V.A.Morozov

The generalized sourcewiseness and the error estimates for
regularization of linear and nonlinear problems

There is considered the problem of evaluating the unlimited operator on
the solutions of origin operator equation. The a priori constraints on the
desired solution are supposed to be given; the operators in the problem
can be nonlinear. The input data (the operators, elements of spaces, set
of constraints and so on) may be given approximately. Both in linear and
nonlinear cases as well there are given the variational conditions
(analogous to source conditions), that made possible non-improvable on
the order error esimates of regularized solutions. In linear case the
obtained theorems are invertible. The robustness of the estimates under
perturbation in the generalized source conditions is also studied.

The obtained results are appliable to a number of problems: mathematical
physics and natural sciences, identification, the experimental data
processing (for both direct and indirect measuring), reconstructing
of noisy images and so on. The special attention is paid to studying of
"large" noise in data. There is selected special classes of ill-posed
problems, possessing the first order estimate of an error.