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Lavrentiev's regularization method in Hilbert spaces revisited
1. | Technische Universität Chemnitz, Reichenhainer Str. 41, 09111 Chemnitz, Germany |
2. | Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria |
References:
[1] |
R. G. Airapetyan and A. G. Ramm, Dynamical systems and discrete methods for solving nonlinear illposed problems, Applied Mathematical Reviews, vol. 1 (Ed.: G. Anastassion). World Scientific, Singapore, 2000, 491-536.
doi: 10.1142/9789812792686_0012. |
[2] |
Y. Alber and I. Ryazantseva, Nonlinear Ill-posed Problems of Monotone Type, Springer, Dordrecht, 2006. |
[3] |
R. Andreev, P. Elbau, M. V. de Hoop, L. Qiu and O. Scherzer, Generalized convergence rates results for linear inverse problems in Hilbert spaces, Numer. Funct. Anal. Optim., 36 (2015), 549-566.
doi: 10.1080/01630563.2015.1021422. |
[4] |
S. W. Anzengruber, B. Hofmann and P. Mathé, Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces, Applicable Analysis, 93 (2014), 1382-1400.
doi: 10.1080/00036811.2013.833326. |
[5] |
I. K. Argyros, Y. J. Cho and S. George, Expanding the applicability of Lavrentiev regularization methods for ill-posed problems, Boundary Value Problems, 114 (2013), 15pp.
doi: 10.1186/1687-2770-2013-114. |
[6] |
A. Bakushinskii and A. Goncharskii, Ill-Posed Problems: Theory and Applications, Kluwer, Dordrecht 1994.
doi: 10.1007/978-94-011-1026-6. |
[7] |
A. B. Bakushinsky and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Springer, Dordrecht, 2004. |
[8] |
A. Bakushinsky and A. Smirnova, A posteriori stopping rule for regularized fixed point iterations, Nonlinear Anal., 64 (2006), 1255-1261.
doi: 10.1016/j.na.2005.06.031. |
[9] |
A. Bakushinsky and A. Smirnova, Iterative regularization and generalized discrepancy principle for monotone operator equations, Numer. Funct. Anal. Optim., 28 (2007), 13-25.
doi: 10.1080/01630560701190315. |
[10] |
R. I. Boţ and B. Hofmann, An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems, Journal of Integral Equations and Applications, 22 (2010), 369-392.
doi: 10.1216/JIE-2010-22-3-369. |
[11] |
D. Düvelmeyer, B. Hofmann and M. Yamamoto, Range inclusions and approximate source conditions with general benchmark functions, Numer. Funct. Anal. Optim., 28 (2007), 1245-1261.
doi: 10.1080/01630560701749649. |
[12] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996, 2nd Edition 2000.
doi: 10.1007/978-94-009-1740-8. |
[13] |
J. Flemming, B. Hofmann and P. Mathé, Sharp converse results for the regularization error using distance functions, Inverse Problems, 27 (2011), 025006 (18pp).
doi: 10.1088/0266-5611/27/2/025006. |
[14] |
S. George, S. Pareth and M. Kunhanandan, Newton Lavrentiev regularization for ill-posed operator equations in Hilbert scales, Appl. Math. Comput., 219 (2013), 11191-11197.
doi: 10.1016/j.amc.2013.05.021. |
[15] |
R. Gorenflo, Yu. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev space, Fract. Calc. Appl. Anal., 18 (2015), 799-820.
doi: 10.1515/fca-2015-0048. |
[16] |
M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, Vol. 169, Birkhäuser Verlag, Basel, 2006.
doi: 10.1007/3-7643-7698-8. |
[17] |
T. Hein and B. Hofmann, Approximate source conditions for nonlinear ill-posed problems - chances and limitations, Inverse Problems, 25 (2009), 035003 (16pp).
doi: 10.1088/0266-5611/25/3/035003. |
[18] |
T. Hohage and F. Weidling, Verification of a variational source condition for acoustic inverse medium scattering problems, Inverse Problems, 31 (2015), 075006 (14pp).
doi: 10.1088/0266-5611/31/7/075006. |
[19] |
B. Hofmann, Approximate source conditions in Tikhonov-Phillips regularization and consequences for inverse problems with multiplication operators, Math. Methods Appl. Sci., 29 (2006), 351-371.
doi: 10.1002/mma.686. |
[20] |
B. Hofmann, On smoothness concepts in regularization for nonlinear inverse problems, in Banach spaces. Chapter 8 in Mathematical and Computational Modeling: With Applications in Natural and Social Sciences, Engineering, and the Arts (Ed.: R. Melnik). John Wiley, New Jersey 2015, pp. 192-221. |
[21] |
B. Hofmann, D. Düvelmeyer and K. Krumbiegel, Approximate source conditions in Tikhonov regularization - new analytical results and some numerical studies, Mathematical Modelling and Analysis, 11 (2006), 41-56. |
[22] |
B. Hofmann, B. Kaltenbacher, C. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.
doi: 10.1088/0266-5611/23/3/009. |
[23] |
B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems, 28 (2012), 104006 (17pp).
doi: 10.1088/0266-5611/28/10/104006. |
[24] |
B. Hofmann, P. Mathé and S. V. Pereverzev, Regularization by projection: Approximation theoretic aspects and distance functions, J. Inverse Ill-Posed Probl., 15 (2007), 527-545.
doi: 10.1515/jiip.2007.029. |
[25] |
J. Janno, Lavrent'ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Problems, 16 (2000), 333-348.
doi: 10.1088/0266-5611/16/2/305. |
[26] |
B. Kaltenbacher, On Broyden's method for nonlinear ill-posed problems, Numerical Functional Analysis and Optimization, 19 (1998), 807-833.
doi: 10.1080/01630569808816860. |
[27] |
M. M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics, Springer, New York, 1967. |
[28] |
F. Liu and M. Z. Nashed, Convergence of regularized solutions of nonlinear ill-posed problems with monotone operators, In: Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Vol. 177, pp. 353-361. Dekker, New York, 1996. |
[29] |
P. Mahale and M. T. Nair, Lavrentiev regularization of nonlinear ill-posed equations under general source condition, J. Nonlinear Anal. Optim., 4 (2013), 193-204. |
[30] |
P. Mathé, The Lepskiĭ principle revisited, Inverse Problems, 22 (2006), L11-L15.
doi: 10.1088/0266-5611/22/3/L02. |
[31] |
R. Plato, Iterative and Other Methods for Linear Ill-Posed Equations, Habilitation Thesis, Techn. Univ. Berlin, 1995. |
[32] |
R. Plato, P. Mathé and B. Hofmann, Optimal rates for Lavrentiev regularization with adjoint source conditions, Preprint 2016-03, Preprintreihe der Fakultät für Mathematik, TU Chemnitz, Germany, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-199010. |
[33] |
J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. 87. Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[34] |
O. Scherzer, H. W. Engl and K. Kunisch, Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM J. Numer. Anal., 30 (1993), 1796-1838.
doi: 10.1137/0730091. |
[35] |
T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, volume 10 of Radon Ser. Comput. Appl. Math., Walter de Gruyter, Berlin/Boston, 2012.
doi: 10.1515/9783110255720. |
[36] |
E. V. Semenova, Lavrentiev regularization and balancing principle for solving ill-posed problems with monotone operators, Comput. Methods Appl. Math., 10 (2010), 444-454.
doi: 10.2478/cmam-2010-0026. |
[37] |
U. Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems, 18 (2002), 191-207.
doi: 10.1088/0266-5611/18/1/313. |
[38] |
U. Tautenhahn, Lavrentiev regularization of nonlinear ill-posed problems, Vietnam J. Math., 32 (2004), 29-41. |
[39] |
U. Tautenhahn and Q. Jin, Tikhonov regularization and a posteriori rules for solving nonlinear ill-posed problems, Inverse Problems, 19 (2003), 1-21.
doi: 10.1088/0266-5611/19/1/301. |
[40] |
F. Werner and T. Hohage, Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data, Inverse Problems, 28 (2012), 104004 (15pp).
doi: 10.1088/0266-5611/28/10/104004. |
show all references
References:
[1] |
R. G. Airapetyan and A. G. Ramm, Dynamical systems and discrete methods for solving nonlinear illposed problems, Applied Mathematical Reviews, vol. 1 (Ed.: G. Anastassion). World Scientific, Singapore, 2000, 491-536.
doi: 10.1142/9789812792686_0012. |
[2] |
Y. Alber and I. Ryazantseva, Nonlinear Ill-posed Problems of Monotone Type, Springer, Dordrecht, 2006. |
[3] |
R. Andreev, P. Elbau, M. V. de Hoop, L. Qiu and O. Scherzer, Generalized convergence rates results for linear inverse problems in Hilbert spaces, Numer. Funct. Anal. Optim., 36 (2015), 549-566.
doi: 10.1080/01630563.2015.1021422. |
[4] |
S. W. Anzengruber, B. Hofmann and P. Mathé, Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces, Applicable Analysis, 93 (2014), 1382-1400.
doi: 10.1080/00036811.2013.833326. |
[5] |
I. K. Argyros, Y. J. Cho and S. George, Expanding the applicability of Lavrentiev regularization methods for ill-posed problems, Boundary Value Problems, 114 (2013), 15pp.
doi: 10.1186/1687-2770-2013-114. |
[6] |
A. Bakushinskii and A. Goncharskii, Ill-Posed Problems: Theory and Applications, Kluwer, Dordrecht 1994.
doi: 10.1007/978-94-011-1026-6. |
[7] |
A. B. Bakushinsky and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Springer, Dordrecht, 2004. |
[8] |
A. Bakushinsky and A. Smirnova, A posteriori stopping rule for regularized fixed point iterations, Nonlinear Anal., 64 (2006), 1255-1261.
doi: 10.1016/j.na.2005.06.031. |
[9] |
A. Bakushinsky and A. Smirnova, Iterative regularization and generalized discrepancy principle for monotone operator equations, Numer. Funct. Anal. Optim., 28 (2007), 13-25.
doi: 10.1080/01630560701190315. |
[10] |
R. I. Boţ and B. Hofmann, An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems, Journal of Integral Equations and Applications, 22 (2010), 369-392.
doi: 10.1216/JIE-2010-22-3-369. |
[11] |
D. Düvelmeyer, B. Hofmann and M. Yamamoto, Range inclusions and approximate source conditions with general benchmark functions, Numer. Funct. Anal. Optim., 28 (2007), 1245-1261.
doi: 10.1080/01630560701749649. |
[12] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996, 2nd Edition 2000.
doi: 10.1007/978-94-009-1740-8. |
[13] |
J. Flemming, B. Hofmann and P. Mathé, Sharp converse results for the regularization error using distance functions, Inverse Problems, 27 (2011), 025006 (18pp).
doi: 10.1088/0266-5611/27/2/025006. |
[14] |
S. George, S. Pareth and M. Kunhanandan, Newton Lavrentiev regularization for ill-posed operator equations in Hilbert scales, Appl. Math. Comput., 219 (2013), 11191-11197.
doi: 10.1016/j.amc.2013.05.021. |
[15] |
R. Gorenflo, Yu. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev space, Fract. Calc. Appl. Anal., 18 (2015), 799-820.
doi: 10.1515/fca-2015-0048. |
[16] |
M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, Vol. 169, Birkhäuser Verlag, Basel, 2006.
doi: 10.1007/3-7643-7698-8. |
[17] |
T. Hein and B. Hofmann, Approximate source conditions for nonlinear ill-posed problems - chances and limitations, Inverse Problems, 25 (2009), 035003 (16pp).
doi: 10.1088/0266-5611/25/3/035003. |
[18] |
T. Hohage and F. Weidling, Verification of a variational source condition for acoustic inverse medium scattering problems, Inverse Problems, 31 (2015), 075006 (14pp).
doi: 10.1088/0266-5611/31/7/075006. |
[19] |
B. Hofmann, Approximate source conditions in Tikhonov-Phillips regularization and consequences for inverse problems with multiplication operators, Math. Methods Appl. Sci., 29 (2006), 351-371.
doi: 10.1002/mma.686. |
[20] |
B. Hofmann, On smoothness concepts in regularization for nonlinear inverse problems, in Banach spaces. Chapter 8 in Mathematical and Computational Modeling: With Applications in Natural and Social Sciences, Engineering, and the Arts (Ed.: R. Melnik). John Wiley, New Jersey 2015, pp. 192-221. |
[21] |
B. Hofmann, D. Düvelmeyer and K. Krumbiegel, Approximate source conditions in Tikhonov regularization - new analytical results and some numerical studies, Mathematical Modelling and Analysis, 11 (2006), 41-56. |
[22] |
B. Hofmann, B. Kaltenbacher, C. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.
doi: 10.1088/0266-5611/23/3/009. |
[23] |
B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems, 28 (2012), 104006 (17pp).
doi: 10.1088/0266-5611/28/10/104006. |
[24] |
B. Hofmann, P. Mathé and S. V. Pereverzev, Regularization by projection: Approximation theoretic aspects and distance functions, J. Inverse Ill-Posed Probl., 15 (2007), 527-545.
doi: 10.1515/jiip.2007.029. |
[25] |
J. Janno, Lavrent'ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Problems, 16 (2000), 333-348.
doi: 10.1088/0266-5611/16/2/305. |
[26] |
B. Kaltenbacher, On Broyden's method for nonlinear ill-posed problems, Numerical Functional Analysis and Optimization, 19 (1998), 807-833.
doi: 10.1080/01630569808816860. |
[27] |
M. M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics, Springer, New York, 1967. |
[28] |
F. Liu and M. Z. Nashed, Convergence of regularized solutions of nonlinear ill-posed problems with monotone operators, In: Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Vol. 177, pp. 353-361. Dekker, New York, 1996. |
[29] |
P. Mahale and M. T. Nair, Lavrentiev regularization of nonlinear ill-posed equations under general source condition, J. Nonlinear Anal. Optim., 4 (2013), 193-204. |
[30] |
P. Mathé, The Lepskiĭ principle revisited, Inverse Problems, 22 (2006), L11-L15.
doi: 10.1088/0266-5611/22/3/L02. |
[31] |
R. Plato, Iterative and Other Methods for Linear Ill-Posed Equations, Habilitation Thesis, Techn. Univ. Berlin, 1995. |
[32] |
R. Plato, P. Mathé and B. Hofmann, Optimal rates for Lavrentiev regularization with adjoint source conditions, Preprint 2016-03, Preprintreihe der Fakultät für Mathematik, TU Chemnitz, Germany, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-199010. |
[33] |
J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. 87. Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[34] |
O. Scherzer, H. W. Engl and K. Kunisch, Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM J. Numer. Anal., 30 (1993), 1796-1838.
doi: 10.1137/0730091. |
[35] |
T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, volume 10 of Radon Ser. Comput. Appl. Math., Walter de Gruyter, Berlin/Boston, 2012.
doi: 10.1515/9783110255720. |
[36] |
E. V. Semenova, Lavrentiev regularization and balancing principle for solving ill-posed problems with monotone operators, Comput. Methods Appl. Math., 10 (2010), 444-454.
doi: 10.2478/cmam-2010-0026. |
[37] |
U. Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems, 18 (2002), 191-207.
doi: 10.1088/0266-5611/18/1/313. |
[38] |
U. Tautenhahn, Lavrentiev regularization of nonlinear ill-posed problems, Vietnam J. Math., 32 (2004), 29-41. |
[39] |
U. Tautenhahn and Q. Jin, Tikhonov regularization and a posteriori rules for solving nonlinear ill-posed problems, Inverse Problems, 19 (2003), 1-21.
doi: 10.1088/0266-5611/19/1/301. |
[40] |
F. Werner and T. Hohage, Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data, Inverse Problems, 28 (2012), 104004 (15pp).
doi: 10.1088/0266-5611/28/10/104004. |
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