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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 |
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. |
show all references
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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