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1 Introduction and Basic Concepts1£®1 Examples oflnverse Problems1£®2 Ill-Posed Problems1£®3 The Worst-Case Error1£®4 Problems2 Regularization Theory for Equations of the First Kind2£®1 A General Regularization Theory2£®2 Tikhonov Regularization2£®3 Landweber Iteration2£®4 A Numerical Example2£®5 The Discrepancy Principle of Morozov2£®6 Landweber's Iteration Method with Stopping Rule2£®7 The Conjugate Gradient Method2£®8 Problems3 Regularization by Discretization3£®1 Projection Methods3£®2 Galerkin Methods3£®2£®1 The Least Squares Method3£®2£®2 The Dual Least Squares Method3£®2£®3 The Bubnov-Galerkin Method for Coercive Operators3£®3 Application to Symm's Integral Equation of the First Kind3£®4 Collocation Methods3£®4£®1 Minimum Norm Collocation3£®4£®2 Collocation of Symm's Equation3£®5 Numerical Experiments for Symm's Equation3£®6 The Backus-Gilbert Method3£®7 Problems4 Inverse Eigenvalue Problems4£®1 Introduction4£®2 Construction of a Fundamental System4£®3 Asymptotics of the Eigenvalues and Eigenfunctions4£®4 Some Hyperbolic Problems4£®5 The Inverse Problem4£®6 A Parameter Identification Problem4£®7 Numerical Reconstruction Techniques4£®8 Problems5 An Inverse Problemin Electrical Impedance Tomography5£®1 Introduction5£®2 The Direct Problem and the Neumann-Dirichlet Operator5£®3 The Inverse Problem5£®4 The Factorization Method5£®5 Problems6 An Inverse Scattering Problem6£®1 Introduction6£®2 The Direct Scattering Problem6£®3 Properties of the Far Field Patterns6£®4 Uniqueness of the Inverse Problem6£®5 The Factorization Method6£®6 Numerical Methods6£®6£®1 A Simplified Newton Method6£®6£®2 A Modified Gradient Method6£®6£®3 The Dual Space Method6£®7 ProblemsA Basic Facts from Functional AnalysisA£®1 Normed Spaces and Hilbert SpacesA£®2 Orthonormal SystemsA£®3 Linear Bounded and Compact OperatorsA£®4 Sobolev Spaces of Periodic FunctionsA£®5 Sobolev Spaces on the Unit DiscA£®6 Spectral Theory for Compact Operators in Hilbert SpacesA£®7 The Frechet DerivativeB Proofs of the Results of Section 2£®7ReferencesIndex