Identification Problems of Wave Phenomena

Theory and Numerics

A. Lorenzi, S. I. Kabanikhin

• Publisher: De Gruyter
• Year: 2014
• Print ISBN: 9783110354980
• eBook ISBN: 9783110943290
• Edition: 1
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

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Table of contents

• 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