| Uniform title | Iterativnye metody reshenii︠a︡ nekorrektnykh zadach. English |
| Series |
Inverse and ill-posed problems series ; 54
Inverse and ill-posed problems series ; v. 54. ^A458753
|
| Contents |
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 |
| Contents |
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 |
| Contents |
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 |
| Contents |
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 ≤ z ≤ 12 -- 15.4. Regularized method for z ≥ 13 -- Problems -- 16. Conclusion. |
| Bibliography note | Includes bibliographical references and index. |
| Access restriction | Available only to authorized users. |
| Technical details | Mode of access: World Wide Web |
| Genre/form | Electronic books. |
| LCCN | 2010038154 |
| ISBN | 9783110250640 (alk. paper) |
| ISBN | 3110250640 (alk. paper) |
