Gelfand-Levitan-Krein method in one-dimensional elasticity inverse problem

In this article we propose the numerical solution of the one dimensional inverse coefficient problem for seismic equation. We use a dynamical version of Gelfand-Levitan-Krein approach for reducing a nonlinear inverse problem for recovering the shear wave’s velocity and the density of the medium to two sequences of the linear integral equations. We propose numerical algorithm for solving these equations based on a fast inversion of a Toeplitz matrix. The proposed numerical methods base on the structure of the problem and therefore improve the efficiency of the algorithms, compared with standard approaches. We present numerical results for solving considered integral equations.

Определение геометро-акустических свойств сварного соединения как решение обратной коэффициентной задачи для скалярного волнового уравнения

The article is devoted to the development of ultrasonic tomographic methods of non-destructive testing of objects in order to determine the geometry of a welded joint and estimate the velocity field in it. The article offers a solution to the inverse coefficient problem for the echosignal registration scheme in the mirror-shadow mode. Numerical simulations were performed for various tomographic schemes on samples with acoustic parameters and geometry corresponding to the real experiment using an antenna array with an operating frequency of 2.25 MHz. Numerical methods have been used to optimize tomographic schemes for various applied problems. It is shown that with the help of the developed tomographic schemes, it is possible not only to detect the boundaries of the welded joint, but also to determine the velocity field inside the control object.

Numerical Solution Methods for a Nonlinear Operator Equation Arising in an Inverse Coefficient Problem

We consider the inverse problem of determining two unknown coefficients in a linear system of partial differential equations using additional information about one of the solution components. The problem is reduced to a nonlinear operator equation for one of the unknown coefficients. The successive approximation method and the Newton method are used to solve this operator equation numerically. Results of calculations illustrating the convergence of numerical methods for solving the inverse problem are presented.

Conditional stability for an inverse coefficient problem of a weakly coupled time-fractional diffusion system with half order by Carleman estimate

Under a priori boundedness conditions of solutions and coefficients, we prove a Hölder stability estimate for an inverse problem of determining two spatially varying zeroth order non-diagonal elements of a coefficient matrix in a one-dimensional fractional diffusion system of half order in time. The proof relies on the conversion of the fractional diffusion system to a system of order 4 in the space variable and the Carleman estimate.