Ill Posed Problems with A Priori Information

Part 1 Unstable problems: base formulations of problems; ill-posed problems examples and its stability analysis; the classification of methods for unstable problems with a priori information. Part 2 Iterative methods for approximation of fixed points and their application to ill-posed problems: basic classes of mappings; convergence theorems for iterative processes; iterations with correcting multipliers; applications to problems of mathematical programming; regularizing properties of iterations; iterative processes with averaging; iterative regularization of variation inequalities and of operator equations with monotone operators; iterative regularization of operator equations in the partially-ordered spaces; iterative schemes based on the Gauss-Newton method. Part 3 Regularization methods for symmetric spectral problems: L-basis of linear operator kernel; analogies of Tikhonov's and Lavent'ev's methods; the variational residual method and the quasisolutions method; regularization of generalized spectral problem. Part 4 The finite-moment problem and systems of operators equations: statement of the problem and convergence of finite-dimensional approximations; iterative methods on the basis of projections; the Fejer processes with correcting multipliers; FMP regularization in Hilbert spaces with reproducing kernels; iterative approximation of solution of linear operator equation system. Part 5 Discrete approximation of regularizing algorithms: discrete convergence of elements and operators; convergence of discrete approximations for Tikhonov's regularizing algorithm; applications to integral and operator equations; interpolation of discrete approximate solutions by splines; discrete approximation of reconstuction of linear operator kernel basis; finite-dimensional approximation of regularized algorithms on discontinuous functions classes. Part 6 Numerical applications: iterative algorithms for solving gravimetry problem; computing schemes for finite-moment problem; methods for experiment data processing in structure investigations of amorphous alloys. Appendix: correction parameters methods for solving integral equations of the first kind.