1 Introduction I Regularization Methods 2 Analysis of regularization methods 2.1 Linear regularization 2.2 Nonlinear Tikhonov regularization 2.3 BV functions, sets of finite perimeter and their relation to level sets 2.4 Level set regularization 2.4.1 Analysis of Level Set Regularization 2.4.2 Towards an Analysis of Level Set Regularization Techniques 2.4.3 Minimizing Concept 2.4.4 Convergence Analysis ffinctiohs 2.5 Ginzburg-Landau regularization 2.5.1 Ginzburg-Landan functional 2.5.2 Regularization with a Ginzburg-Landau functional 3 Numerical implementation of regularizatiou methods 3.1 Continuous regularizatinn and a connection to iterative regularization 3.1.1 Continuous regularization 3.1.2 Convex analysis 3.1.3 Connection between continuous and iterative regu]arization 3.2 Implementation of the level set regularization 3.2.1 Numerical solution 3.2.2 Iterative Regularization and the Relation to Dynamic Level Set Methods 3.3 Implementation of the Ginzhurg-Landan regularization method 3.4 Numerical examples 3.4.1 The inverse conductivity problem 3.4.2 Implementation 3.4.3 Results and Discussion II A parabolic-elliptic problem 4 The direct problem 4.1 Analysis of the parabolic-elliptic problem 4.2 Numerical realization of the parabolic-elliptic problem 4.2.1 Reformulation of the direct problem 4.2.2 Convergence analysis of the reformulated problem 4.2.3 Implementation of the reformulated problem 4.2.4 Numerical examples 5 The inverse problem 5.1 The Factorization Method 5.1.1 The Factorization of A0 - A1 5.1.2 Range characterization 5.1.3 Characterization of the inclusion 5.2 Implementation of the inverse problem 5.2.1 Implementation by regularization 5.2.2 Implementation applying the Picard criterion 5.2.3 Numerical examples A Additional results:constraint ill-posed operator equation A.1 A modified level set regularization method A.2 Ginzburg-Landau Regularization A.3 Implementation of the constraint operator equation A.4 Examples of the numerical implementation B Additional results:Image processing