Volynets I. Optimal order accuracy strategy for solving ill-posed problems

The thesis is dedicated to development of optimal order accuracy methods for solving wide classes of ill-posed problems. A modified projection scheme based on the idea of hiberbolic cross is proposed for ill-posed problems with finitely smoothing operators and smooth solutions. It is shown that the approach guaranties optimal order of accuracy and uses essentially less amount of discrete information compare with known methods. Using Fakeev - Lardy regularization method and 1-method alghorithms based on the adaptive discretization strategy are proposed. It is shown that the alghorithms guarantee optimal order of accuracy and are economical in the sence of amount of discrete information. For semi-discrete ill-posed problems appearing in Sobolev scales two alghorithms are constructed. It is shown that the alghorithms guarantee optimal order of accuracy while use the balancing principle for choosing regularization parameter.