Adaptive frame based regularization methods for linear ill-posed inverse problems [Elektronische Ressource] / von Mariya Zhariy

Adaptive Frame Based Regularization Methodsfor Linear Ill-Posed Inverse Problemsvon Mariya ZhariyDissertationzur Erlangung des akademischen GradesDoktorin der Naturwissenschaften– Dr. rer. nat. –Vorgelegt im Fachbereich III der Universit¨ at Bremenim Oktober 2008Betreuer: Prof. Dr. Gerd TeschkeProf. Dr. Ronny RamlauAbstractThis thesis is concerned with the development and analysis of adaptive regularization methods for solvinglinear inverse ill-posed problems. Based on nonlinear approximation theory, the adaptivity concept hasbecome popular in the field of well-posed problems, especially in the solution of elliptic PDE’s. Undercertain conditions on the smoothness of the solution and the compressibility of the operator it has beenshown that the nonlinear approximation guarantees a more efficient approximation with respect to thesparsity of the solution and to the computational effort.In the area of inverse problems, the sparse approximation approach has been applied in solving de-noising and de-blurring problems as well as in general regularization. An essential cost reduction hasbeen achieved by newly developed strategies like domain decomposition and specific projection meth-ods. However, the option of adaptive application, leading simultaneously to cost reduction and sparseapproximation, has not been taken into account yet.