Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems

✍ Scribed by Sergey I. Kabanikhin; Abdigany D. Satybaev; Maxim A. Shishlenin

• Publisher: De Gruyter
• Year: 2013
• Tongue: English
• Leaves: 188
• Series: Inverse and Ill-Posed Problems Series; 48
• Edition: Reprint 2013
• Category: Library

✦ Synopsis

The authors consider dynamic types of inverse problems in which the additional information is given by the trace of the direct problem on a (usually time-like) surface of the domain. They discuss theoretical and numerical background of the finite-difference scheme inversion, the linearization method, the method of Gel'fand-Levitan-Krein, the boundary control method, and the projection method and prove theorems of convergence, conditional stability, and other properties of the mentioned methods.

✦ Table of Contents

Main definitions and notations
Introduction
Chapter 1. Finite-difference scheme inversion (FDSI)
1.1. Introduction
1.2. Volterra operator equations
1.3. Definitions and examples
1.4. Convergence of FDSI
1.5. Numerical examples
Chapter 2. Linearized multidimensional inverse problem for the wave equation
2.1. Introduction
2.2. Problem formulation
2.3. Linearization
2.4. Analyzing the structure of the solution to one-dimensional direct problem
2.5. Existence theorem for the direct problem
2.6. Uniqueness of solutions to the inverse problem and regularization
2.7. Numerical examples
Chapter 3. Methods of I. M. GePfand, B. M. Levitan and M. G. Krein
3.1. Introduction
3.2. Gel’fand-Levitan-Krein (GLK) equation for one-dimensional inverse problem
3.3. Multidimensional analog of GLK-equations
3.4. Gel’fand-Levitan method for wave equation
3.5. Discrete analog of the Gel'fand-Levitan equation
3.6. Multidimensional discrete analog
3.7. Numerical examples
Chapter 4. Boundary control method (BC method)
4.1. Introduction. Statement of the problem
4.2. BC method in one-dimensional case
4.3. BC method for 2D acoustic inverse problem
4.4. Numerical examples
Chapter 5. Projection method
5.1. Introduction
5.2. Projection method for solving inverse problem for the wave equation
5.3. Projection method for solving inverse acoustic problem
5.4. Numerical examples
Appendix A
Appendix B
Bibliography