On a posteriori accuracy estimates for solutions of linear ill-posed problems and extra-optimal regularizing algorithms

A new scheme of an a posteriori accuracy estimate for solutions of linear ill-posed problems is proposed along with an algorithm of its calculation. A new notion of an extra-optimal regularizing algorithm is introduced as a method for solving ill-posed problems with an a posteriori accuracy estimate optimal in order. An example of an optimal-in-order method being not extra-optimal is discussed. The developed theory is illustrated by a numerical experiment.