Regularization of Ill-Posed Problems by Iteration Methods

2000
ISBN: 978-0-7923-6131-2
Verlag: Springer Netherlands

Buch, Englisch, Band 499, 342 Seiten, HC runder Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 1490 g

Reihe: Mathematics and Its Applications

Iteration regularization, i.e., utilization of iteration methods of any form for the stable approximate solution of ill-posed problems, is one of the most important but still insufficiently developed topics of the new theory of ill-posed problems. In this monograph, a general approach to the justification of iteration regulari­ zation algorithms is developed, which allows us to consider linear and nonlinear methods from unified positions. Regularization algorithms are the 'classical' iterative methods (steepest descent methods, conjugate direction methods, gradient projection methods, etc.) complemented by the stopping rule depending on level of errors in input data. They are investigated for solving linear and nonlinear operator equations in Hilbert spaces. Great attention is given to the choice of iteration index as the regularization parameter and to estimates of errors of approximate solutions. Stabilizing properties such as smoothness and shape constraints imposed on the solution are used. On the basis of these investigations, we propose and establish efficient regularization algorithms for stable numerical solution of a wide class of ill-posed problems. In particular, descriptive regularization algorithms, utilizing a priori information about the qualitative behavior of the sought solution and ensuring a substantial saving in computational costs, are considered for model and applied problems in nonlinear thermophysics. The results of calculations for important applications in various technical fields (a continuous casting, the treatment of materials and perfection of heat-protective systems using laser and composite technologies) are given.

Zielgruppe

Research

Autoren/Hrsg.

Gol'dman, N. L.

Gilyazov, S.F.

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik
• Mathematik | Informatik Mathematik Mathematische Analysis Moderne Anwendungen der Analysis
• Technische Wissenschaften Verkehrstechnik | Transportgewerbe Luft- und Raumfahrttechnik, Luftverkehr
• Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Wirtschaftswissenschaften Wirtschaftssektoren & Branchen Fertigungsindustrie Luftfahrtindustrie
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Technische Wissenschaften Verkehrstechnik | Transportgewerbe Fahrzeugtechnik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Computeranwendungen in der Mathematik

Inhaltsverzeichnis

1 Regularizing Algorithms for Linear Ill-Posed Problems: Unified Approach.- 2 Iteration Steepest Descent Methods for Linear Operator Equations.- 3 Iteration Conjugate Direction Methods for Linear Operator Equations.- 4 Iteration Steepest Descent Methods for Nonlinear Operator Equations.- 5 Iteration Methods for Ill-Posed Constrained Minimization Problems.- 6 Descriptive Regularization Algorithms on the Basis of the Conjugate Gradient Projection Method.