Discretization of equations Gelfand-Levitan-Krein and regularization algorithms

Abstract

The paper considers the initial-boundary-value inverse problem of acoustics for onedimensional and multidimensional cases. The inverse problems are to reconstruct the coefficients using one-dimensional and multidimensional analogues of the Gelfand-Levitan-Krein integral equations. It is known that such equations are linear integral Fredholm equations of the first kind, which are ill-posed. The aim of the work is to find a numerical solution of the Gelfand-Levitan-Krein equation using iterative regularizing algorithms. Using the specifics of these equations (the kernel of the equation depends on the difference of arguments) it is possible to create highly efficient iterative regularizing algorithms. The implemented algorithms can be successfully applied in solving such problems as reconstruction of blurred and defocused images, inverse problem of gravimetric, linear programming problem with inaccurately given matrix of constraints, inverse problem of Geophysics, inverse problems of computed tomography, etc. The main results of the work are the discretization of the one-dimensional and multidimensional Gelfand-Levitan-Krein equation and the construction of iterative regularization algorithms.