Variational smoothness assumptions in convergence rate theory—an overview

Variational smoothness assumptions are a concept for measuring abstract smoothness of solutions to operator equations. Such assumptions are useful for analyzing regularization methods, especially for proving convergence rates in Banach spaces and in more general settings. We collect results from different papers published by several authors during the last five years. The aim is to present an overview of this relatively new concept to the interested reader without going too deep into the details. 1 Ill-posed problems, regularization, convergence rates A frequently used model for practical problems are equations F (x) = y, x ∈ dom(F ) ⊆ X, y ∈ Y (1.1) between Banach spaces X and Y . The exact right-hand side y is typically unknown. Instead, only noisy data yδ ∈ Y satisfying ‖y − yδ‖ ≤ δ are available. The noise level is quantified by δ ≥ 0. For solving such equations numerically the mapping F : dom(F ) → Y has to have nice properties. ∗Chemnitz University of Technology, Department of Mathematics, D-09107 Chemnitz, Germany, jens.flemming@mathematik.tu-chemnitz.de.

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