Regularization Methods for Ill-Posed Problems

Abstract

In this chapter, we outline the mathematical theory of direct regularization methods for in general nonlinear and ill-posed inverse problems. One focus is on Tikhonov regularization in Hilbert spaces with quadratic misfit and penalty terms. Moreover, recent results of an extension of the theory to Banach spaces are presented concerning the variational regularization with convex penalty term. Five examples of parameter identification problems in integral and differential equations are given in order to show how to apply the theory of this chapter to specific inverse and ill-posed problems.