Ill-Posed Problems with a Priori Information.

Intro -- Introduction -- CHAPTER 1. UNSTABLE PROBLEMS -- 1 Base formulations ofproblems -- 1.1. Operator equations and systems -- 1.2. The eigen-subspace determination of a linear operator -- 2 Ill-posed problems examples and its stability analysis -- 2.1. The problem of gravimetry -- 2.2. Integral equations in structure investigations of disorder materials -- 2.3. The computerized tomography -- 3 The classification of methods for unstable problems with a priori information -- 3.1. Tikhonov's method -- 3.2. The compact imbedding method -- 3.3. Linear iterative processes -- 3.4. α-processes -- 3.5. The descriptive regularization -- 3.6. Iterative processes with quasi-contractions -- 3.7. The iterative regularization method -- 3.8. Combined methods -- 3.9. Method of the regularization and penalties -- 3.10. Methods of the mathematical programming -- CHAPTER 2. ITERATIVE METHODS FOR APPROXIMATION OF FIXED POINTS AND THEIR APPLICATION TO ILL-POSED PROBLEMS -- 1 Basic classes of mappings -- 1.1. Quasi-nonexpansive and pseudo-contractive mappings -- 1.2. Existence of fixed points -- 2 Convergence theorems for iterative processes -- 2.1. Strong convergence of iterations for quasi-contractions -- 2.2. Weak convergence of iterations for pseudo-contractions -- 3 Iterations with correcting multipliers -- 3.1. Stability of fixed points from parameter -- 3.2. Strong iterative approximation of fixed points -- 3.3. Generalization of results to quasi-nonexpansive operators -- 4 Applications to problems of mathematical programming -- 4.1. Setting of a problem and definition of well-posedness -- 4.2. Prox-algorithm for minimization of convex functional -- 4.3. Fejer processes for convex inequalities system -- 4.4. Iterative processes for solution of operator equations with a priori information.

4.5. The gradient projection method for convex functional -- 4.6. Minimization of quadratic functional -- 5 Regularizing properties of iterations -- 5.1. Iterations with perturbed data and construction of regularizing algorithm -- 5.2. Disturbance analysis for the Fejer processes -- 5.3. Analysis of solution stability in the projection gradient method -- 6 Iterative processes with averaging -- 6.1. Formulation of the method and preliminary results -- 6.2. The convergence theorem -- 6.3. Stability with respect to perturbations. Weak regularization -- 6.4. The Mann iterative processes -- 7 Iterative regularization of variational inequalities and of operator equations with monotone operators -- 7.1. Formulation of problem -- 7.2. The method of successive approximation in well-posed case -- 7.3. Convergence of the iteratively regularized method of successive approximations -- 7.4. Strong convergence of the Mann processes -- 8 Iterative regularization of operator equations in the partially ordered spaces -- 8.1. Preliminary information -- 8.2. The convergence of iterations for monotonically decomposable operators -- 8.3. Explicit iterative processes for operator equations of the first kind -- 8.4. Monotone processes of Newton's type -- 9 Iterative schemes based on the Gauss-Newton method -- 9.1. The two-step method -- 9.2. Iteratively regularized schemes of the Gauss-Newton method -- CHAPTER 3. REGULARIZATION METHODS FOR SYMMETRIC SPECTRAL PROBLEMS -- 1 L-basis of linear operator kernel -- 1.1. Definition of L-basis and its properties -- 1.2. Measure of nearness between orthonormal bases -- 2 Analogies of Tikhonov's and Lavrent 'ev's methods -- 2.1. Tikhonov's method -- 2.2. Regularizing properties of Tikhonov's method -- 2.3. The Lavrent'ev method -- 3 The variational residual method and the quasisolutions method.

3.1. The residual method for linear operator kernel determination -- 3.2. Residual principle proof for determination of regularization parameter -- 3.3. Ivanov's quasisolutions method -- 3.4. Quasisolutions principle proof for choice of regularization parameter -- 4 Regularization of generalized spectral problem -- 4.1. Gershgorin's domains for generalized spectral problem -- 4.2. Regularization method -- CHAPTER 4. THE FINITE MOMENT PROBLEM AND SYSTEMS OF OPERATORS EQUATIONS -- 1 Statement of the problem and convergence offinite-dimensional approximations -- 1.1. Statement of the infinite moment problem -- 1.2. The convergence theorem of approximations -- 2 Iterative methods on the basis of projections -- 2.1. Convergence of iterations for exact data -- 2.2. Convergence of iterations in the presence of noise -- 3 The Fejerprocesses with correcting multipliers -- 3.1. The finite moment problem in the form of inequalities -- 3.2. Finite dimensional approximation of normal solution -- 3.3. Application to integral equations of the first kind -- 4 FMP regularization in Hilbert spaces with reproducing kernels -- 4.1. Definition of reproducing kernels and their properties -- 4.2. Representation of normal solution in the space W°12[-1,1] -- 4.3. Construction of the orthogonal polynomial system -- 4.4. Computation of the resolving system matrix -- 4.5. Regularized solution -- 4.6. Analysis of solution's sensitivity -- 4.7. Application to inversion of the Laplace transform -- 5 Iterative approximation of solution of linear operator equation system -- 5.1. Problem formulation and construction of the method -- 5.2. Auxiliary results -- 5.3. Convergence theorems for exact and perturbed data -- CHAPTER 5. DISCRETE APPROXIMATION OF REGULARIZING ALGORITHMS -- 1 Discrete convergence of elements and operators.

1.1. Strong and weak convergence of elements -- 1.2. Interpolation operators -- 1.3. Convergence theorems for operators -- 1.4. Discrete convergence in uniform convex spaces -- 2 Convergence of discrete approximations for Tikhonov's regularizing algorithm -- 2.1. Convergence of regularized solutions -- 2.2. Finite-dimensional approximation. Sufficient conditions of convergence -- 3 Applications to integral and operator equations -- 3.1. Mechanical quadrature method -- 3.2. Collocation method -- 3.3. Projection methods -- 3.4. Nonlinear integral equations -- 3.5. Discretization of Volterra equations. Self-regularization -- 4 Interpolation of discrete approximate solutions by splines -- 4.1. Piecewise constant and piecewise linear interpolation -- 4.2. Parabolic and cubic splines -- 4.3. Approximation of a priori set -- 5 Discrete approximation of reconstruction of linear operator kernel basis -- 5.1. Discrete measures of nearness -- 5.2. Finite-dimensional approximation of Tikhonov's method -- 5.3. Finite-dimensional approximation of the residual method -- 5.4. Discrete approximation of Ivanov's quasisolutions method -- 6 Finite-dimensional approximation of regularized algorithms on discontinuous functions classes -- 6.1. Finite-dimensional approximation of function of unbounded operator -- 6.2. Discrete approximation of Tikhonov's method with special stabilizer -- 6.3. Regularizing algorithms on classes of discontinuous functions -- CHAPTER 6. NUMERICAL APPLICATIONS -- 1 Iterative algorithms for solving gravimetry problem -- 1.1. Regularization and discretization of base equation -- 1.2. Reconstruction of model solution -- 2 Computing schemes for finite moment problem -- 2.1. Decomposition by means of Legendre polynomials and iterations with projections -- 2.2. Quadrature approximation and iterations with correcting multipliers.

2.3. Numerical solution of the finite moment problem in the space with a reproducing kernel -- 3 Methods for experiment data processing in structure investigations of amorphous alloys -- 3.1. Solution of EXAFS-equation by Tikhonov variational method -- 3.2. Approximation algorithms for the kernel of an integral operator -- 3.3. A priori information accounting for EXAFS -- 3.4. Uniqueness for the diffraction equation -- 3.5. Iterative algorithm for solving the diffraction equation -- 3.6. Algorithm for solving an integral equations system -- APPENDIX. CORRECTION PARAMETERS METHODS FOR SOLVING INTEGRAL EQUATIONS OF THE FIRST KIND -- 1. The error model and problem statement -- 2. Algorithms of the parameter correction -- 3. The discussion. The results of numerical experiments -- Bibliography.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.