An inner-outer regularizing method for ill-posed problems

Paola Favati; Grazia Lotti; Ornella Menchi; Francesco Romani

doi:10.3934/ipi.2014.8.409

Article Contents

2014, Volume 8, Issue 2: 409-420. Doi: 10.3934/ipi.2014.8.409

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An inner-outer regularizing method for ill-posed problems

1.
IIT - CNR Via G. Moruzzi 1, 56124 Pisa
2.
Dipart. di Matematica e Informatica, University of Parma, Viale G. Usberti 53/A, 43100 Parma
3.
Dipart. di Informatica, University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa

Received: August 31, 2013

Revised: January 31, 2014

Published: April 2014

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Abstract

Conjugate Gradient is widely used as a regularizing technique for solving linear systems with ill-conditioned coefficient matrix and right-hand side vector perturbed by noise. It enjoys a good convergence rate and computes quickly an iterate, say $x_{k_{opt}}$, which minimizes the error with respect to the exact solution. This behavior can be a disadvantage in the regularization context, because also the high-frequency components of the noise enter quickly the computed solution, leading to a difficult detection of $k_{opt}$ and to a sharp increase of the error after the $k_{opt}$th iteration. In this paper we propose an inner-outer algorithm based on a sequence of restarted Conjugate Gradients, with the aim of overcoming this drawback. A numerical experimentation validates the effectiveness of the proposed algorithm.

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Mathematics Subject Classification: 65F22, 65F10.

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