Ill-posed Problems with A Priori Information (Inverse and Ill-Posed Problems 3)
By: V.V. Vasin (author), A.L. Ageev (author)Hardback
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This work deals with economic and flexible regular methods for solving ill-posed (unstable) problems. These methods take into account additional information on the desired solution. The authors have considered the following unstable problems: linear and nonlinear integral equations, the finite-moment problem, mathematical programming problems, spectral problems and variational inequalities. This book should be of interest to researchers in the field of mathematics and physics, as well as to those working in the field of inverse problems and applications.
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.
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- ID: 9789067641913
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