Chapter
3.7. The iterative regularization method
pp.：
28 – 29
3.8. Combined methods
pp.：
29 – 30
3.9. Method of the regularization and penalties
pp.：
30 – 30
3.10. Methods of the mathematical programming
pp.：
30 – 31
CHAPTER 2. ITERATIVE METHODS FOR APPROXIMATION OF FIXED POINTS AND THEIR APPLICATION TO ILL-POSED PROBLEMS
pp.：
31 – 33
§ 1 Basic classes of mappings
pp.：
33 – 33
1.1. Quasi-nonexpansive and pseudo-contractive mappings
pp.：
33 – 34
1.2. Existence of fixed points
pp.：
34 – 35
§ 2 Convergence theorems for iterative processes
pp.：
35 – 39
2.2. Weak convergence of iterations for pseudo-contractions
pp.：
39 – 41
2.1. Strong convergence of iterations for quasi-contractions
pp.：
39 – 39
§ 3 Iterations with correcting multipliers
pp.：
41 – 42
3.1. Stability of fixed points from parameter
pp.：
42 – 43
3.2. Strong iterative approximation of fixed points
pp.：
43 – 44
3.3. Generalization of results to quasi-nonexpansive operators
pp.：
44 – 45
§ 4 Applications to problems of mathematical programming
pp.：
45 – 47
4.1. Setting of a problem and definition of well-posedness
pp.：
47 – 48
4.2. Prox-algorithm for minimization of convex functional
pp.：
48 – 50
4.3. Fejer processes for convex inequalities system
pp.：
50 – 52
4.4. Iterative processes for solution of operator equations with a priori information
pp.：
52 – 56
4.5. The gradient projection method for convex functional
pp.：
56 – 57
4.6. Minimization of quadratic functional
pp.：
57 – 58
§ 5 Regularizing properties of iterations
pp.：
58 – 64
5.2. Disturbance analysis for the Fejer processes
pp.：
64 – 66
5.1. Iterations with perturbed data and construction of regularizing algorithm
pp.：
64 – 64
5.3. Analysis of solution stability in the projection gradient method
pp.：
66 – 68
§ 6 Iterative processes with averaging
pp.：
68 – 70
6.1. Formulation of the method and preliminary results
pp.：
70 – 70
6.2. The convergence theorem
pp.：
70 – 72
6.3. Stability with respect to perturbations. Weak regularization
pp.：
72 – 73
6.4. The Mann iterative processes
pp.：
73 – 74
§ 7 Iterative regularization of variational inequalities and of operator equations with monotone operators
pp.：
74 – 76
7.2. The method of successive approximation in well-posed case
pp.：
76 – 77
7.1. Formulation of problem
pp.：
76 – 76
7.3. Convergence of the iteratively regularized method of successive approximations
pp.：
77 – 78
7.4. Strong convergence of the Mann processes
pp.：
78 – 83
§8 Iterative regularization of operator equations in the partially ordered spaces
pp.：
83 – 85
8.2. The convergence of iterations for monotonically decomposable operators
pp.：
85 – 86
8.1. Preliminary information
pp.：
85 – 85
8.3. Explicit iterative processes for operator equations of the first kind
pp.：
86 – 88
8.4. Monotone processes of Newton’s type
pp.：
88 – 91
§ 9 Iterative schemes based on the Gauss-Newton method
pp.：
91 – 93
9.1. The two-step method
pp.：
93 – 93
9.2. Iteratively regularized schemes of the Gauss-Newton method
pp.：
93 – 95
CHAPTER 3. REGULARIZATION METHODS FOR SYMMETRIC SPECTRAL PROBLEMS
pp.：
95 – 101
1.1. Definition of L-basis and its properties
pp.：
101 – 101
1.2. Measure of nearness between orthonormal bases
pp.：
101 – 104
§ 1 L-basis of linear operator kernel
pp.：
101 – 101
§ 2 Analogies of Tikhonov’s and Lavrent ’ev’s methods
pp.：
104 – 105
2.2. Regularizing properties of Tikhonov’s method
pp.：
105 – 107
2.1. Tikhonov’s method
pp.：
105 – 105
2.3. The Lavrent’ev method
pp.：
107 – 110
§ 3 The variational residual method and the quasisolutions method
pp.：
110 – 111
3.1. The residual method for linear operator kernel determination
pp.：
111 – 111
3.2. Residual principle proof for determination of regularization parameter
pp.：
111 – 113
3.3. Ivanov’s quasisolutions method
pp.：
113 – 116
3.4. Quasisolutions principle proof for choice of regularization parameter
pp.：
116 – 118
§ 4 Regularization of generalized spectral problem
pp.：
118 – 119
4.2. Regularization method
pp.：
119 – 123
4.1. Gershgorin’s domains for generalized spectral problem
pp.：
119 – 119
CHAPTER 4. THE FINITE MOMENT PROBLEM AND SYSTEMS OF OPERATORS EQUATIONS
pp.：
123 – 128
1.2. The convergence theorem of approximations
pp.：
128 – 129
1.1. Statement of the infinite moment problem
pp.：
128 – 128
§ 1 Statement of the problem and convergence offinite-dimensional approximations
pp.：
128 – 128
§ 2 Iterative methods on the basis of projections
pp.：
129 – 133
2.1. Convergence of iterations for exact data
pp.：
133 – 134
2.2. Convergence of iterations in the presence of noise
pp.：
134 – 135
§ 3 The Fejerprocesses with correcting multipliers
pp.：
135 – 137
3.2. Finite dimensional approximation of normal solution
pp.：
137 – 138
3.1. The finite moment problem in the form of inequalities
pp.：
137 – 137
3.3. Application to integral equations of the first kind
pp.：
138 – 142
§ 4 FMP regularization in Hilbert spaces with reproducing kernels
pp.：
142 – 142
4.2. Representation of normal solution in the space W°12[-1,1]
pp.：
142 – 144
4.1. Definition of reproducing kernels and their properties
pp.：
142 – 142
4.3. Construction of the orthogonal polynomial system
pp.：
144 – 146
4.4. Computation of the resolving system matrix
pp.：
146 – 147
4.5. Regularized solution
pp.：
147 – 148
4.6. Analysis of solution’s sensitivity
pp.：
148 – 149
4.7. Application to inversion of the Laplace transform
pp.：
149 – 151
§ 5 Iterative approximation of solution of linear operator equation system
pp.：
151 – 153
5.2. Auxiliary results
pp.：
153 – 154
5.1. Problem formulation and construction of the method
pp.：
153 – 153
5.3. Convergence theorems for exact and perturbed data
pp.：
154 – 157
CHAPTER 5. DISCRETE APPROXIMATION OF REGULARIZING ALGORITHMS
pp.：
157 – 161
§ 1 Discrete convergence of elements and operators
pp.：
161 – 161
1.2. Interpolation operators
pp.：
161 – 165
1.1. Strong and weak convergence of elements
pp.：
161 – 161
1.3. Convergence theorems for operators
pp.：
165 – 166
1.4. Discrete convergence in uniform convex spaces
pp.：
166 – 168
§2 Convergence of discrete approximations for Tikhonov’s regularizing algorithm
pp.：
168 – 172
2.1. Convergence of regularized solutions
pp.：
172 – 172
2.2. Finite-dimensional approximation. Sufficient conditions of convergence
pp.：
172 – 173
§ 3 Applications to integral and operator equations
pp.：
173 – 177
3.1. Mechanical quadrature method
pp.：
177 – 178
3.2. Collocation method
pp.：
178 – 180
3.3. Projection methods
pp.：
180 – 182
3.4. Nonlinear integral equations
pp.：
182 – 183
3.5. Discretization of Volterra equations. Self-regularization
pp.：
183 – 185
§ 4 Interpolation of discrete approximate solutions by splines
pp.：
185 – 187
4.1. Piecewise constant and piecewise linear interpolation
pp.：
187 – 187
4.2. Parabolic and cubic splines
pp.：
187 – 192
4.3. Approximation of a priori set
pp.：
192 – 195
§ 5 Discrete approximation of reconstruction of linear operator kernel basis
pp.：
195 – 197
5.2. Finite-dimensional approximation of Tikhonov’s method
pp.：
197 – 198
5.1. Discrete measures of nearness
pp.：
197 – 197
5.3. Finite-dimensional approximation of the residual method
pp.：
198 – 199
5.4. Discrete approximation of Ivanov’s quasisolutions method
pp.：
199 – 200
§ 6 Finite-dimensional approximation of regularized algorithms on discontinuous functions classes
pp.：
200 – 202
6.1. Finite-dimensional approximation of function of unbounded operator
pp.：
202 – 202
6.2. Discrete approximation of Tikhonov's method with special stabilizer
pp.：
202 – 205
6.3. Regularizing algorithms on classes of discontinuous functions
pp.：
205 – 206
CHAPTER 6. NUMERICAL APPLICATIONS
pp.：
206 – 211
1.1. Regularization and discretization of base equation
pp.：
211 – 211
1.2. Reconstruction of model solution
pp.：
211 – 213
§ 1 Iterative algorithms for solving gravimetry problem
pp.：
211 – 211
§2 Computing schemes for finite moment problem
pp.：
213 – 218
2.2. Quadrature approximation and iterations with correcting multipliers
pp.：
218 – 220
2.1. Decomposition by means of Legendre polynomials and iterations with projections
pp.：
218 – 218
2.3. Numerical solution of the finite moment problem in the space with a reproducing kernel
pp.：
220 – 221
§ 3 Methods for experiment data processing in structure investigations of amorphous alloys
pp.：
221 – 224
3.1. Solution of EXAFS-equation by Tikhonov variational method
pp.：
224 – 224
3.2. Approximation algorithms for the kernel of an integral operator
pp.：
224 – 226
3.3. A priori information accounting for EXAFS
pp.：
226 – 228
3.4. Uniqueness for the diffraction equation
pp.：
228 – 229
3.5. Iterative algorithm for solving the diffraction equation
pp.：
229 – 231
APPENDIX. CORRECTION PARAMETERS METHODS FOR SOLVING INTEGRAL EQUATIONS OF THE FIRST KIND
pp.：
231 – 236
3.6. Algorithm for solving an integral equations system
pp.：
231 – 231
2. Algorithms of the parameter correction
pp.：
236 – 239
1. The error model and problem statement
pp.：
236 – 236
3. The discussion. The results of numerical experiments
pp.：
239 – 242
Bibliography
pp.：
242 – 247
Ill-Posed Problems with A Priori Information
( Inverse and Ill-Posed Problems Series )
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