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Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. (English) Zbl 1287.65043
Summary: Minimization-based noise level-free parameter choice rules for the selection of the regularization parameter in linear ill-posed problems are studied. Abstract convergence results for spectral filter regularization operators using a qualitative condition on the (deterministic) data noise are proven. Furthermore, under source conditions on the exact solution, suboptimal convergence rates and, under certain additional regularity conditions, optimal order convergence rates are shown. The abstract results are examined in more detail for several known parameter choice rules: the quasi-optimality rules (both continuous and discrete) and the Hanke-Raus-rules, together with some specific regularization methods: Tikhonov regularization, Landweber iteration, and spectral cutoff.

MSC:
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
47A52 Linear operators and ill-posed problems, regularization
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
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