Unsaturable methods for solving severely ill-posed problems

Considered here is the problem of an approximate solution of severely ill-posed problems represented in the form of linear operator equations of the first kind with approximately known right-hand sides and operators. For a class of the problems two methods of solving were constructed which consist in combination of Morozov's discrepancy principle and a finite-dimensional version of the ordinary Tikhonov regularisation. It is shown that the methods provide for the optimal order of accuracy without saturation. The efficiency of the theoretical results is checked by test example.