Extra-optimal methods for solving ill-posed problems: survey of theory and examples

A new direction in methods for solving ill-posed problems, namely, the theory of regularizing algorithms with approximate solutions of extra-optimal quality is surveyed. A distinctive feature of these methods is that they are optimal not only in the order of accuracy of resulting approximate solutions, but also with respect to a user-specified quality functional. Such functionals can be specified, for example, as an a posteriori estimate of the quality (accuracy) of approximate solutions, a posteriori estimates of various linear functionals of these solutions, and estimates of their mathematical entropy and multidimensional variations of chosen types. The relationship between regularizing algorithms that are extra-optimal and optimal in the order of quality is studied. Issues concerning the practical derivation of a posteriori estimates for the quality of approximate solutions are addressed, and numerical algorithms for finding such estimates are described. The exposition is illustrated by results of numerical experiments.