Oversmoothing Tikhonov regularization for ill-posed inverse problems

Inverse problems arise when causes cannot be measured directly but must be concluded from observed effects.
The inaccuracies arising from measurements of the effects can lead to significant deviations in determining the causes due to the typically inherent ill-posedness in inverse problems. Regularization methods overcome this ill-posedness by finding an approximation of the solution that is stable with respect to the measured data. The regularization parameter should be chosen optimally to achieve a balance between stability and approximation, minimizing the deviation of the regularized solution from the actual solution. This thesis examines Tikhonov regularization for solving nonlinear ill-posed inverse problems.
The considered Tikhonov functional has an oversmoothing penalty term, such that minimization of the Tikhonov functional determines regularized solutions that are, in a certain sense, smoother than the actual solution of the inverse problem. Research on oversmoothing Tikhonov regularization has rapidly advanced, focusing on convergence rates under various conditions.
Extensions to nonlinear operator equations and exploration of different source conditions and parameter choice strategies have enriched this field. This work contributes by generalizing results to a mixed source condition and providing convergence rates for a priori strategies and for the discrepancy principle as methods to select the regularization parameter. Another focus is on oversmoothing Tikhonov regularization in the finite-dimensional setting, where discretization is achieved through projection methods. This is an area that has yet to be thoroughly explored in this context.