Нart L. Projection-iteration methods for solving operator equations and infinite-dimensional optimization problems

The object of the research is the theory of approximate methods and algorithms for solving operator equations and problems of infinite-dimensional optimization and its applications. The subject of the research is projection-iteration methods and algorithms for solving operator equations of the first kind and infinite-dimensional problems of conditional minimization of functionals and their applications. Research methods are the apparatus of mathematical and functional analysis, optimization methods, numerical methods, theory of optimal control, theory of algorithms and computation. The purpose of the research is to develop a theoretical apparatus of projection-iteration methods and algorithms for solving operator equations of the first kind (including ill-posed ones) and infinite-dimensional extremal constraint problems, constructing, based on grounded methods, and numerical implementation of new effective algorithms for solving some practically important classes of differential and integral equations and optimal control problems. The stability and rate of convergence of projection-iterative methods for solving linear operator equations of the first kind in Banach spaces are studied for the first time. The projection-iteration methods for solving ill-posed operator equations in Hilbert space are developed and theoretically substantiated. New efficient computational schemes of projection-iteration methods and algorithms are developed and theoretically substantiated for solving nonlinear parametric operator equations and infinite-dimensional problems of constrained minimization of functionals. General conditions of approximation, stability, convergence, estimates of error and degree of convergence of the proposed methods are obtained. New numerical algorithms based on the finite-differences method and the straight lines method, are developed; the software that implements them, is created. The analysis of efficiency of the proposed methods and algorithms is made in solving some practically important classes of nonlinear, ill-posed and optimization problems. The proposed approach, numerical algorithms and software systems that implement them, can be used in research and design organizations for the design and calculation of structural elements of new technology and controlled systems.