Iterative methods for ill-posed problems an introduction
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 grad...
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| Sprache: | eng |
| Veröffentlicht: |
Berlin u.a.
De Gruyter
2011
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| Schriftenreihe: | Inverse and ill-posed problems series
54 |
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Tag hinzufügen | Zusammenfassung: | 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 ≤ z ≤ 12 -- 15.4.Regularized method for z ≥ 13 -- Problems -- 16.Conclusion. |
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| Beschreibung: | Orig.-Ausg. ersch.: Moskva : Nauka, 1989 Literaturverz. S. [132] - 136 |
| Beschreibung: | XI, 136 S. Ill., graph. Darst. 25 cm |
| ISBN: | 3110250640 3-11-025064-0 9783110250640 978-3-11-025064-0 |