Regularizing properties of a class of matrices including the optimal and the superoptimal preconditioners

Cipolla, Stefano, Di Fiore, Carmine, Durastante, Fabio and Zellini, Paolo (2018) Regularizing properties of a class of matrices including the optimal and the superoptimal preconditioners. Numerical Linear Algebra with Applications, 26 (2), [e2225]. (doi:10.1002/nla.2225).

Record type: Article

Abstract

In this work, given a positive definite matrix A, we introduce a class of matrices related to A, which is obtained by suitably combining projections of its powers onto algebras of matrices simultaneously diagonalized by a unitary transform. After a detailed theoretical study of some spectral properties of the matrices of this class, which suggests their use as regularizing preconditioners, we prove that such matrices can be cheaply computed when the matrix A has a Toeplitz structure. We provide numerical evidence of the advantages coming from the employment of the proposed preconditioners when used in regularizing procedures.

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Accepted/In Press date: 15 November 2018

e-pub ahead of print date: 17 December 2018

Published date: 17 December 2018

Additional Information: Funding Information: Gruppo Nazionale per il Calcolo Scientifico dell'Istituto Nazionale di Alta Matematica (INdAM-GNCS); 2018 INdAM-GNCS project “Tecniche innovative per problemi di algebra lineare”; MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, Grant/Award Number: CUP E83C18000100006

Keywords: optimal preconditioning, regularizing preconditioners, superoptimal preconditioning