Direct methods of solving multidimensional inverse hyperbolic problems


    • Kabanikhin, S. I.
    • Satybaev, A. D. (Abdigany D.)
    • Shishlenin, M. A. (Maxim A.)


Direct methods of solving multidimensional inverse hyperbolic problems

S.I. Kabanikhin, A.D. Satybaev, and M.A. Shishlenin

(Inverse and ill-posed problems series)

VSP, c2005

大学図書館所蔵 件 / 8



Includes bibliographical references (p. [169]-179)



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 methodand prove theorems of convergence, conditional stability, and other properties of the mentioned methods.


  • 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
  • vi Direct Methods of Solving Multidimensional Inverse Problems
  • Chapter 3. Methods of I.M. Gel'fand, B.M. Levitan and M. G. Krein
  • 3.1. Introduction
  • 3.2. Gel'fand-Levitan-Krein (GLK) equation for one-dimensional inverse problem

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