Iteration regularization, i.e., utilization of iteration methods of any
form for the stable approximate solution of ill-posed problems, is one
of the most important but still insufficiently developed topics of the
new theory of ill-posed problems. In this monograph, a general approach
to the justification of iteration regulari- zation algorithms is
developed, which allows us to consider linear and nonlinear methods from
unified positions. Regularization algorithms are the 'classical'
iterative methods (steepest descent methods, conjugate direction
methods, gradient projection methods, etc.) complemented by the stopping
rule depending on level of errors in input data. They are investigated
for solving linear and nonlinear operator equations in Hilbert spaces.
Great attention is given to the choice of iteration index as the
regularization parameter and to estimates of errors of approximate
solutions. Stabilizing properties such as smoothness and shape
constraints imposed on the solution are used. On the basis of these
investigations, we propose and establish efficient regularization
algorithms for stable numerical solution of a wide class of ill-posed
problems. In particular, descriptive regularization algorithms,
utilizing a priori information about the qualitative behavior of the
sought solution and ensuring a substantial saving in computational
costs, are considered for model and applied problems in nonlinear
thermophysics. The results of calculations for important applications in
various technical fields (a continuous casting, the treatment of
materials and perfection of heat-protective systems using laser and
composite technologies) are given.
