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

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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
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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 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.

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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.

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