Ill-Posed Problems with A Priori Information

Transcription

1 INVERSE AND ILL-POSED PROBLEMS SERIES Ill-Posed Problems with A Priori Information V.V.Vasin andalageev HIV SPIII Utrecht, The Netherlands, 1995

2 CONTENTS Introduction 1 CHAPTER 1. UNSTABLE PROBLEMS 1 Base formulations of problems Operator equations and Systems The eigen-subspace determination of a linear Operator 7 2 Ill-posedproblems examples and its stability analysis The problem ofgravimetry Integral equations in structure investigations of disorder materials The computerized tomography 13 3 The Classification of methods for unstable problems with a priori Information Tikhonov's method The compact imbedding method Linear iterative processes a-processes The descriptive regularization Iterative processes with quasi-contractions The iterative regularization method Combined methods Method of the regularization and penalties Methods ofthe mathematical programming 21 CHAPTER 2. ITERATIVE METHODS FOR APPROXIMATION OF FIXED POINTS AND THEIR APPLICATION TO ILL-POSED PROBLEMS 1 Basic classes ofmappings Quasi-nonexpansive and pseudo-contractive mappings Existence of fixed points 25 2 Convergence theorems for iterative processes Strong convergence of iterations for quasi-contractions Weak convergence of iterations for pseudo-contractions 31 3 Iterations with correcting multipliers Stability of fixed points from parameter Strong iterative approximation of fixed points Generalization of results to quasi-nonexpansive Operators 35

3 VI Contents 4 Applications to problems of mathematicalprogramming Setting of a problem and definition of well-posedness Prox-algorithm for minimization of convex functional Fejer processes for convex inequalities System Iterative processes for Solution of Operator equations with a priori information The gradient projection method for convex functional Minimization of quadratic functional 48 5 Regularizing properties ofiterations Iterations with perturbed data and construction of regularizing algorithm Disturbance analysis for the Fejer processes Analysis of Solution stability in the projection gradient method 58 6 Iterative processes with averaging Formulation of the method and preliminary results The convergence theorem Stability with respect to perturbations. Weak regularization The Mann iterative processes 64 7 Iterative regularization ofvariational inequalities and of Operator equations with monotone Operators Formulation of problem The method of successive approximation in well-posed case Convergence of the iteratively regularized method of successive approximations Strong convergence of the Mann processes 73 8 Iterative regularization of Operator equations in the partially ordered Spaces Preliminary information The convergence ofiterations for monotonically decomposable Operators Explicit iterative processes for Operator equations of the first kind Monotone processes of Newton's type 81 9 Iterative schemes based on the Gauss-Newton method The two-step method Iteratively regularized schemes of the Gauss-Newton method 85 CHAPTER 3. REGULARIZATION METHODS FOR SYMMETRIC SPECTRAL PROBLEMS 1 L-basis of linear Operator kernel Definition of L-basis and its properties Measure of nearness between orthonormal bases 94

4 Contents VII 2 Analogies oftikhonov 's and Lavrent 'ev 's methods Tikhonov's method Regularizing properties of Tikhonov's method The Lavrent'ev method The variational residual method and the quasisolutions method The residual method for linear Operator kernel determination Residual principle proof for determination of regularization Parameter Ivanov's quasisolutions method Quasisolutions principle proof for choice of regularization Parameter Regularization of generalized spectral problem Gershgorin's domains for generalized spectral problem Regularization method 113 CHAPTER 4. THE FINITE MOMENT PROBLEM AND SYSTEMS OF OPERATORS EQUATIONS 1 Statement ofthe problem and convergence offinitedimensional approximations Statement ofthe infinite moment problem The convergence theorem of approximations Iterative methods on the basis ofprojections Convergence of iterations for exact data Convergence of iterations in the presence of noise The Fejer processes with correcting multipliers The finite moment problem in the form of inequalities Finite dimensional approximation of normal Solution Application to integral equations ofthe first kind FMP regularization in Hubert Spaces with reproducing kerneis Definition of reproducing kerneis and their properties Representation of normal Solution in the space fp^j-l,!] Construction ofthe orthogonal polynomial System Computacion ofthe resolving System matrix Regularized Solution Analysis of solution's sensitivity Application to inversion ofthe Laplace transform Iterative approximation of Solution of linear Operator equation System Problem formulation and construction ofthe method Auxiliary results Convergence theorems for exact and perturbed data 147

5 VIU contents CHAPTER 5. DISCRETE APPROXIMATION OF REGULARIZING ALGORITHMS 1 Discrete convergence ofelements and Operators Strong and weak convergence of elements Interpolation Operators Convergence theorems for Operators Discrete convergence in uniform convex Spaces Convergence of discrete approximations for Tikhonov 's regularizing algorithm Convergence of regularized Solutions Finite-dimensional approximation. Sufficient conditions of convergence Applications to integral and Operator equations Mechanical quadrature method Collocation method Projection methods Nonlinear integral equations Discretization of Volterra equations. Self-regularization Interpolation of discrete approximate Solutions by splines Piecewise constant and piecewise linear interpolation Parabolic and cubic splines Approximation of a priori set Discrete approximation of reconstruction of linear Operator kernel basis Discrete measures of nearness Finite-dimensional approximation of Tikhonov's method Finite-dimensional approximation of the residual method Discrete approximation of Ivanov's quasisolutions method Finite-dimensional approximation ofregularized algorithms on discontinuous functions classes Finite-dimensional approximation of function of unbounded Operator Discrete approximation of Tikhonov's method with special stabilizer Regularizing algorithms on classes of discontinuous functions 196 CHAPTER Ö.NUMERICAL APPLICATIONS 1 Iterative algorithms for solving gravimetry problem Regularization and discretization of base equation Reconstruction of model Solution Computing schemes for finite moment problem Decomposition by means of Legendre polynomials and iterations with projections 208

6 Contents ix 2.2. Quadrature approximation and iterations with correcting multipliers Numerical Solution of the finite moment problem in the space with a reproducing kernel Methods for experiment data processing in structure investigations ofamorphous alloys Solution of EXAFS-equation by Tikhonov variational method Approximation algorithms for the kernel of an integral Operator A priori information accounting for EXAFS Uniqueness for the diffraction equation Iterative algorithm for solving the diffraction equation Algorithm for solving an integral equations System 221 APPENDIX. CORRECTION PARAMETERS METHODS FOR SOLVING INTEGRAL EQUATIONS OF THE FIRST KIND 1. The error model and problem Statement Algorithms of the parameter correction The discussion. The results of numerical experiments 232 Bibliography 237