Volume 15, Issue 8

The analysis of EXAFS data aimed at extracting information on the local atomic structure of materials is discussed. To identify the structure of interest, one has to solve a system of Fredholm integral equations of the first kind. The situation is considered in which three functions, that characterize the structure under study, are to be determined from one equation. In practice, in this case a nonuniqueness arises in the reconstruction of the unknown functions within a prescribed accuracy, so that additional information has to be used. For a broad range of materials, the functions that are reconstructed possess a rather specific form; namely, these functions appear to be a sum of several narrow peaks that can be approximated by δ functions. This qualitative information about the functions is non-classical. The proposed new analysis procedure is based on the classical theory of ill-posed problems and on using the available non-classical information. The efficiency of the algorithms proposed is illustrated by applying them to model and experimental data of a test compound consisting of uranium, silicon and oxygen, whose structure was previously identified by other authors.

An inverse spectral problem is studied for the non-selfadjoint matrix Sturm–Liouville differential equation on the half-line. We give a formulation of the inverse problem, prove the corresponding uniqueness theorem and provide a constructive procedure for the solution of the inverse problem by the method of spectral mappings. The obtained results are natural generalizations of the classical results in inverse problem theory for scalar Sturm–Liouville operators.

This paper is concerned with deep-lying profiles identification of stress corrosion cracking related to eddy current testing. A nondestructive testing is described by quasi-steady state vector potential problems in three dimensions. An analytical approach is proposed with the background knowledge of domain identification problem. Applying the method of mapping to the problem treated here, a computational method is proposed for recovering stress corrosion crack shape.

Tikhonov regularization is one of the widely used procedures for the regularization of nonlinear as well as linear ill-posed problems. The error analysis carried out in most of the works that appeared in last few years on Tikhonov regularization of nonlinear ill-posed problems are under Hölder type source conditions on the unknown solution which is known to be applicable only for mildly ill-posed problems. In this paper we consider Tikhonov regularization of nonlinear ill-posed problems and derive order optimal error estimate under a general source condition together with an a posteriori parameter rule proposed by Scherzer et al., which is applicable for severely ill-posed problems as well.

The paper is devoted to the approximation of a control element for an abstract parabolic equation. The presentation is given for general approximation schemes, which includes finite element methods, finite difference schemes, projection methods. We apply an iteration-approximation method for the investigation of inverse problems for parabolic equations.

Tikhonov regularization with a stabilizer being the sum of total variation and the norm of space L p [ a ; b ] (1 < p ≤ ∞) is considered. For one-dimensional case we prove that, along with the convergence of regularized solutions, in L p we have uniform convergence over intervals where the solution to be found is continuous. Results of numerical experiments on reconstruction of model solutions, differing in smoothness, of the one-dimensional first-kind Fredholm integral equation are reported.