# American Institute of Mathematical Sciences

May  2014, 8(2): 409-420. doi: 10.3934/ipi.2014.8.409

## An inner-outer regularizing method for ill-posed problems

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

Received  September 2013 Revised  February 2014 Published  May 2014

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.
Citation: Paola Favati, Grazia Lotti, Ornella Menchi, Francesco Romani. An inner-outer regularizing method for ill-posed problems. Inverse Problems & Imaging, 2014, 8 (2) : 409-420. doi: 10.3934/ipi.2014.8.409
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