Abstract
Ill-posed inverse problems are one of the most important mathematical problems due to their wide applications in many branches of physics, engineering, medicine, etc. Frequently, the solution of these problems requires regularization techniques to achieve stability. One of the commonly used periodicization methods is due to Tikhonov. In this thesis, our goal is to study and investigate generalizations of Tikhonov regularization for linear ill-posed problems. In Chapter 1 and Chapter 2, we provide details of the mathematical concepts that are related to ill-posed inverse problems and their regularization techniques. In Chapter 3, we present and discuss the Tikhonov regularization method for solving linear ill-posed inverse problems. While in Chapter 4, we focus on general Tikhonov regularization and we discuss sufficient conditions on generalized regularization terms that guarantee existence, uniqueness and stability of the solution. Finally, we present some examples of penalizers which are given by differentorms such as the bound variation norm, the Sobolev norm and powers of seminorms associated to closed operators...
