Part 1 Regularization of linear operator equations: classification of ill-posed problems and the concept of the optimal method; the estimate from below for Dopt; the error of the regularization method; the algorithmic peculiarities of the generalized residual principle; the error of the quasi-solutions method; the regularization method with the parameter a chosen by the residual; the projection regularization method; on the choice of the optimal regularization parameter; optimal methods for solving unstable problems with additional information on the operator A; on the regularization of operator equations of the first kind with the approximately given operator and on the choice of the regularization parameter; the generalized reesidual principle; the optimum of the generalized residual principle. Part 2 Finite-dimensional methods of constructing regularized solutions: the notion of t-uniform convergence of linear operators; the general scheme of finite-dimensional approximation in the regularization method; application of the general scheme to the projection and finite difference methods; the general scheme of finite-dimensional approximation in the generalized residual method; the iterative method for determining the finite-dimensional approximation in the generalized residual principle; the general scheme of finite-dimensional approximations in the quasi-solution method; the necessary and sufficient conditions for the convergence of finite-imenaional approximations in the regularized method; on the discretization of the variational problems (1.11.5); finite-dimensional approximation of regularized solutions; application. Part 3 Regulariztion of non-linear operator equations: approximate solution of non-linear operator equations with a disturbed operator by the regularization method; approximate solution of implicit operator equations of the first kind by the regularization method; optimal by the order method for solving non-linear unstable problems.
