Chapter 1 Inverse Problems

1.1 The inverse problem of gravimetry

1.2 The inverse conductivity problem

1.3 Inverse scattering

1.4 Tomography and the inverse seismic problem

1.5 Inverse spectral problems

Chapter 2 Ill-Posed Problems and Regularization

2.1 Well- and ill-posed problems

2.2 Conditional correctness:Regularization

2.3 Construction of regularizers

2.4 Convergence of regularization algorithms

2.5 Iterative algorithms

Chapter 3 Uniqueness and Stability in the Cauchy Problem

3.1 The backward parabolic equation

3.2 General Carleman estimates and the Cauchy problem

3.3 Elliptic and parabolic equations

3.4 Hyperbolic and Schr？dinger equations

3.5 Systems of partial differential equations

3.6 Open problems

Chapter 4 Elliptic Equations:Single Boundary Measurements

4.0 Results on elliptic boundary value problems

4.1 Inverse gravimetry

4.2 Reconstruction of lower-order terms

4.3 The inverse conductivity problem

4.4 Methods of the theory of one complex variable

4.5 Linearization of the coefficients problem

4.6 Some problems of detection of defects

4.7 Open problems

Chapter 5 Elliptic Equations:Many Boundary Measurements

5.0 The Dirichlet-to-Neumann map

5.1 Boundary reconstruction

5.2 Reconstruction in Ω

5.3 Completeness of products of solutions of PDE

5.4 Recovery of several coefficients

5.5 The plane case

5.6 Nonlinear equations

5.7 Discontinuous conductivities

5.8 Maxwell's and elasticity systems

5.9 Open problems

Chapter 6 Scattering Problems

6.0 Direct Scattering

6.1 From A to near field

6.2 Scattering by a medium

6.3 Scattering by obstacles

6.4 Open problems

Chapter 7 Integral Geometry and Tomography

7.1 The Radon transform and its inverse

7.2 The energy integral methods

7.3 Boman's counterexample

7.4 The transport equation

7.5 Open problems

Chapter 8 Hyperbolic Problems

8.0 Introduction

8.1 The one-dimensional case

8.2 Single boundary measurements

8.3 Many measurements:use of beam solutions

8.4 Many measurements:methods of boundary control

8.5 Recovery of discontinuity of the speed of propagation

8.6 Open problems

Chapter 9 Inverse parabolic problems

9.0 Introduction

9.1 Final overdetermination

9.2 Lateral overdetermination:single measurements

9.3 The inverse problem of option pricing

9.4 Lateral overdetermination:many measurements

9.5 Discontinuous principal coefficient and recovery of a domain

9.6 Nonlinear equations

9.7 Interior sources

9.8 Open problems

Chapter 10 Some Numerical Methods

10.1 Linearization

10.2 Variational regularization of the Cauchy problem

10.3 Relaxation methods

10.4 Layer-stripping

10.5 Range test algorithms

10.6 Discrete methods

Appendix.Functional Spaces

References

Index