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 & 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. 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.

[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, (1996).

[5]

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

[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.

[7]

A. S. Leonov, On the accuracy of Tikhonov regularizing algorithms and quasioptimal selection of a regularization parameter,, Soviet Math. Dokl., 44 (1992), 711.

[8]

A. Neubauer, On converse and saturation results for regularization methods,, in, (1994), 262.

[9]

A. Neubauer, On converse and saturation results for Tikhonov regularization of linear ill-posed problems,, SIAM J. Numer. Anal., 34 (1997), 517. doi: 10.1137/S0036142993253928.

[10]

A. N. Tikhonov and V. Arsenin, "Solutions of Ill-Posed Problems,'', Wiley, (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.

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. 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.

[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, (1996).

[5]

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

[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.

[7]

A. S. Leonov, On the accuracy of Tikhonov regularizing algorithms and quasioptimal selection of a regularization parameter,, Soviet Math. Dokl., 44 (1992), 711.

[8]

A. Neubauer, On converse and saturation results for regularization methods,, in, (1994), 262.

[9]

A. Neubauer, On converse and saturation results for Tikhonov regularization of linear ill-posed problems,, SIAM J. Numer. Anal., 34 (1997), 517. doi: 10.1137/S0036142993253928.

[10]

A. N. Tikhonov and V. Arsenin, "Solutions of Ill-Posed Problems,'', Wiley, (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.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[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]

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

[8]

Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036

[9]

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

[10]

Samir EL Mourchid. On a hypercylicity criterion for strongly continuous semigroups. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 271-275. doi: 10.3934/dcds.2005.13.271

[11]

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

[12]

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

[13]

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

[14]

Ming-Yong Lai, Chang-Shi Liu, Xiao-Jiao Tong. A two-stage hybrid meta-heuristic for pickup and delivery vehicle routing problem with time windows. Journal of Industrial & Management Optimization, 2010, 6 (2) : 435-451. doi: 10.3934/jimo.2010.6.435

[15]

Ke Ruan, Masao Fukushima. Robust portfolio selection with a combined WCVaR and factor model. Journal of Industrial & Management Optimization, 2012, 8 (2) : 343-362. doi: 10.3934/jimo.2012.8.343

[16]

Reinhard Bürger. A survey of migration-selection models in population genetics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 883-959. doi: 10.3934/dcdsb.2014.19.883

[17]

Ying Hao, Fanwen Meng. A new method on gene selection for tissue classification. Journal of Industrial & Management Optimization, 2007, 3 (4) : 739-748. doi: 10.3934/jimo.2007.3.739

[18]

Sebastian Bonhoeffer, Pia Abel zur Wiesch, Roger D. Kouyos. Rotating antibiotics does not minimize selection for resistance. Mathematical Biosciences & Engineering, 2010, 7 (4) : 919-922. doi: 10.3934/mbe.2010.7.919

[19]

Renato Bruni, Gianpiero Bianchi, Alessandra Reale. A combinatorial optimization approach to the selection of statistical units. Journal of Industrial & Management Optimization, 2016, 12 (2) : 515-527. doi: 10.3934/jimo.2016.12.515

[20]

P. Magal, G. F. Webb. Mutation, selection, and recombination in a model of phenotype evolution. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 221-236. doi: 10.3934/dcds.2000.6.221

2016 Impact Factor: 1.094

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]