Theory of linear ill-posed problems and its applications /

This monograph is a revised and extended version of the Russian edition from 1978. It includes the general theory of linear ill-posed problems concerning e. g. the structure of sets of uniform regularization, the theory of error estimation, and the optimality method. As a distinguishing feature the...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Ivanov, Valentin Konstantinovich
Otros Autores: Vasin, V. V., Tanana, Vitaliĭ Pavlovich
Formato: Electrónico eBook
Idioma:Inglés
Ruso
Publicado: Utrecht ; Boston : VSP, 2002.
Edición:Second edition].
Colección:Inverse and ill-posed problems series.
Temas:
Operator equations.
Integral operators.
Numerical analysis > Improperly posed problems.
Équations à opérateurs.
Opérateurs intégraux.
Analyse numérique > Problèmes mal posés.
MATHEMATICS > Calculus.
MATHEMATICS > Mathematical Analysis.
Integral operators
Numerical analysis > Improperly posed problems
Operator equations
Inkorrekt gestelltes Problem
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Preface to the Second Edition
  • Preface
  • Contents
  • Introduction
  • Chapter 1. Well-posedness of problems
  • 1.1. Problem formulation. Hadamardâ€?s concept of well-posedness
  • 1.2. Examples of ill-posed problems
  • 1.3. Tikhonovâ€?s concept of well-posedness. Sets of well-posedness
  • 1.4. Stability theorems and their applications
  • 1.5. Normal solvability of operator equations
  • 1.6. Quasisolutions on compact and boundedly compact sets
  • Chapter 2. Regularizing family of operators
  • 2.1. Pointwise and uniform regularization of operator equations
  • 2.2. Geometric theorems on structure of boundedly compact sets2.3. Uniform regularization of equations with completely continuous operators
  • 2.4. Structure of sets of uniform regularization in Hilbert spaces
  • 2.5. Sets of uniform regularization for continuous operators
  • Chapter 3. Basic techniques for constructing regularizing algorithms
  • 3.1. Reduction to operator equations of the second kind
  • 3.2. Method of quasisolutions
  • 3.3. Tikhonovâ€?s method of regularization
  • 3.4. Method of residual
  • 3.5. On relations between variational methods
  • 3.6. Generalized method of residual3.7. Method based on the Picard theorem
  • 3.8. Iterative methods
  • 3.9. Regularization of the Fredholm integral equations of the first kind
  • 3.10. Regularization methods for differential equations
  • Chapter 4. Optimality and stability of methods for solving ill-posed problems. Error estimation
  • 4.1. Classification of ill-posed problems and the concept of an optimal method
  • 4.2. Lower estimate for error of the optimal method
  • 4.3. Error of the regularization method
  • 4.4. Algorithmic peculiarities of the generalized method of residual4.5. Error of the quasisolution method
  • 4.6. The regularization method with the parameter a satisfying the residual principle
  • 4.7. Investigation of the simplest scheme of the Lavrent'ev method
  • 4.8. The method of projective regularization
  • 4.9. Calculation of the module of continuity
  • Chapter 5. Determination of values of unbounded operators
  • 5.1. A unified approach to the solution of ill-posed problems
  • 5.2. Multivalued linear operators and their properties
  • 5.3. Determination of normal values of linear operators by variational methods5.4. The best approximation of unbounded operators
  • 5.5. Optimal regularization of the problem of evaluating a derivative in the space C(â€?â?ž, â?ž)
  • Chapter 6. Finite-dimensional approximation of regulirizing algorithms
  • 6.1. The concept of r-uniform convergence of linear operators
  • 6.2. A general scheme of the finite-dimensional approximation
  • 6.3. Application of the general scheme
  • 6.4. Projection method

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