American Institute of Mathematical Sciences

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

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

Stefan Kindermann 1, and  Andreas Neubauer 2,

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 $T$*$T$.
Keywords: heuristic parameter selection., Quasioptimality criterion.
Mathematics Subject Classification: 47A5.
Citation: Stefan Kindermann, Andreas Neubauer. On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Problems & 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.  Google Scholar

[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.  Google Scholar

[3]

F. Bauer and S. Kindermann, The quasi-optimality criterion for classical inverse problems, Inverse Problems, 24 (2008). Google Scholar

[4]

H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996.  Google Scholar

[5]

M. Hanke and P. C. Hansen, Regularization methods for large-scale problems, Surveys Math. Indust., 3 (1993), 253-315.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[8]

A. Neubauer, On converse and saturation results for regularization methods, in "Beiträge zur Angewandten Analysis und Informatik,'' Shaker, Aachen, (1994), 262-270.  Google Scholar

[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.  Google Scholar

[10]

A. N. Tikhonov and V. Arsenin, "Solutions of Ill-Posed Problems,'' Wiley, New York, 1977.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

F. Bauer and S. Kindermann, The quasi-optimality criterion for classical inverse problems, Inverse Problems, 24 (2008). Google Scholar

[4]

H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996.  Google Scholar

[5]

M. Hanke and P. C. Hansen, Regularization methods for large-scale problems, Surveys Math. Indust., 3 (1993), 253-315.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[8]

A. Neubauer, On converse and saturation results for regularization methods, in "Beiträge zur Angewandten Analysis und Informatik,'' Shaker, Aachen, (1994), 262-270.  Google Scholar

[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.  Google Scholar

[10]

A. N. Tikhonov and V. Arsenin, "Solutions of Ill-Posed Problems,'' Wiley, New York, 1977.  Google Scholar

[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. Google Scholar

[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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Control Theory, 2013, 2 (4) : 741-747. doi: 10.3934/eect.2013.2.741
[14]

Sebastian Springer, Heikki Haario, Vladimir Shemyakin, Leonid Kalachev, Denis Shchepakin. Robust parameter estimation of chaotic systems. Inverse Problems & Imaging, 2019, 13 (6) : 1189-1212. doi: 10.3934/ipi.2019053
[15]

Azmy S. Ackleh, Jeremy J. Thibodeaux. Parameter estimation in a structured erythropoiesis model. Mathematical Biosciences & Engineering, 2008, 5 (4) : 601-616. doi: 10.3934/mbe.2008.5.601
[16]

Tolulope Fadina, Ariel Neufeld, Thorsten Schmidt. Affine processes under parameter uncertainty. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 5-. doi: 10.1186/s41546-019-0039-1
[17]

Tzu-Li Chen, James T. Lin, Shu-Cherng Fang. A shadow-price based heuristic for capacity planning of TFT-LCD manufacturing. Journal of Industrial & Management Optimization, 2010, 6 (1) : 209-239. doi: 10.3934/jimo.2010.6.209
[18]

Mostafa Abouei Ardakan, A. Kourank Beheshti, S. Hamid Mirmohammadi, Hamed Davari Ardakani. A hybrid meta-heuristic algorithm to minimize the number of tardy jobs in a dynamic two-machine flow shop problem. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 465-480. doi: 10.3934/naco.2017029
[19]

Liangliang Sun, Fangjun Luan, Yu Ying, Kun Mao. Rescheduling optimization of steelmaking-continuous casting process based on the Lagrangian heuristic algorithm. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1431-1448. doi: 10.3934/jimo.2016081
[20]

Mao Chen, Xiangyang Tang, Zhizhong Zeng, Sanya Liu. An efficient heuristic algorithm for two-dimensional rectangular packing problem with central rectangle. Journal of Industrial & Management Optimization, 2020, 16 (1) : 495-510. doi: 10.3934/jimo.2018164

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (153)
  • HTML views (0)
  • Cited by (45)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences