Ricerc@Sapienza

In this paper, we deal with matrix-free preconditioners for nonlinear conjugate gradient (NCG) methods. In particular, we review proposals based on quasi-Newton updates, and either satisfying the secant equation or a secant-like equation at some of the previous iterates. Conditions are given proving that, in some sense, the proposed preconditioners also approximate the inverse of the Hessian matrix. In particular, the structure of the preconditioners depends both on low-rank updates along with some specific parameters. The low-rank updates are obtained as by-product of NCG iterations.

We introduce a class of positive definite preconditioners for the solution of large symmetric indefinite linear systems or sequences of such systems, in optimization frameworks. The preconditioners are iteratively constructed by collecting information on a reduced eigenspace of the indefinite matrix by means of a Krylov-subspace solver. A spectral analysis of the preconditioned matrix shows the clustering of some eigenvalues and possibly the nonexpansion of its spectrum.

Our understanding of decompression physiopathology has slowly improved during this last decade and some uncertainties have disappeared. A better understanding of anatomy and functional aspects of Patent Foramen Ovale (PFO) have slowly resulted in a more liberal approach toward the medical fitness to dive for those bearing a PFO.