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Morozov’s discrepancy principle under general source conditions. (English) Zbl 1037.65057

The authors study ill-posed problems \(Ax=b\) in a Hilbert space setting where instead of the exact data \(y\) some noisy data \(y^\delta\) are given such that \(\| y-y^\delta|\leq \delta\). They are interested in order optimality results of the regularized approximation of the minimum norm solution of the problem under the general source condition \[ x^+=[\varphi (A^*A)]^{1/2}v \quad {\text{with}}\quad \| v\| \leq E. \] Earlier U. Tautenhahn [Numer. Funct. Anal. Optimization 19, 377–398 (1998; Zbl 0907.65049)] has shown that the classical Tikhonov regularization method combined with a special choice of the regularization parameter \(\alpha\) defined by the function \(\phi\) is optimal. Here the authors investigate regularization methods represented in the general form \[ x^{\delta}_{\alpha}=g_{\alpha}(A^*A)A^{*}y^{\delta} \] and combined with Morozov’s discrepancy principle (independent of the function \(\varphi\)). They have found some analytical properties of the functions \(\varphi\) and \(d_{\alpha}\) that provide the order optimality of this simple regularization technique.

MSC:

65J10 Numerical solutions to equations with linear operators
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47A52 Linear operators and ill-posed problems, regularization

Citations:

Zbl 0907.65049
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References:

[1] Engl, H. W., Hanke, M. and A. Neubauer: Regularization of Inverse Problems. Dordrecht: Kluwer 1996.
[2] Groetsch, C. W.: The Theory of Tikhonov Regularization for Fredholm Equa- tions of the First Kind. Boston: Pitman 1984. · Zbl 0545.65034
[3] Hanke, M. and P. C. Hansen: Regularization methods for large-scale problems. Surv. Math. Ind. 3 (1993), 253 - 315. · Zbl 0805.65058
[4] Hofmann, B.: Regularization for Applied Inverse and Ill-Posed Problems (Teubner- Texte zur Mathematik: Vol. 85). Leipzig: B. G. Teubner Verlagsges. 1986. · Zbl 0606.65038
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