Conjugate Gradient Type Methods for Ill-Posed Problems

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 mor
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 Most of the results are as yet unpublished, or obscured in the Russian literature.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 gradien type methods to real orthogonal polynomials, and elementar properties of these polynomials. These prerequisites are provided i a first chapter. Applications to image reconstruction and invers heat transfer problems are pointed out, and exemplarily numerica results are shown for these applications.