On regularization procedures with linear accuracy estimates of approximations

We consider numerical methods for stable approximation of solutions to irregular nonlinear equations with general smooth operators in the Hilbert space. The known variational procedures and iterative regularization methods deliver approximations with accuracy estimates greater in order than error levels in the input data. In the paper for certain components of the desired solution we establish the possibility of obtaining approximations with linear accuracy estimates relative to the error level. These components correspond to the projections of the solution onto proper subspaces of the symmetrized derivative for the operator of the problem.