Pointwise extra-optimal regularizing algorithms

Pointwise a posteriori accuracy estimates for approximate solutions to multidimensional inverse ill-posed problems, i.e. for functions of several variables, are considered. The estimates are constructed for given values of the argument of an approximate solution found by a regularizing algorithm (RA). A technique for calculation of pointwise a posteriori estimates is proposed. A new notion of a pointwise extra-optimal regularizing algorithm is introduced as a method for the solution of ill-posed problems with a posteriori pointwise accuracy estimate optimal in order for every given argument. A number of examples of pointwise extra-optimal regularizing algorithms are discussed. The proposed theory is illustrated by numerical experiments.