Cargando…
Iterative methods for ill-posed problems : an introduction /
Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the assoc...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés Ruso |
| Publicado: |
Berlin ; New York :
De Gruyter,
©2011.
|
| Colección: | Inverse and ill-posed problems series ;
v. 54. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Machine generated contents note: 1. The regularity condition. Newton's method
- 1.1. Preliminary results
- 1.2. Linearization procedure
- 1.3. Error analysis
- Problems
- 2. The Gauss
- Newton method
- 2.1. Motivation
- 2.2. Convergence rates
- Problems
- 3. The gradient method
- 3.1. The gradient method for regular problems
- 3.2. Ill-posed case
- Problems
- 4. Tikhonov's scheme
- 4.1. The Tikhonov functional
- 4.2. Properties of a minimizing sequence
- 4.3. Other types of convergence
- 4.4. Equations with noisy data
- Problems
- 5. Tikhonov's scheme for linear equations
- 5.1. The main convergence result
- 5.2. Elements of spectral theory
- 5.3. Minimizing sequences for linear equations.
- 5.4. A priori agreement between the regularization parameter and the error for equations with perturbed right-hand sides
- 5.5. The discrepancy principle
- 5.6. Approximation of a quasi-solution
- Problems
- 6. The gradient scheme for linear equations
- 6.1. The technique of spectral analysis
- 6.2. A priori stopping rule
- 6.3. A posteriori stopping rule
- Problems
- 7. Convergence rates for the approximation methods in the case of linear irregular equations
- 7.1. The source-type condition (STC)
- 7.2. STC for the gradient method
- 7.3. The saturation phenomena
- 7.4. Approximations in case of a perturbed STC
- 7.5. Accuracy of the estimates
- Problems
- 8. Equations with a convex discrepancy functional by Tikhonov's method
- 8.1. Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional.
- 8.2. An illustrative example
- Problems
- 9. Iterative regularization principle
- 9.1. The idea of iterative regularization
- 9.2. The iteratively regularized gradient method
- Problems
- 10. The iteratively regularized Gauss
- Newton method
- 10.1. Convergence analysis
- 10.2. Further properties of IRGN iterations
- 10.3. A unified approach to the construction of iterative methods for irregular equations
- 10.4. The reverse connection control
- Problems
- 11. The stable gradient method for irregular nonlinear equations
- 11.1. Solving an auxiliary finite dimensional problem by the gradient descent method
- 11.2. Investigation of a difference inequality
- 11.3. The case of noisy data
- Problems
- 12. Relative computational efficiency of iteratively regularized methods
- 12.1. Generalized Gauss
- Newton methods
- 12.2. A more restrictive source condition.
- 12.3. Comparison to iteratively regularized gradient scheme
- Problems
- 13. Numerical investigation of two-dimensional inverse gravimetry problem
- 13.1. Problem formulation
- 13.2. The algorithm
- 13.3. Simulations
- Problems
- 14. Iteratively regularized methods for inverse problem in optical tomography
- 14.1. Statement of the problem
- 14.2. Simple example
- 14.3. Forward simulation
- 14.4. The inverse problem
- 14.5. Numerical results
- Problems
- 15. Feigenbaum's universality equation
- 15.1. The universal constants
- 15.2. Ill-posedness
- 15.3. Numerical algorithm for 2 & le; z & le; 12
- 15.4. Regularized method for z & ge; 13
- Problems
- 16. Conclusion.