Volume 8, Issue 2

- In this paper, some methods of solution of multidimensional inverse problems are developed. The results of constructive solutions of inverse problems are presented. In particular, inverse problems for differential equations of the second order are considered. Furthermore, problems of integral geometry and problems of determination of convex surfaces are considered. In the last case, the functionals of them are given in the plane. This paper also concerns solution of some other inverse problems.

- Dainelli’s formulae, that offer the real two-dimensional autonomous force fields giving rise to a preassigned family of real orbits f ( x, y ) = c , are allowed to provide complex (autonomous and nonautonomous) forces producing both real and complex families of orbits. Numerical and analytical examples reveal that complex motion is taking place, in real time, on the given family . The transition from real to complex and sometimes back to real motion is described. Complex potentials generating a given family of orbits are also derived by inverse problem considerations.

- We suggest two numerical methods for solving a 2D-emission tomography problem. One of them bases on the approach of best approximation, the other is the algorithm of approximate inverse, which represents a stable regularization scheme for linear ill-posed problems. The mathematical model of emission tomography is given by the exponential ray transform (ERT). The aim of this paper is on the one hand a comparison of both methods by applying them on the inversion of the ERT with constant absorption, and on the other hand the investigation of the reconstruction accuracy, if we have some refraction or absorption within the object under consideration. Several numerical test series quantify the influence of refraction and absorption to the solution and answer some questions, which come from related problems.

- An iterated version of the Lavrentiev’s method, in the setting of a Banach space, is suggested for obtaining stable approximate solutions for the ill-posed operator equation Au = v , when the data A and v are known only approximately. In the setting of a Hilbert space with appropriate a priori parameter choice, the suggested procedure yields order optimal error estimates. An iterated version of Tikhonov regularization yielding order optimal error estimate is a special case of the procedure. The assumption on the approximating operators show that the finite dimensional system arising out of it would be of smaller size for larger iterates. This aspect is compared with an assumption of [3] for a degenerate kernel method for integral equations of the first kind.

- A new method for finding small inhomogeneities from surface scattering data is proposed and mathematically justified. The method allows one to find small holes and cracks in metallic and other objects from the observation of the acoustic field scattered by the objects.

- In this paper we discuss an inverse problem in which we try to determine the refractive index corresponding to the scattering of electromagnetic waves by an inhomogeneous orthotropic medium, which is a special case of anisotropic medium, from far field data. In particular we shall be treating the use of the dual space method (due to Colton and Monk) to reconstruct the refractive index.