. Leitão, On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations, Inverse Probl. Imaging, 4 (2010), 335-350. doi: 10.3934/ipi.2010.4.335. [4] H. Bui and Q. Nguyen, Thermomechanical Couplings in Solids, North-Holland Publishing Co., Amsterdan, 1987. [5] M. Burger and B. Kaltenbacher, Regularizing Newton-Kaczmarz methods for nonlinear ill-posed problems, SIAM J. Numer. Anal., 44 (2006), 153-182. doi: 10.1137/040613779. [6] M. Crouzeix, Numerical range and functional calculus in Hilbert space, J. Funct. Anal., 244 (2007), 668-690. doi: 10.1016/j.jfa.2006.10.013. [7] 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. [8] A. De Cezaro, J. Baumeister and A. Leitão, Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations, Inverse Probl. Imaging, 5 (2011), 1-17. doi: 10.3934/ipi.2011.5.1. [9] T. Elfving and T. Nikazad, Properties of a class of block-iterative methods, Inverse Problems, 25 (2009), 115011, 13. doi: 10.1088/0266-5611/25/11/115011. [10] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996. [11] M. Haltmeier, A. Leitão and E. Resmerita, On regularization methods of EM-Kaczmarz type, Inverse Problems, 25 (2009), 075008, 17pp. doi: 10.1088/0266-5611/25/7/075008. [12] M. Haltmeier, Convergence analysis of a block iterative version of the loping Landweber-Kaczmarz iteration, Nonlinear Anal., 71 (2009), e2912-e2919. doi: 10.1016/j.na.2009.07.016. [13] M. Haltmeier, R. Kowar, A. Leitão and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations. II. Applications, Inverse Probl. Imaging, 1 (2007), 507-523. doi: 10.3934/ipi.2007.1.507. [14] 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. doi: 10.3934/ipi.2007.1.289. [15] M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37. doi: 10.1007/s002110050158. [16] M. Hanke and W. Niethammer, On the acceleration of Kaczmarz's method for inconsistent linear systems, Linear Algebra Appl., 130 (1990), 83-98, Linear algebra in image reconstruction from projections. doi: 10.1016/0024-3795(90)90207-S. [17] S. Kaczmarz, Angenäherte Auflösung von Systemen linearer Gleichungen, Bull. Internat. Acad. Polon. Sci. A, 1937 (1937), 355-357. [18] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276. [19] N. Kalton, S. Montgomery-Smith, K. Oleszkiewicz and Y. Tomilov, Power-bounded operators and related norm estimates, J. London Math. Soc. (2), 70 (2004), 463-478. doi: 10.1112/S0024610704005514. [20] T. Kato, A generalization of the Heinz inequality, Proc. Japan Acad., 37 (1961), 305-308. doi: 10.3792/pja/1195523678. [21] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition. [22] R. Kowar and O. Scherzer, Convergence analysis of a Landweber-Kaczmarz method for solving nonlinear ill-posed problems, in Ill-posed and inverse problems, VSP, Zeist, (2002), 253-270. [23] Y. Lyubich, Spectral localization, power boundedness and invariant subspaces under Ritt's type condition, Studia Math., 134 (1999), 153-167. [24] S. McCormick, An iterative procedure for the solution of constrained nonlinear equations with application to optimization problems, Numer. Math., 23 (1975), 371-385. doi: 10.1007/BF01437037. [25] 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. [26] K.-H. Meyn, Solution of underdetermined nonlinear equations by stationary iteration methods, Numer. Math., 42 (1983), 161-172. doi: 10.1007/BF01395309. [27] B. Nagy and J. Zemánek, A resolvent condition implying power boundedness, Studia Math., 134 (1999), 143-151. [28] M. Z. Nashed, Continuous and semicontinuous analogues of iterative methods of Cimmino and Kaczmarz with applications to the inverse Radon transform, in Mathematical aspects of computerized tomography (Oberwolfach, 1980), vol. 8 of Lecture Notes in Med. Inform., Springer, Berlin, (1981), 160-178. doi: 10.1007/978-3-642-93157-4_14. [29] F. Natterer, The Mathematics of Computerized Tomography, SIAM, Philadelphia, PA, 2001. doi: 10.1137/1.9780898719284. [30] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. doi: 10.1137/1.9780898718324. [31] O. Nevanlinna, Convergence of Iterations for Linear Equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8547-8. [32] R. Plato and U. Hämarik, On pseudo-optimal parameter choices and stopping rules for regularization methods in Banach spaces, Numer. Funct. Anal. Optim., 17 (1996), 181-195. doi: 10.1080/01630569608816690. [33] R. Plato, Iterative and Other Methods for Linear Ill-Posed Equations, Habilitation thesis, Department of Mathematics, TU Berlin, 1995. [34] A. Rieder, On the regularization of nonlinear ill-posed problems via inexact Newton iterations, Inverse Problems, 15 (1999), 309-327. doi: 10.1088/0266-5611/15/1/028. [35] O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl., 194 (1995), 911-933. doi: 10.1006/jmaa.1995.1335.
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