1 Introduction and Basic Concepts 1.1 Examples oflnverse Problems 1.2 Ill-Posed Problems 1.3 The Worst-Case Error 1.4 Problems 2 Regularization Theory for Equations of the First Kind 2.1 A General Regularization Theory 2.2 Tikhonov Regularization 2.3 Landweber Iteration 2.4 A Numerical Example 2.5 The Discrepancy Principle of Morozov 2.6 Landweber's Iteration Method with Stopping Rule 2.7 The Conjugate Gradient Method 2.8 Problems 3 Regularization by Discretization 3.1 Projection Methods 3.2 Galerkin Methods 3.2.1 The Least Squares Method 3.2.2 The Dual Least Squares Method 3.2.3 The Bubnov-Galerkin Method for Coercive Operators 3.3 Application to Symm's Integral Equation of the First Kind 3.4 Collocation Methods 3.4.1 Minimum Norm Collocation 3.4.2 Collocation of Symm's Equation 3.5 Numerical Experiments for Symm's Equation 3.6 The Backus-Gilbert Method 3.7 Problems 4 Inverse Eigenvalue Problems 4.1 Introduction 4.2 Construction of a Fundamental System 4.3 Asymptotics of the Eigenvalues and Eigenfunctions 4.4 Some Hyperbolic Problems 4.5 The Inverse Problem 4.6 A Parameter Identification Problem 4.7 Numerical Reconstruction Techniques 4.8 Problems 5 An Inverse Problemin Electrical Impedance Tomography 5.1 Introduction 5.2 The Direct Problem and the Neumann-Dirichlet Operator 5.3 The Inverse Problem 5.4 The Factorization Method 5.5 Problems 6 An Inverse Scattering Problem 6.1 Introduction 6.2 The Direct Scattering Problem 6.3 Properties of the Far Field Patterns 6.4 Uniqueness of the Inverse Problem 6.5 The Factorization Method 6.6 Numerical Methods 6.6.1 A Simplified Newton Method
6.6.2 A Modified Gradient Method 6.6.3 The Dual Space Method 6.7 Problems A Basic Facts from Functional Analysis A.1 Normed Spaces and Hilbert Spaces A.2 Orthonormal Systems A.3 Linear Bounded and Compact Operators A.4 Sobolev Spaces of Periodic Functions A.5 Sobolev Spaces on the Unit Disc A.6 Spectral Theory for Compact Operators in Hilbert Spaces A.7 The Frechet Derivative B Proofs of the Results of Section 2.7 References Index
[