Volume 18, Issue 9

This minisymposium, held at the 5th International Conference “Inverse Problems: Modeling and Simulation”, May 24–29, 2010 in Antalya, Turkey, was dedicated to the 60th birthday of Michael V. Klibanov.

We consider a class of finite difference schemes for approximating solutions to ill-posed Cauchy problems for first order linear operator differential equations in a Hilbert space. Both the operator and the initial state in the problems are supposed to be noisy. Using an appropriate coordination between the mesh width and error levels, we improve previous error estimates for approximations generated by the schemes.

The inverse problem for the acoustic equation is considered. We propose a method of reconstruction of the density approximating 2D inverse acoustic problem by a finite system of one dimensional inverse acoustic problems. The 2D analogy of the Gel'fand–Levitan–Krein method is established. The inverse acoustic problem is formulated and the short outline of the history and development in this field are given in Section 1. In Section 2 we consider the 2D analogy of the Gel'fand–Levitan–Krein equation. The N-approximation of the Gel'fand–Levitan–Krein equation is obtained for inverse acoustic problem in Section 3. The numerical results are presented in Section 4.

In this paper we consider the thermoacoustic inverse problem of identifying the oscillatory heat release from pressure measurements. We consider the spatially onedimensional and time harmonic case. Three different regularization methods for the stable solution of this ill-posed inverse problem are proposed: Lavrent'ev's method, regularization by discretization, and a method based on an explicit formula combined with regularized numerical differentiation. For these methods, the results of numerical experiments are documented.

Recovery of magnetic target parameters from magnetic sensor measurements has attracted wide interests and found many practical applications. However, difficulties present in identifying the permanent magnetization due to the complications of magnetization distributions over the ship body, and errors and noises of measurement data degrade the accuracy and quality of the parameter identification. In this paper, we use a two step sequential solutions to solve the inversion problem. In the first step, a numerical model is built and used to determine the induced magnetization of the ship. In the second step, we solve a type of continuous magnetization inversion problem by solving 2D Fredholm integral equation of the 1st kind. We use parallel computing which allows solve the inverse problem with high accuracy. In additional, Tikhonov regularization has been applied in solving the inversion problems. The proposed methods have been validated using simulation data with added noises.

The aim of the paper is to substantiate a general scheme of constructing twostage iterative processes for solving irregular nonlinear operator equations in a Hilbert space with a priori conditions on the solution. The processes use the technique of sequential approximate minimization of locally strongly convex Tikhonov functionals.