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Optimal rates for Lavrentiev regularization with adjoint source conditions

Authors: Robert Plato, Peter Mathé and Bernd Hofmann
Journal: Math. Comp. 87 (2018), 785-801
MSC (2010): Primary 65N20; Secondary 45Q05
Published electronically: June 13, 2017
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Abstract: There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive, then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato [J. Math. Soc. Japan 13(1961), no. 3, 247-274], we establish power type convergence rates for this case. By measuring the optimality of such rates in terms of limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of positive semidefinite selfadjoint operators.

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  • [1] Yakov Alber and Irina Ryazantseva, Nonlinear Ill-posed Problems of Monotone Type, Springer, Dordrecht, 2006. MR 2213033
  • [2] John B. Conway, A Course in Operator Theory, Graduate Studies in Mathematics, vol. 21, American Mathematical Society, Providence, RI, 2000. MR 1721402
  • [3] Heinz W. Engl, Martin Hanke, and Andreas Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1408680
  • [4] Jens Flemming, Bernd Hofmann, and Peter Mathé, Sharp converse results for the regularization error using distance functions, Inverse Problems 27 (2011), no. 2, 025006, 18. MR 2754351,
  • [5] Rudolf Gorenflo, Yuri Luchko, and Masahiro Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal. 18 (2015), no. 3, 799-820. MR 3351501,
  • [6] Rudolf Gorenflo and Masahiro Yamamoto, Operator-theoretic treatment of linear Abel integral equations of first kind, Japan J. Indust. Appl. Math. 16 (1999), no. 1, 137-161. MR 1676342,
  • [7] C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Research Notes in Mathematics, vol. 105, Pitman, Boston, MA, 1984.
  • [8] Markus Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser Verlag, Basel, 2006. MR 2244037
  • [9] Bernd Hofmann, Barbara Kaltenbacher, and Elena Resmerita, Lavrentiev's regularization method in Hilbert spaces revisited, Inverse Probl. Imaging 10 (2016), no. 3, 741-764. MR 3562269,
  • [10] Bernd Hofmann and Peter Mathé, Analysis of profile functions for general linear regularization methods, SIAM J. Numer. Anal. 45 (2007), no. 3, 1122-1141. MR 2318806,
  • [11] Tosio Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), 246-274. MR 0138005
  • [12] M.A. Krasnosel'skiĭ, P.P. Zabreĭko, E.I. Pustyl'nik, and P.E. Sobolevskiĭ, Integral Operators in Spaces of Summable Functions, 1 ed., Noordjoff Int. Publ., Leyden, 1976.
  • [13] M. M. Lavrentiev, Some Ill-Posed Problems of Mathematical Physics (Russian), Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1962.
  • [14] Fengshan Liu and M. Zuhair Nashed, Convergence of regularized solutions of nonlinear ill-posed problems with monotone operators, Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., vol. 177, Dekker, New York, 1996, pp. 353-361. MR 1371608
  • [15] Alan McIntosh, On the comparability of $ A^{1/2}$ and $ A^{\ast 1/2}$, Proc. Amer. Math. Soc. 32 (1972), 430-434. MR 0290169
  • [16] Albrecht Pietsch, Operator Ideals, Mathematische Monographien [Mathematical Monographs], vol. 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 519680
  • [17] R. Plato, Iterative and Parametric Methods for Linear Ill-posed Equations, Habilitation thesis, Department of Mathematics, Technical University Berlin, 1995.
  • [18] Robert Plato, On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations, Numer. Math. 75 (1996), no. 1, 99-120. MR 1417865,
  • [19] Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. Edited and with a foreword by S. M. Nikolskiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
  • [20] Hiroki Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. Translated from the Japanese by N. Mugibayashi and H. Haneda. MR 533824
  • [21] U. Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems 18 (2002), no. 1, 191-207. MR 1893590,

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Additional Information

Robert Plato
Affiliation: Department of Mathematics, University of Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany

Peter Mathé
Affiliation: Weierstrass Institute Berlin, Mohrenstr. 39, 10117 Berlin, Germany

Bernd Hofmann
Affiliation: Faculty of Mathematics, TU Chemnitz, 09107 Chemnitz, Germany

Received by editor(s): March 14, 2016
Received by editor(s) in revised form: August 4, 2016, and October 23, 2016
Published electronically: June 13, 2017
Additional Notes: The research of the third author was supported by the German Research Foundation (DFG) under grant HO 1454/8-2.
Article copyright: © Copyright 2017 American Mathematical Society

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