On an iterative method for a class of integral equations of the first kind

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Abstract
In this paper, we investigate an iterative method which has been proposed [1] for the numerical solution of a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right‐hand side. Integral equations of this special type occur in experimental physics, astronomy, medical tomography and other fields where density functions cannot be measured directly, but are related to observable functions via integral equations. In order to take into account the non‐negativity of density functions, the proposed iterative scheme was defined in such a way that only non‐negative solutions can be approximated. The first part of the paper presents a motivation for the iterative method and discusses its convergence. In particular, it is shown that there is a connection between the iterative scheme and a certain concave functional associated with integral equations of this type. This functional can be interpreted as a generalization of the log‐likelihood function of a model from emission tomography. The second part of the paper investigates the convergence properties of the discrete analogue of the iterative method associated with the discretized equation. Sufficient conditions for local convergence are given; and it is shown that, in general, convergence is very slow. Two numerical examples are presented.

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