Seminar in Numerical Analysis: Thorsten Hohage (Universität Göttingen)
Inverse problems usually consist in finding causes for observed effects. The essential difficulty in solving inverse problems is ill-posedness: causes typically do not depend continuously on their effects although vice versa effects typically depend continuously on causes. To avoid infinite amplification of measurement errors, regularization methods have to be employed to solve inverse problems numerically. The aim of regularization theory is to analyze the convergence and speed of convergence of such methods as the noise level tends to 0.
Over the last years Variational Source Conditions (VSCs) have become a standard assumption for the analysis of these methods. Compared to spectral source conditions they have a number of advantages: They can be used for general nonquadratic penalty and data fidelity terms, lead to simpler proofs, are often not only sufficient, but even necessary for certain convergence rates, and they do not involve the derivative of the forward operator (and hence do not require restrictive assumptions such as a tangential cone condition). However, so far only few sufficient conditions for VSCs for specific inverse problems are known.
To overcome this drawback, we propose a general strategy for the verification of VSCs, which consists of two sufficient conditions: One of them describes the smoothness of the solution, and the other one the degree of ill-posedness of the operator. For a number of important linear inverse problems this leads to equivalent characterizations of VSCs in terms of Besov spaces and necessary and sufficient conditions for rates of convergence. We also discuss the application of our strategy to nonlinear parameter identification and inverse medium scattering problems where it provides sufficient conditions for VSCs in terms of standard function spaces.
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