Features of Application of the Consecutive Projections Method in Deconvolution Problems

Abstract

The method of consecutive projections (MCP) is popular due to the simplicity of its implementation and efficient use of memory. The main idea of the method is that a convex set is represented as a finite or infinite intersection of a set of simple convex (elementary) sets. Then it is projected on the sets external to the current point. The projection on these elementary sets is very simple because they are usually semispaces. It is proved that the iterative process of consecutive projections converges, and its modifications that ensure final convergence are developed. Three subproblems are solved within the MCP at each iteration: find an elementary set for projection, determine the direction, and calculate the step length in this direction. In this paper, we make several simple propositions that make it possible to combine these three problems and accelerate the convergence of the method for solving a special class of problems called deconvolution problems.