Myleiko H. Numerical solving severely ill-posed problems. Approximation and information aspects

In the thesis to construct a stable approximation of severely ill-posed problems two methods are proposed. The approaches consist in combination of the stop rule according to balancing principle with the standard Tikhonov method and its iterated version correspondingly. It is established that proposed algorithms provide optimal order of accuracy on the wide classes of equations under consideration. In addition, for Fredholm's integral equations of the first kind with finite-smooth kernels an efficient projection scheme is proposed. Based on this scheme an efficient projection method is developed. The order estimates of the minimal radius of the Galerkin information and the minimal radius of computional efforts are obtained. It is established that the optimal orders of these values are achieved under proposed method.