Adaptive Frame Based Regularization Methods for Linear Ill-Posed Inverse Problems.

Abstract

This thesis is concerned with the development and analysis of adaptiveregularization methods for solving linear inverse ill-posed problems.Based on nonlinear approximation theory, the adaptivity concept has becomepopular in the field of well-posed problems, especially in the solution of elliptic PDE's.Under certain conditions on the smoothness of the solution and the compressibility of the operatorit has been shown that the nonlinear approximation guarantees a more efficient approximationwith respect to the sparsity of the solution and to the computational effort.In the area of inverse problems, the sparse approximation approachhas been applied in solving de-noising and de-blurring problems as well as in general regularization.An essential cost reduction has been achieved by newlydeveloped strategies like domain decomposition and specific projection methods.However, the option of adaptive application, leading simultaneously to cost reduction and sparse approximation,has not been taken into account yet.In our research we combine the advantages of the nonlinear approximation with theclassical regularization and parameter identification strategies, like Tikhonov and Landwebermethods. We modify the classical approach for solving the ill-posed inverse problemsin order to obtain a sparse approximation with essentially reduced numerical effort.The main novelty of the developed regularization methods is the adaptive operator approximation.In general, we have shown that it is possible to construct adaptive regularization methods,which yield the same convergence rates as the conventional regularization methods, but aremuch more efficient with respect to the numerical costs.The analytic results of this work have been confirmed and complemented by numerical experiments,that illustrate the sparsity of the obtained solution and desired convergence rates.