REGULARIZATION FOR NON LINEAR INVERSE PROBLEMS VIA STABILITY CONSTRAINTS

Abstract

Mathematical models of the inverse problems are, in general, ill-posed. This is due to their solutions’ instability with perturbations in the data. Various applications of inverse problems are in non-destructive testing, medical imaging etc. Mathematically, we express an inverse problem via an operator equation in Hilbert spaces, where the operator involved can be either linear or nonlinear. Due to the nice structure of Hilbert spaces in comparison to Banach spaces, the convergence analysis of various regularization methods has been extensively studied in Hilbert spaces. However, for a number of problems such as parameter identification problems, certain inverse scattering problems, X-ray diffractometry etc., a Banach space setting would be more appropriate. Consequently, various well known regularization methods have been extended in Banach space settings from the Hilbert space settings. Meanwhile, variational regularization methods (e.g., Tikhonov regularization), iterative methods such as Landweber method, iteratively regularized Landweber iteration method, inexact Newton- Landweber iteration method etc., have been extensively studied in Banach spaces. Typically, for the approximate solutions obtained via regularization methods, stability, convergence to the exact solution and convergence rates are of primary interest. To deduce the convergence rates, some additional information related to the exact solution (also known as smoothness) needs to be employed. Various smoothness concepts have been defined in the literature such as source conditions, variational inequalities, approximate source conditions, conditional stability estimates in Hilbert scales etc., and convergence rates have been obtained by incorporating these smoothness concepts. Most of these rates are in form of Bregman distance which is a distance function well suited to Banach space settings. In the present thesis, the concept of conditional stability estimates is incorporated in Banach spaces to deduce the convergence rates for solving inverse problems (primarily nonlinear) via variational regularization. In addition to this, the convergence rates for finite dimensional variational regularization are investigated by engaging the special form of conditional stability estimates known as Hölder stability estimates as well as the recently developed smoothness concept of variational inequalities. A novel smoothness concept termed as i ii approximate Hölder stability estimates is introduced with which convergence rates for variational regularization methods are deduced. There are a lot of ill-posed inverse problems which satisfy a conditional stability estimate in terms of a norm which can not be defined in terms of Bregman distance. Motivated by this fact, the convergence rates for such inverse problems, which satisfy a special type of stability estimates are investigated. Tikhonov regularization with sparsity constraints is one of the classical regularization techniques. The problem of determination of convergence rates is also considered for this sparsity regularization scheme and convergence rates are obtained. The method of obtaining the convergence rates via conditional stability estimates is carried forward to iterative methods in Banach spaces. The convergence analysis of various iterative methods such as iteratively regularized Landweber iteration method, iteratively regularized Gauss-Newton method is discussed and convergence rates are obtained provided the inverse problems under consideration fulfill a conditional stability estimate. A two-point gradient method is introduced for solving ill-posed problems in Banach spaces. Basically, we formulated this method by combining the iteratively regularized Landweber iteration method with an extrapolation strategy. This method allows to incorporate several functionals which are helpful in the reconstruction of various desired features of the required solution. For most of the work done in this thesis, examples are provided in order to verify the abstract assumptions considered to obtain the convergence rates.