Ill-Posed Problems with a Priori Information.

By: Vasin, V. VContributor(s): Ageev, A. LSeries: Inverse and Ill-Posed Problems SerPublisher: Berlin/Boston : De Gruyter, Inc., 1995Copyright date: ©1995Description: 1 online resource (264 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783110900118Subject(s): Approximation theory.;Differential equations, Partial -- Improperly posed problems.;Iterative methods (Mathematics)Genre/Form: Electronic books. Additional physical formats: Print version:: Ill-Posed Problems with a Priori InformationDDC classification: 515.353 LOC classification: QA377.V334 1995ebOnline resources: Click to View
Contents:
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.
Summary: The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology.
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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.

The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology.

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