So-called local a posteriori accuracy estimates for approximate
solutions of ill-posed inverse problems are under
investigation. We present an approach to local a posteriori
estimation along with the numerical algorithm for calculating
the estimators. We analyze the local optimality in order of
obtained a posteriori accuracy estimates and introduce a new
important notion of locally extra-optimal regularizing
algorithm as a method for the solution of ill-posed problems
having optimal in order local a posteriori accuracy estimates.
Finally, we demonstrate the results of numerical experiments on
applications of locally extra-optimal regularizing methods,
constructed on the base of Tikhonov regularization, for
calculating local a posteriori accuracy estimates.