Identification Problems of Wave Phenomena

Part 1 Statement of the direct and inverse problems -examples: inverse problems of mathematical physics; inverse problem for the wave equation; the equation of plane waves - the d'Alembert formula; the Cauchy problem; the d'Alembert operator with smooth initial data; the Kirchoff and Poisson formulae; Huygens principle; time-like and space-like surfaces; inverse problems with smooth initial data; inverse problem for the acoustic equation. Part 2 Volterra operator equations: main definitions; local well-posedness; well-posedness for sufficiently small data; well-posedness in the neighbourhood of the exact solution. Part 3 Inverse problems for Maxwell's equations: reduction of inverse problems for Maxwell's equations to a Volterra operator equation; local well-posedness and global uniqueness; well-posedness in the neighbourhood of the exact solution. Part 4 Linearization and Newton-Kantorovich method: linearization of Volterra operator equations; the linearized inverse problem for the wave equation; the Newton-Kantorovich method. Part 5 The Gel'fand-Levitan method: Gel'fand-Levitan's approach to multidimensional inverse problems; discrete inverse problems; discrete direct problems; an auxiliary problem; a necessary condition for the existence of the global solution to the discrete inverse problem; sufficient conditions for the existence of the global solution to the discrete inverse problem. Part 6 Regularization: Volterra regularization. Part 7 The methods of the optimal control: discrete inverse problem; special representation for the solution to the discrete direct problem; uniqueness of the stationary point. Part 8 Inversion of finite-difference schemes: convergence of the method of inversion of finite-difference schemes; Picard and Caratheodory successive approximations. Part 9 Strongly ill-posed problems: a strongly ill-posed problem for the Laplace equation; conditional continuous dependence on the data; approximate solutions to the Cauchy problem for the Laplace equation; approximate solutions to the non-characteristic problem for the multidimensional heat equation; existence of solutions satisfying operator inequalities. Part 10 An identification problem related to first-order scalar semilinear equations: the scalar inverse problem. Part 11 An identification problem for a first-order integro-differential equation: the identification problem and its equivalence to a system of integral equations; existence and uniqueness; proof of lemma 11.2.1. Part 12 An inverse hyperbolic integro-differential problem arising in geophysics: introduction and statement of the main results; the transformed inverse problem; equivalence problem (12.2.18)-(12.2.22) with a fixed-point system; solving the fixed-point system (12.3.9)-(12.3.12); estimating the solution (z,p,q) to problem (12.3.9)-(12.3.12); proof of theorem 12.1.1. Part 13 Integro-differential identification problems related to the one-dimensional wave equation: statement of the identification probl