Pereverzev S. V.

We present a survey of results on the optimal discretization of ill-posed problems obtained in the Institute of Mathematics of the Ukrainian National Academy of Sciences.

We consider a new version of the projection-iterative method for the solution of operator equations of the first kind. We show that it is more economical in the sense of amount of used discrete information.

We determine the exact order of the minimum radius of information in the logarithmic scale for Fredholm integral equations of the second kind with periodic analytic kernels and free terms. We show that the information complexity of the solution of Fredholm equations with analytic kernels is greater in order than the complexity of the approximation of analytic functions. This distinguishes the analytic case from the case of finite smoothness.

We propose a new scheme of discretization of the Lavrent’ev method for operator equations of the first kind with self-adjoint nonnegative operators of certain “smoothness.” This scheme is more economical in the sense of the amount of used discrete information as compared with traditional approaches.

We prove that the application of so-called adaptive direct methods to approximation of Fredholm equations of the first kind leads to a more economical way of finite-dimensional approximation as compared with traditional approaches.

For classes of weakly singular integral equations of the second kind whose kernels have a power singularity, we find the optimal order of the rate of convergence of projection-iterative methods. Moreover, iterative methods of the Sokolov type are considered and, for weakly singular equations with differentiable coefficients, we present estimates of the rate of convergence of such methods.

We establish the exact power order of information complexity for integral equations whose kernels have power singularities and free terms belong to the corresponding Hölder space.

We consider some classes of Fredholm equations with integral operators acting in spaces of periodic analytic functions. For these classes, we establish the exact order of the optimal rates of convergence for some versions of the method of iterative projections and KP methods.