Lavrentiev's regularization method in Hilbert spaces revisited

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August 2016, 10(3): 741-764. doi: 10.3934/ipi.2016019

Lavrentiev's regularization method in Hilbert spaces revisited

Bernd Hofmann ^1, , Barbara Kaltenbacher ^2, and Elena Resmerita ^2,

1.	Technische Universität Chemnitz, Reichenhainer Str. 41, 09111 Chemnitz, Germany
2.	Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria

Received June 2015 Revised March 2016 Published August 2016

Abstract
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In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the derivative. There are plenty of qualitative and quantitative convergence results in the literature, both in Hilbert and Banach spaces. Our aim here is mainly to contribute to convergence rates results in Hilbert spaces based on some types of error estimates derived under various source conditions and to interpret them in some settings. In particular, we propose and investigate new variational source conditions adapted to these Lavrentiev-type techniques. Another focus of this paper is to exploit the concept of approximate source conditions.

Keywords: source conditions, Lavrentiev regularization, convergence analysis, regularization parameter choice., monotone operators.

Mathematics Subject Classification: Primary: 65J22; Secondary: 45Q05, 35R30, 47H0.

Citation: Bernd Hofmann, Barbara Kaltenbacher, Elena Resmerita. Lavrentiev's regularization method in Hilbert spaces revisited. Inverse Problems and Imaging, 2016, 10 (3) : 741-764. doi: 10.3934/ipi.2016019

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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