Inverse problems
"Inverse problems are problems that consist of finding an unknown property of an object or medium through the observation of a response from this object or medium to a probing signal. Thus, the theory of inverse problems yields a theoretical basis for remote sensing and non-destructive evaluation. For example, if an acoustic plane wave is scattered by an obstacle, and one observes the scattered field from the obstacle, or in some exterior region, then the inverse problem is to find the shape and material properties of the obstacle. Such problems are important in the identification of flying objects (airplanes, missiles etc.), objects immersed in water (submarines, fish), and in many other situations." "This book presents the theory of inverse spectral and scattering problems and of many other inverse problems for differential equations in an essentially self-contained way. An outline of the theory of ill-posed problems is given because inverse problems are often ill-posed. There are many novel features in this book. The concept of property C, introduced by the author, is developed and used as the basic tool for a study of a wide variety of one- and multi-dimensional inverse problems, making the theory easier and shorter."--Jacket
1 online resource (xx, 442 pages)
9780387231952, 9780387232188, 9780387321837, 9786610189885, 0387231951, 0387232184, 0387321837, 6610189889
58532660
Cover
Table of Contents
Foreword
Preface
1. Introduction
1.1 Why are inverse problems interesting and practically important?
1.2 Examples of inverse problems
1.3 Ill-posed problems
1.4 Examples of Ill-posed problems
2. Methods of solving ill-posed problems
2.1 Variational regularization
2.2 Quasisolutions, quasinversion, and Backus-Gilbert method
2.3 Iterative methods
2.4 Dynamical system method (DSM)
2.5 Examples of solutions of ill-posed problems
2.6 Projection methods for ill-posed problems
3. One-dimensional inverse scattering and spectral problems
3.1 Introduction
3.2 Property C for ODE
3.3 Inverse problem with I-function as the data
3.4 Inverse spectral problem
3.5 Inverse scattering on half-line
3.6 Inverse scattering problem with fixed-energy phase shifts as the data
3.7 Inverse scattering with "incomplete data"
3.8 Recovery of quarkonium systems
3.9 Krein's method in inverse scattering
3.10 Inverse problems for the heat and wave equations
3.11 Inverse problem for an inhomogeneous Schrèodinger equation
3.12 An inverse problem of ocean acoustics
3.13 Theory of ground-penetrating radars
4. Inverse obstacle scattering
4.1 Statement of the problem
4.2 Inverse obstacle scattering problems
4.3 Stability estimates for the solution to IOSP
4.4 High-frequency asymptotics of the scattering amplitude and inverse
4.5 Remarks about numerical methods for finding S from the scattering
4.6 Analysis of a method for identification of obstacles
5. Stability of the solutions to 3D Inverse scattering problems with fixed-energy
5.1 Introduction
5.2 Inverse potential scattering problem with fixed-energy data
5.3 Inverse geophysical scattering with fixed-frequency
5.4 Proofs of some estimates
5.5 Construction of the Dirichlet-to-Neumann map from the scattering data
5.6 Property C
5.7 Necessary and sufficient condition for scatterers to be spherically
5.8 The Born inversion
5.9 Uniqueness theorems for inverse spectral problems
6. Non-uniqueness and uniqueness results
6.1 Examples of nonuniqueness for an inverse problem of geophysics
6.2 A uniqueness theorem for inverse boundary value problems for parabolic
6.3 Property C and an inverse problem for a hyperbolic equation
6.4 Continuation of the data
7. Inverse problems of potential theory and other inverse source problems
7.1 Inverse problem of potential theory
7.2 Antenna synthesis problems
7.3 Inverse source problem for hyperbolic equations
8. Non-overdetermined inverse problems
8.1 Introduction
8.2 Assumptions
8.3 The problem and the result
8.4 Finding Wj(s) from W2j(s)
8.5 Appendix
9. Low-frequency inversion
9.1 Derivation of the basic equation. Uniqueness results
9.2 Analytical solution of the basic equation
9.3 Characterization of the low-frequency data
9.4 Problems of numerical implementation
9.5 Half-spaces with different properties
9.6 Inversion of the data given on a sphere
9.7 Inversion of the data given on a cylinder
9.8 Two-dimensional inverse problems
9.9 One-dimensional inversion
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