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

Previous Article
Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise
IPI Home
This Issue
Next Article
Local stability for soft obstacles by a single measurement

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

Abstract
Related Papers

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. 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]	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
[5]	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
[6]	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
[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]	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]	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
[12]	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
[13]	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
[14]	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
[15]	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
[16]	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, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018164
[17]	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
[18]	Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics & Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020
[19]	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
[20]	Emanuela Caliceti, Sandro Graffi. An existence criterion for the $\mathcal{PT}$-symmetric phase transition. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1955-1967. doi: 10.3934/dcdsb.2014.19.1955

2017 Impact Factor: 1.465

American Institute of Mathematical Sciences