An Introduction to the Mathematical Theory of Inverse Problems

1 Introduction and Basic Concepts.- 1.1 Examples of Inverse Problems.- 1.2 III-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 Scattering Problem.- 5.1 Introduction.- 5.2 The Direct Scattering Problem.- 5.3 Properties of the Far Field Patterns.- 5.4 Uniqueness of the Inverse Problem.- 5.5 Numerical Methods.- 5.5.1 A Simplified Newton Method.- 5.5.2 A Modified Gradient Method.- 5.5.3 The Dual Space Method.- 5.6 Problems.- A Basic Facts from Functional Analysis.- A. l 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 Spectral Theory for Compact Operators in Hilbert Spaces.- A.6 The Frechet Derivative.- B Proofs of the Results of Section 2.7.- References.