Regularization of Linear Ill-posed Problems in Two Steps: Combination of Data Smoothing and Reconstruction Methods

Abstract

This thesis is a contribution to the field of "ill-posed inverse problems". During the last ten years a new development in this field has taken place: Besides operator-adapted methods for the solution of inverse problems also methods adjusted to smoothness propertiesof functions are studied. The intention of this thesis is to present and analyze "two-step methods" for the solution of linear ill-posed problems. It is the fundamental idea of a two-step method to perform first a data estimation step of probably noisy data and then to perform a reconstruction step to solve the inverse problem using the data estimate. Besides the general description of two-step methods two realizations are analyzed. On the one hand classical regularization methods like the ones proposed by Tikhonov or Landweber are interpreted as two-step methods. On the other hand the combination of wavelet shrinkage and classical regularization methods is analyzed. This yields an order optimal method which is, by the use of wavelet shrinkage, adapted to smoothness properties of functions in Sobolev and Besov spaces and, by the use of the singular system, adapted to the operator under consideration.