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.

Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming

Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.

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.