Introduction
Chapter 1. Statements of the direct and inverse problems. Examples
1.1. Introduction
1.2. Inverse Problems of Mathematical Physics
1.3. Inverse problem for the wave equation
1.4. The equation of plane waves. The d’Alembert formula
1.5. The Cauchy problem
1.6. The d’Alembert operator with smooth initial data
1.7. The Kirchhoff and Poisson formulae
1.8. Huygens principle
1.9. Time-like and space-like surfaces
1.10. Inverse problems with smooth initial data
1.11. Inverse problem for the acoustic equation
Chapter 2. Volterra operator equations
2.1. Main definitions
2.2. Local well-posedness
2.3. Well-posedness for sufficiently small data
2.4. Well-posedness in the neighborhood of the exact solution
Chapter 3. Inverse problems for Maxwell’s equations
3.1. Introduction
3.2. Reduction of inverse problem for Maxwell’s equations to a Volterra operator equation
3.3. Local well-posedness and global uniqueness
3.4. Well-posedness in the neighborhood of the exact solution
Chapter 4. Linearization and Newton-Kantorovich method
4.1. Linearization of Volterra operator equations
4.2. he linearized inverse problem for the wave equation
4.3. The Newton-Kantorovich method
Chapter 5. The Gel’fand - Levitan Method
5.1. Introduction
5.2. Gel’fand-Levitan’s approach to multidimensional inverse problems
5.3. Discrete inverse problems
5.4. Discrete direct problems
5.5. An auxiliary problem
5.6. A necessary condition for the existence of the global solution to the discrete inverse problem
5.7. Sufficient conditions for the existence of the global solution to the discrete inverse problem
Chapter 6. Regularization
6.1. Introduction
6.2. Volterra regularization
Chapter 7. The method of the optimal control
7.1. Introduction
7.2. Discrete inverse problem
7.3. Special representation for the solution to the discrete direct problem
7.4. Uniqueness of the stationary point
Chapter 8. Inversion of finite-difference schemes
8.1. Convergence of the method of inversion of finite-difference schemes
8.2. Picard and Caratheodory successive approximations
Chapter 9. Strongly ill-posed problems
9.1. A strongly ill-posed problem for the Laplace equation
9.2. Conditional continuous dependence on the data
9.3. Approximate solutions to the Cauchy problem for the Laplace equation
9.4. Approximate solutions to the non-characteristic problem for the multidimensional heat equation
9.5. Existence of solutions satisfying operator inequalities
Chapter 10. Identification problems related to first-order scalar semilinear equations
10.1. The scalar inverse problem
Chapter 11. An identification problem for a first-order integro- differential equation
11.1. Introduction
11.2. The identification problem and its equivalence to a system of integral equations
11.3. Existence and uniqueness
11.4. Proof of Lemma 11.2.1
Chapter 12. An inverse hyperbolic integro-differential problem arising in Geophysics
12.1. Introduction and statement of the main result
12.2. The transformed inverse problem
12.3. Equivalence of problem (12.2.18)—(12.2.22) with a fixed-point system
12.4. Solving the fixed-point system (12.3.9)—(12.3.12)
12.5. Estimating the solution (z,p,q) to problem (12.3.9)—(12.3.12)
12.6. Proof of Theorem 12.1.1
Chapter 13. Integro-differential identification problems related to the one-dimensional wave equation
13.1. Introduction
13.2. Statement of the identification problem
13.3. The existence and uniqueness theorem
Chapter 14. Lavrent’ev regularization of solutions to linear integro-differential inverse problems
14.1. Introduction
14.2. Well-posedness of the linear inverse problem when g Є C3([0, T])
14.3. A convergence theorem
14.4. An algorithm for a numerical solution
14.5. Basic properties of the cost function and its gradient
Chapter 15. A stability result for the identification of a nonlinear term in a semilinear hyperbolic integro-differential equation
15.1. Introduction
15.2. Statements of the main results
15.3. Proof of Theorem 15.2.1
15.4. Statement of the stability result
15.5. Proof of Theorem 15.4.1
Chapter 16. Inverse problems in Electromagnetoelasticity
16.1. Formulation of the direct and inverse problems
16.2. The optimization method for solving the inverse problem
16.3. The finite-difference schemes
Bibliography