Identification Problems of Wave Phenomena —— Theory and Numerics

----- 波动现象的识别问题：理论与数值

ISBN: 9783110354980 出版年：2014 页码：352 Lorenzi, A ,Kabanikhin, S I De Gruyter

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