Novel Error Measures and Source Conditions of Regularization Methods for Inverse Problems (SCIP)

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Regularization methods are designed to limit the reconstruction errors in inverse problems. The basic principle of regularization is to limit the investigations in the reconstruction process to solutions which respect certain a-priori information, such as a maximal and minimal magnitude, smoothness, or certain conservation principles.Current regularization theory focuses on problems where the a-priori information can be represented as bounds of convex functionals, and then techniques from the mathematical field of Convex Analysis can be used to prove theoretical properties of the regularized solutions. Recently developed and more efficient regularization methods cannot be analyzed with such techniques, and in fact require novel measures for evaluating the efficiency. The development of such measures and conditions which guarantee the efficiency of modern regularization methods is the overall topic of this proposal which consists of five work packages.In the first and fundamental work package, the focus is on the verification of new convergence rates results for non-convex Tikhonov regularization. The second work package deals with the consequences of over smoothing penalties occurring in general Tikhonov regularization for a Hilbert space or Banach space setting. In the third work package the cross connections between source conditions and the convergence of level sets are under consideration. The fourth work package, however, deals with the interplay of variational source conditions and conditional stability estimates. New aspects of the Lavrentiev regularization with explicit and implicit forward operators are in the focus of the final fifth work package.