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The present thesis addresses both theoretical as well as numerical aspects of the treatment of nonlinear inverse problems. The first part considers Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. Sufficient as well as necessary conditions to establish convergence are introduced and convergence rate results are given for various parameter choice rules under a two sided nonlinearity constraint. Ultimately, both a posteriori as well as certain a priori parameter choice rules lead to identical converce rates. The theoretical results are supported and augmented by extensive numerical case studies. In particular it is shown, that the localization of the above mentioned nonlinearity constraint is not trivial. Incorrect localization will prevent convergence of the regularized to the exact solution. The second part of the thesis considers two open problems in inverse option pricing and electrical impedance tomography. While regularization through discretization is sufficient to overcome ill-posedness of the latter, the first requires a more sophisticated approach. It is shown, that the recovery of time dependent volatility and interest rate functions from observed option prices is everywhere locally ill-posed. This motivates Tikhonov-type or variational regularization with two parameters and penalty terms to simultaneously recover these functions. Two parameter choice rules using the L-hypersurface as well as a combination of L-curve and quasi-optimality are introduced. The results are again supported by extensive numerical case studies.

The dissetation deals with the inverse problem of identification of local volatilities from given option price data. The used separation between purely time- and purely price-dependent volatilities enables a detailed mathematical analysis of the corresponding inverse problems. Those are formulated in proper Banach spaces (Hilbert spaces) as operator equations. The unique solvability of these equations are examined. Because the solutions doesn't depend continuously form the given data, possibilities of regularization are discussed. In particular the nonlinear Tikhonov regularization and its applicability to the corresponding problems plays the leading part in these investigations. Detailed numerical studies illustrate these considerations and top this disseration off. Die Dissertation beschäftigt sich mit dem inversen Problem der Identifikation lokaler Volatilitäten aus gegebenen Optionspreisen. Die dabei benutzte Trennung zwischen rein zeit- und rein preisabhängigen Volatilitäten erlaubt eine tiefgehende mathematische Analyse der entsprechend formulierten inversen Probleme. Diese werden in geeigneten Banachräumen (Hilberträumen) als Operatorgleichung angegeben und auf die eindeutige Lösbarkeit hin untersucht. Da sich die Lösungen als instabil gegenüber Störungen in den Daten erweisen, werden Möglichkeiten der Regularisierung diskutiert. Insbesondere steht dabei die Untersuchung der Anwendbarkeit der Theorie der nichtlinearen Tikhonov-Regularisierung auf die entsprechenden Probleme im Vordergrund. Ausführliche numerische Studien illustrieren die diese Überlegungen und runden die Arbeit ab.