Numerical approach to estimate the error of ill-posed problems

The modulus of continuity of the inverse operator leads to minimization of nonconvex functions. In practice, the modulus of continuity can be calculated for a very narrow class of problems. The main difficulty in the calculation is the commutation members of the task operators. Since this condition in a real application is rarely executed, it became necessary in numerical algorithms for error estimation. In this paper we consider a numerical algorithm to evaluate the approximate solution of operator equations of the first kind obtained by the method of residuals not taking into account the mo¬dulus of continuity of the inverse operator. It is shown that this error estimate is not worse than the estimation using the modulus of continuity of the inverse operator. The proposed approach can significantly extend the class of problems to which it is applicable and to obtain the accuracy of the estimation, not inferior to that which could be obtained by using the modulus of continuity of the inverse operator.