Investigation of the bounded perturbation resilience of a modified picard type iterative algorithm and its application to linear inverse problems

In recent years, a method called "superiorization" has been gaining increasing attention in order to increase the efficiency of iterative algorithms. This method is based on the principle of allowing perturbations in iterative algorithms to determine whether the perturbed algorithm fulfills the task of the original algorithm or obtains a superior output. The aim of this study is to apply the superiorization method to linear inverse problems by taking advantage of the relationship between fixed point problems, convex minimization problems and linear inverse problems. For this purpose, a modified Picard type iterative algorithm, which was shown in previous studies to solve the convex minimization problem and to be resilient to perturbations, has been utilized. In this thesis, the result that this iterative algorithm is resilient to perturbations is examined in detail. In order to apply the superiorization method to linear inverse problems, it is first shown that this iterative algorithm, which solves the convex minimization problem, also solves linear inverse problems. Then, using the result that this iterative algorithm is resilient to perturbations, the superiorization method is applied to linear inverse problems. Keywords: Superiorization method; Convex minimization problem; Fixed point problem; Linear inverse problem; Iterative algorithms.