Projection-iteration implementation of explicit variation type methods of solving ill-posed operator equations

Authors

DOI:

https://doi.org/10.20535/SRIT.2308-8893.2017.1.09

Keywords:

ill-posed equation, operator, space, sequence, approximation, iteration method, discrepancy, error, projection-iteration method, convergence, estimation

Abstract

Projection-iteration regularizing methods based on explicit variation type methods (steepest descent and minimal residual methods) are investigated for solving ill-posed linear operator equations in a Hilbert space which do not satisfy the third condition of the correctness of the problem by Hadamard (stability). The proposed approach is to replace the original ill-posed equation by a sequence of simpler equations that approximate it defined in finite-dimensional subspaces of the original space. Then, only few approximations for each of the "approximate" equations are constructed using an explicit variation method, and the last of them is used as the initial approximation in the iterative process for the next "approximate" equation. The theorems on the convergence of the projection-iteration methods are proved, error estimates are obtained. The recommendations on the choice of the regularizing number of iterations are given.

Author Biography

Liudmyla L. Hart, Oles Honchar Dnipro National University, Dnipro

Liudmyla Lavrentiivna Hart,

Candidate of Physical and Mathematical Sciences, Associate Professor, Head of the Research Laboratory of the optimization of complex systems of the Department of Computational Mathematics and Mathematical Cybernetics of the Applied Mathematics Faculty of Oles Honchar Dnipro National University, Dnipro, Ukraine.

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Published

2017-03-21

Issue

Section

Mathematical methods, models, problems and technologies for complex systems research