Conjugate Gradient Type Methods for Ill-Posed Problems: 1st Edition (Hardback) book cover

Conjugate Gradient Type Methods for Ill-Posed Problems

1st Edition

By Martin Hanke

Chapman and Hall/CRC

144 pages

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Hardback: 9780582273702
pub: 1995-04-26
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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 Most of the results are as yet unpublished, or obscured in the Russian literature.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.

Table of Contents

1. Preface and Notation

2. Conjugate gradient type methods

3. Regularizing properties of MR and CGNE

4. Regularizing properties of CG and CGME

5. On the number of iterations

6. A minimal residual method for indefinite problems



About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
MATHEMATICS / Functional Analysis