偏微分方程逆问题

Preface
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 type estimates and the Cauchy problem
3.3 Elliptic and parabolic equations
3.4 Hyperbolic and Schr6dinger equations
3.5 Open problems
Chapter 4 Elliptic Equations:cSingle Boundary Measurements
4.0 Results on elliptic boundary value problems
4.1 Inverse gravimetry
4.2 Reconstruction of lower-ordercterms
4.3 The inverse conductivity problem
4.4 Methodscof the theory of one complex variable
4.5 Linearization of the coefficients problem
4.6 Some problems of nondestructive evaluation
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
5.3 Completeness of products of solutions of PDE
5.4 Thecplane case
5.5 Recovery of several coefficients
5.6 Nonlinear equations
5.7 Discontinuous conductivities
5.9 Open problems
Chapter 6 Scattering Problems
6.0 Dire tcscattering
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 Geometrycand Tomography
7.1 The Radon transform and its inverse
7.2 The energy integrals methods
7.3 Boman'sccounterexample
7.4 The transport equation
7.5 Open problems
Chapter 8 Hyperboli Equations
8.0 Introduction
8.1 Thecone-dimensional case
8.2 Single boundary measurements
8.3 Many measurements:cuse of beam solutions
8.4 Many measurements:cmethods of boundary contro
8.5 Recovery of discontinuity of thecspeed of propagation
8.6 Opencproblems
Chapter 9 Parabolic Equations
9.0 Introduction
9.1 Final overdetermination
9.2 Lateral overdetermination:csingle measurements
9.3 Lateral overdetermination:cmany measurements
9.4 Discontinuous principal coefficient
9.5 Nonlinear equations
9.6 Interior sources
9.7 Open problems
Chapter 10 Some Numerical Methods
10.1 Linearization
10.2 Variational regularizationcof the Cauchycproblem
10.3 Relaxation methods
10.4 Layer-stripping
10.5 Discrete methods
Appendix.cFun tional Spaces
References
Index