An adaptive regularizing method for ill-posed problems

Conjugate Gradient is widely used as a regularizing technique for solving linear systems with ill-conditioned coe±cient matrix and right-hand side vector perturbed by noise. It enjoys a good convergence rate and computes quickly an iterate, say xkopt , 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 diffcult detection of kopt and to a sharp increase of the error after the koptth iteration. In this paper we propose an adaptive 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.