Approximated solution algorithms for Urysohn-type equations

Abstract

Abstract Problems of remote sensing and gravity, magnetic, seismic, geological prospecting use systems of indirect data measurement, which can be modeled with non-linear Urysohn equations. The paper presents equations of Urysohn type and their operator analogues. A number of algorithms for their solution has been developed based on the availability of a priori information, asymptotic properties of integral operators, and specific features of a model. It has been formulated a theorem on constructing a solution of the original equation based on the neighboring one with an error estimate. Both the original and neighboring equations are taken as the regularized equations. The proposed approach allows for a variety of algorithms, depending on the type of regularization and iteration schemes, in particular, a modified version of the Levenberg-Marquardt algorithm. Additionally, the algorithm for searching characteristic points of a given function based on the asymptotics of an integral Urysohn operator is provided.