The optimality of regularizing algorithms for solving stochastic operator equations

Abstract

The problem of solving operator equations which are a generalization of integral equations of the first kind of the convolution type with an approximately specified right-hand side, is considered. Methods are developed for obtaining asymptotic estimates, unimprovable in order of the level of noise, of the optimum accuracy and accuracy of the regularizing algorithms of Tikhonov's type on the parametric classes of solutions, noises and equations. It is proved that the regularization method produces an order-optimum algorithm for constructing quasioptimal approximations.