Carleman Estimates for Coefficient Inverse Problems and Numerical Applications

Chapter 1Introduction
1.1. Some historical remarks
1.2. Mathematical background
1.3. The concept of overdetermination
1.4. Uniqueness results in one dimension
1.5. Uniqueness results in high dimensions
1.6. A brief overview of uniqueness results
Chapter 2. Carleman estimates and ill-posed Cauchy problems
2.1. A second order partial differential operator 2.2. Examples of Carleman estimates
2.3. Uniqueness and the Holder stability
2.4. The Lipschitz stability for a hyperbolic Cauchy problem
2.5. Error estimates in the method of quasireversibility
Chapter 3. Global uniqueness results in high dimensions
3.1. Estimating a Volterra-like operator
3.2. An inverse problem for a hyperbolic equation
3.3. Some inverse problems for a parabolic equation
3.4. Inverse problems for a elliptic equation
3.5. The global uniqueness in a 2D inverse conductivity problem
Chapter 4. The global uniqueness of a nonlinear parabolic problem
4.1. Problem formulation
4.2. Statement of the main result
4.3. An estimate of an integral
4.4. The integro-differential inequality
4.5. Domains
4.6. Notations
4.7. A Carleman estimate
4.8. Proof of the main result
Chapter 5. On the numerical solution of coefficient inverse problems
5.1. Some traditional methods
5.2. Using the Dirichlet-to-Neumann map
5.3. Convexification
5.4. Strict convexity of J_{\lambda, \kappa}(q)
Chapter 6. Some globally convergent convexification algorithms
6.1. Model problems in one dimension
6.2. The recurrence minimization method
6.3. Two numerical methods for nonlinear convex programming
6.4. Error estimates
6.5. Unifying framework
Chapter 7. Some applied problems
7.1. Magnetotelluric sounding of layered media 7.2. Magnetotelluric sounding of 2D inhomogeneous media
7.3. Space electric sounding of 2D inhomogeneous media
7.4. Near-infrared optical sensing of layered biotissues
7.5. Computational time reversal
Bibliography