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Main Title: IDR explained
Author(s): Gutknecht, Martin H.
Type: Research Paper
Abstract: The Induced Dimension Reduction (IDR) method is a Krylov space method for solving linear systems that was developed by Peter Sonneveld around 1979. It was only noticed by few people, and mainly as the forerunner of Bi-CGSTAB, which was introduced a decade later. In 2007 Sonneveld and van Gijzen reconsidered IDR and generalized it to IDR(s), claiming that IDR(1) \approx IDR is equally fast but preferable to the closely related Bi-CGSTAB, and that IDR(s) with s > 1 may be much faster than Bi-CGSTAB. It also turned out that when s > 1, IDR(s) is related to ML(s)BiCGSTAB of Yeung and Chan, and that there is quite some flexibility in the IDR approach. This approach differs completely from traditional approaches to Krylov space methods, and therefore it requires an extra effort to get familiar with it and to understand the connections as well as the differences to better known Krylov space methods. This expository paper aims at providing some help in this and to make the method understandable even to non-experts. After presenting the history of IDR and related methods we summarize some of the basic facts on Krylov space methods. Then we present the original IDR(s) in detail and put it into perspective with other methods. Specifically, we analyze the differences between the IDR method published 1980, IDR(1) and Bi-CGSTAB. At the end, we discuss a recently proposed ingenious variant of IDR(s) whose residuals fulfill extra orthogonality conditions. There we dwell on details that have been left out in the publications of van Gijzen and Sonneveld.
Subject(s): Krylov space method
iterative method
induced dimension reduction
large nonsymmetric linear system
Issue Date: 2-Apr-2009
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 65F10 Iterative methods for linear systems
65F50 Sparse matrices
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2009, 13
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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