Spline-regularization method for solving inverse problems: theory and algorithms

The regularization method, based on spline approximation for solving operator equations of the first kind on discrete data, is considered. The technique of spline approximation consists in preliminary smoothing the noised right-hand side of the equation by Recursive Smoothing method and in the subsequent application of a collocation scheme. Here the new theoretical results on justification of existence and uniqueness of collocation spline are given. In particular, the justifications of some new collocation schemes for singular two-dimensional integral equations, typical for electroencephalography inverse problems, are obtained. The effective algorithms for proposed regularization method are constructed and realized as software package in MATLAB system. It can be used for the applied electrodynamics and aerodynamics problems, electroencephalography inverse problems, heat-conduction coefficient inverse problems and for the identifying of characteristics of the porous media of confined aquifers.