Эффективные алгоритмы вычисления глобальной и локальной апостериорной оценки точности решений линейных некорректных задач

Abstract

We consider extremal problems introduced and investigated earlier by the author for calculating global and local a posteriori error estimates of approximate solutions to ill-posed inverse problems. For linear inverse problems in Hilbert spaces, they consist in maximization of quadratic functionals with two quadratic constraints. The article shows how under certain conditions these problems can be reduced to a problem of maximization of a special (written analytically) differentiable functional with one constraint. New algorithms for calculating global and local a posteriori error estimates based on the solution of these problems are proposed. Their effectiveness is illustrated by numerical experiments on a posteriori error estimation of solutions to the model two-dimensional inverse problem of potential continuation. Experiments show that the proposed algorithms give a posteriori error estimates close to the true error values. Proposed algorithms for global a posteriori error estimation turn out to be more rapid (3 to 5 times) than the previously known algorithms.