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The impact of conditional stability estimates on variational regularization and the distinguished case of oversmoothing penalties

Banff International Research Station for Mathematical Innovation and Discovery

Conditional stability estimates require additional regularization for obtaining stable approximate solutions if the validity area of such estimates is not completely known. The focus of this talk is on the Tikhonov regularization under conditional stability estimates for non-linear ill-posed problems in Hilbert scales, where the case that the penalty is oversmoothing plays a prominent role. This oversmoothing problem has been studied early for linear forward operators, most notably in the seminal paper by Natterer 1984. The a priori parameter choice used there, just providing order optimal convergence rates, has in the oversmoothing case the unexpected property that the quotient of the noise level square and the regularization parameter tends to infinity when the noise level tends to zero. We provide in this talk some new convergence rate results for nonlinear problems and moreover case studies that enlighten the interplay of conditional stability and regularization. In particular, there occur pitfalls for oversmoothing penalties, because convergence can completely fail and the stabilizing effect of conditional stability may be lost.

Mathematics; Functional analysis; Calculus of variations and optimal control; optimization; Numerical analysis

video/mp4

Author affiliation: Technische Universität Chemnitz

2019-12-23

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72918

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/