## Conjugate Gradient Type Methods for Ill-Posed ProblemsThe 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 Most of the results are as yet unpublished, or obscured in the Russian literature.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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### Contents

Preface | 1 |

Conjugate Gradient Type Methods | 7 |

Regularizing Properties of MR and CGNE | 35 |

Regularizing Properties of CG and CGME | 57 |

On the Number of Iterations | 73 |

A Minimal Residual Method for Indefinite Problems | 91 |

127 | |

133 | |

### Common terms and phrases

A_,fc A+,fc Algorithm analysis approximation asymptotics bound CG and CGME CG CGME CGNE chapters compact operator computed conjugate gradient method conjugate gradient type Consequently Consider Corollary corresponding defined denote discrepancy principle diverge eigenvalues estimate example function gradient type methods H Brezis hence heuristic stopping rule ill-posed problems implementation implies indefinite problems inequality inner product Inserting iterates xk iteration count iteration error iteration polynomials Jacobi polynomial k(ys Krylov subspace Lemma matrix method with parameter minimal MR-II noise opt notation Note obtains orthogonal polynomials partial differential equations perturbed pk-i pk+l point spread function proof of Theorem Proposition 2.1 regularizing properties remark following residual polynomials pk respectively rewritten right-hand side roots of pk satisfies Assumption 3.6 Section 2.1 selfadjoint semidefinite sequence shown solution spectrum steps stopping index k(S Stopping Rule 3.10 Stopping Rule 4.7 Theorem 3.4 Tikhonov regularization Txsk yields