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 Most of the results are as yet unpublished,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.
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