This book introduces the reader to the area of inverse problems. A relatively new branch of Applied Mathematics, the study of inverse problems is of vital interest to many areas of science and technology such as geophysical exploration, system identification, nondestructive testing and ultrasonic tomography.
The aim of this book is twofold: in the first part, the reader is exposed to the basic notions and difficulties encountered with ill-posed problems. Basic properties of regularization methods for linear ill-posed problems are studied by means of several simple analytical and numerical examples.
The second part of the book presents two special nonlinear inverse problems in detail - the inverse spectral problem and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness and continuous dependence on parameters. Then some theoretical results as well as numerical procedures for the inverse problems are discussed. The choice of material and its presentation in the book are new, thus making it particularly suitable for graduate students. Basic knowledge of real analysis is assumed.

Inhalt

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.