Conjugate Gradient Type Methods for Ill-Posed Problems (Pitman Research Notes in Mathematics Series)

Martin Hanke

ISBN 10: 0582273706 / ISBN 13: 9780582273702

Published by Longman, 1995

About this Item

Paperback, x + 134 pages, NOT ex-library. Good condition, faint staining to lower edges, gently bent lower outer corners; unmarked text, free of inscriptions and stamps. -- This monograph provides a detailed examination of conjugate gradient type methods and their application to linear ill-posed problems, focusing on their effectiveness as regularisation tools. Such problems, which lack continuous dependence on the data and often require careful handling of noise and incompleteness, appear throughout the natural and technical sciences. Examples include astronomical and geophysical imaging, signal analysis, computerised tomography and inverse heat transfer. The text presents a unifying analysis covering a family of these iterative methods, among them minimal residual approaches and the classical conjugate gradient algorithm. It explores their regularising properties in depth, establishes sharp convergence results and discusses practical aspects such as the selection of stopping rules. The analytical framework relies on the close connection between conjugate gradient iterations and the theory of real orthogonal polynomials, supplying the necessary background in an opening section. Building on foundational contributions, including those of Nemirovskii, the book makes available results that were previously scattered or published mainly in Russian, and introduces new techniques particularly for the Hestenes-Stiefel variant. Extensions to self-adjoint indefinite operator equations are also addressed. Numerical illustrations for image reconstruction and inverse heat transfer demonstrate the methods' performance in concrete settings. With the growing importance of inverse problems in contemporary fields such as medical imaging, remote sensing, machine learning regularisation and geophysical data inversion, this work offers enduring theoretical foundations for the design of stable computational algorithms. It serves as a useful reference for mathematicians, engineers and scientists engaged in developing or applying iterative solutions to unstable inverse problems in high-dimensional or noisy environments.

Seller Inventory # 013347

Bibliographic Details

• Title: Conjugate Gradient Type Methods for Ill-Posed Problems (Pitman Research Notes in Mathematics Series)
• Publisher: Longman
• Publication Date: 1995
• Language: English
• ISBN 10: 0582273706
• ISBN 13: 9780582273702
• Binding: Soft cover
• Edition: 1st Edition
• Condition: Good

About this title

Synopsis

The conjugate gradient method is a powerful tool for the iterative solution of self-adjoint operator equations in Hilbert space.This volume summarizes and extends the developments of the past decade concerning the applicability of the conjugate gradient method (and some of its variants) to ill posed problems and their regularization. Such problems occur in applications from almost all natural and technical sciences, including astronomical and geophysical imaging, signal analysis, computerized tomography, inverse heat transfer problems, and many more This Research Note presents a unifying analysis of an entire family of conjugate gradient type methods. Most of the results are as yet unpublished, or obscured in the Russian literature. Beginning with the original results by Nemirovskii and others for minimal residual type methods, equally sharp convergence results are then derived with a different technique for the classical Hestenes-Stiefel algorithm. In the final chapter some of these results are extended to selfadjoint indefinite operator equations. The main tool for the analysis is the connection of conjugate gradient type methods to real orthogonal polynomials, and elementary properties of these polynomials. These prerequisites are provided in a first chapter. Applications to image reconstruction and inverse heat transfer problems are pointed out, and exemplarily numerical results are shown for these applications.

About the Author

Hanke\, Martin