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 & 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

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