Oversmoothing regularization models in light of local ill-posedness phenomena

点击翻译按钮获取中文摘要

In the past fifteen years, the role of smoothness in regularization theory aimed at the stable approximate solution of ill-posed operator equations in a Hilbert or Banach space setting has substantially grown. For such operator equations that represent inverse problems with applications in natural sciences, engineering, imaging and finance, the concept of smoothness is twofold: smoothness of elements in abstract function spaces to be reconstructed from noisy data (solution smoothness) on the one hand and smoothness of the linear or nonlinear forward operator in the model (operator smoothness) on the other hand. There is a strong interplay between both occurring varieties of smoothness. Successful regularization approaches are preferably adapted to expected smoothness and nonlinearity situations, but such expectations can fail. A typical example is the variational (Tikhonov-type) regularization with oversmoothing penalties, occurring when the penalty functional overestimates the actual smoothness such that the solution elements attain no finite penalty values. For oversmoothing penalties in Hilbert scale models, substantial convergence and rates results were recently achieved with the decisive participation of both applicants and their coauthors. The refinement and improvement of these results as well their extension to Banach space models and sparsity promoting regularization are challenging goals of this project with focus on classes of nonlinear inverse problems, where the character and degree of ill-posedness can be locally distributed. In this context, we also consider specific classes of inverse problems like the deautoconvolution problem for 2D-images and specific approaches like the data driven regularization as an aspect of deep learning, where missing components of the forward operator have to be compensated. Overall, by using analytical methods, discretization approaches and numerical experiments, the project intends to deliver a deeper understanding of methods for the treatment of oversmoothing models with their opportunities and limitations in light of occurring local ill-posedness phenomena in order to benefit from this for the selection of optimal regularization procedures and appropriate choices of the regularization parameters.