Accelerating the induced dimension reduction method using spectral information
type: Item
instance of: scholarly article
title: Accelerating the induced dimension reduction method using spectral information
zbMATH Open document ID: 1408.65011
author: R. Astudillo
author: J. M. de Gier
author: Martin B. van Gijzen
published in: Journal of Computational and Applied Mathematics
publication date: 2018-10-10T00:00:00Z
review text: The induced dimension reduction method (IDR(s)) [\textit{P. Sonneveld} and \textit{M. B. van Gijzen}, SIAM J. Sci. Comput. 31, No. 2, 1035--1062 (2008; Zbl 1190.65053)] is a short-recurrences Krylov method to solve non-symmetric systems of linear equations. In this work, the method is accelerated using spectral information. A Hessenberg relation is constructed from the IDR(s) residual recurrences formulas, and the eigenpairs ot the involved Hessenberg matrix are used to approximate the eigenvalues and eigenvectors. The inverses of the Ritz values are used as choices of the method's free scalars, thus leading to a self-contained variant of the Ritz-IDR(s) method [\textit{V. Simoncini} and \textit{D. B. Szyld}, SIAM J. Sci. Comput. 32, No. 4, 1898--1912 (2010; Zbl 1219.65039)] without call to an external eigensolver. In a number of numerical experiments this requires less computational time than a non-self-contained variant of the Ritz-IDR(s) method. In addition, Ritz vectors corresponding to small magnitude Ritz values are used to augment the search Krylov subspaces. The corresponding augmented methods are used to speed-up IDR(s) for the solution of sequences of systems of linear equations with identical matrices but different right-hand sides that are not available simultaneously. Spectral information gained when solving the first system is recycled in subsequent systems through augmentation with the corresponding approximate invariant subspaces. On both diffusion- and convection-dominated time-dependent convection-diffusion-reaction problems, the recycling shows a clear benefit in both the needed amount of matrix-vector products and CPU time.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F50
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 6950466
zbMATH Keywords: induced dimension reduction method
zbMATH Keywords: deflated and augmented Krylov method
zbMATH Keywords: sequence of systems of linear equations
zbMATH Keywords: sparse system of linear equations
describes a project that uses: MatrixMarket
describes a project that uses: Algorithm 913
MaRDI profile type: Publication
OpenAlex ID: W2809037386
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Wikidata QID: Q129625005
DOI: 10.1016/J.CAM.2018.06.014
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description: scientific article; zbMATH DE number 6950466