# American Institute of Mathematical Sciences

May  2008, 2(2): 291-299. doi: 10.3934/ipi.2008.2.291

## On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization

 1 Industrial Mathematics Institute, Johannes Kepler University Linz, A-4040 Linz, Austria 2 Industrial Mathematics Institute, Johannes Kepler University Linz A-4040 Linz, Austria

Received  December 2007 Revised  March 2008 Published  April 2008

In this paper we derive convergence and convergence rates results of the quasioptimality criterion for (iterated) Tikhonov regularization. We prove convergence and suboptimal rates under a qualitative condition on the decay of the noise with respect to the spectral family of $T$$T$*. Moreover, optimal rates are obtained if the exact solution satisfies a decay condition with respect to the spectral family of $T$*$T$.
Citation: Stefan Kindermann, Andreas Neubauer. On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Problems and Imaging, 2008, 2 (2) : 291-299. doi: 10.3934/ipi.2008.2.291
##### References:
 [1] M. A. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions, Trans. Am. Math. Soc., 320 (1990), 727-735. doi: 10.2307/2001699. [2] A. B. Bakushinskii, Remarks on the choice of regularization parameter from quasioptimality and relation tests, (Russian) Zh. Vychisl. Mat. i Mat. Fiz., 24 (1984), 1258-1259. [3] F. Bauer and S. Kindermann, The quasi-optimality criterion for classical inverse problems, Inverse Problems, 24 (2008). [4] H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. [5] M. Hanke and P. C. Hansen, Regularization methods for large-scale problems, Surveys Math. Indust., 3 (1993), 253-315. [6] A. S. Leonov, On the choice of regularization parameters by means of the quasi-optimality and ratio criteria, Soviet Math. Dokl., 19 (1978), 537-540. [7] A. S. Leonov, On the accuracy of Tikhonov regularizing algorithms and quasioptimal selection of a regularization parameter, Soviet Math. Dokl., 44 (1992), 711-716. [8] A. Neubauer, On converse and saturation results for regularization methods, in "Beiträge zur Angewandten Analysis und Informatik,'' Shaker, Aachen, (1994), 262-270. [9] A. Neubauer, On converse and saturation results for Tikhonov regularization of linear ill-posed problems, SIAM J. Numer. Anal., 34 (1997), 517-527. doi: 10.1137/S0036142993253928. [10] A. N. Tikhonov and V. Arsenin, "Solutions of Ill-Posed Problems,'' Wiley, New York, 1977. [11] A. N. Tikhonov, V. B. Glasko and Y. Kriksin, On the question of quasioptimal choice of a regularized approximation, Soviet Math. Dokl., 20 (1979), 1036-1040.

show all references

##### References:
 [1] M. A. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions, Trans. Am. Math. Soc., 320 (1990), 727-735. doi: 10.2307/2001699. [2] A. B. Bakushinskii, Remarks on the choice of regularization parameter from quasioptimality and relation tests, (Russian) Zh. Vychisl. Mat. i Mat. Fiz., 24 (1984), 1258-1259. [3] F. Bauer and S. Kindermann, The quasi-optimality criterion for classical inverse problems, Inverse Problems, 24 (2008). [4] H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. [5] M. Hanke and P. C. Hansen, Regularization methods for large-scale problems, Surveys Math. Indust., 3 (1993), 253-315. [6] A. S. Leonov, On the choice of regularization parameters by means of the quasi-optimality and ratio criteria, Soviet Math. Dokl., 19 (1978), 537-540. [7] A. S. Leonov, On the accuracy of Tikhonov regularizing algorithms and quasioptimal selection of a regularization parameter, Soviet Math. Dokl., 44 (1992), 711-716. [8] A. Neubauer, On converse and saturation results for regularization methods, in "Beiträge zur Angewandten Analysis und Informatik,'' Shaker, Aachen, (1994), 262-270. [9] A. Neubauer, On converse and saturation results for Tikhonov regularization of linear ill-posed problems, SIAM J. Numer. Anal., 34 (1997), 517-527. doi: 10.1137/S0036142993253928. [10] A. N. Tikhonov and V. Arsenin, "Solutions of Ill-Posed Problems,'' Wiley, New York, 1977. [11] A. N. Tikhonov, V. B. Glasko and Y. Kriksin, On the question of quasioptimal choice of a regularized approximation, Soviet Math. Dokl., 20 (1979), 1036-1040.
 [1] Ross Callister, Duc-Son Pham, Mihai Lazarescu. Using distribution analysis for parameter selection in repstream. Mathematical Foundations of Computing, 2019, 2 (3) : 215-250. doi: 10.3934/mfc.2019015 [2] K. Schittkowski. Optimal parameter selection in support vector machines. Journal of Industrial and Management Optimization, 2005, 1 (4) : 465-476. doi: 10.3934/jimo.2005.1.465 [3] J. Mead. $\chi^2$ test for total variation regularization parameter selection. Inverse Problems and Imaging, 2020, 14 (3) : 401-421. doi: 10.3934/ipi.2020019 [4] R. S. Johnson. A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1497-1522. doi: 10.3934/cpaa.2012.11.1497 [5] Xiangyu Gao, Yong Sun. A new heuristic algorithm for laser antimissile strategy optimization. Journal of Industrial and Management Optimization, 2012, 8 (2) : 457-468. doi: 10.3934/jimo.2012.8.457 [6] Roman Czapla, Vladimir V. Mityushev. A criterion of collective behavior of bacteria. Mathematical Biosciences & Engineering, 2017, 14 (1) : 277-287. doi: 10.3934/mbe.2017018 [7] Kemal Kilic, Menekse G. Saygi, Semih O. Sezer. Exact and heuristic methods for personalized display advertising in virtual reality platforms. Journal of Industrial and Management Optimization, 2019, 15 (2) : 833-854. doi: 10.3934/jimo.2018073 [8] Axel Kohnert, Johannes Zwanzger. New linear codes with prescribed group of automorphisms found by heuristic search. Advances in Mathematics of Communications, 2009, 3 (2) : 157-166. doi: 10.3934/amc.2009.3.157 [9] Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial and Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1 [10] Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems and Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036 [11] Jürgen Scheurle, Stephan Schmitz. A criterion for asymptotic straightness of force fields. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 777-792. doi: 10.3934/dcdsb.2010.14.777 [12] Samir EL Mourchid. On a hypercylicity criterion for strongly continuous semigroups. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 271-275. doi: 10.3934/dcds.2005.13.271 [13] Hans Weinberger. The approximate controllability of a model for mutant selection. Evolution Equations and Control Theory, 2013, 2 (4) : 741-747. doi: 10.3934/eect.2013.2.741 [14] Angelo Antoci, Marcello Galeotti, Mauro Sodini. Environmental degradation and indeterminacy of equilibrium selection. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5755-5767. doi: 10.3934/dcdsb.2021179 [15] Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics and Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020 [16] F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74 [17] Vitaly Bergelson, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Florian K. Richter. A generalization of Kátai's orthogonality criterion with applications. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2581-2612. doi: 10.3934/dcds.2019108 [18] Emanuela Caliceti, Sandro Graffi. An existence criterion for the $\mathcal{PT}$-symmetric phase transition. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1955-1967. doi: 10.3934/dcdsb.2014.19.1955 [19] Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439 [20] Sebastian Springer, Heikki Haario, Vladimir Shemyakin, Leonid Kalachev, Denis Shchepakin. Robust parameter estimation of chaotic systems. Inverse Problems and Imaging, 2019, 13 (6) : 1189-1212. doi: 10.3934/ipi.2019053

2021 Impact Factor: 1.483