• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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ETNA - Electronic Transactions on Numerical Analysis



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Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2,
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
https://verlag.oeaw.ac.at, e-mail: bestellung.verlag@oeaw.ac.at
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Inexact linear solves in the low-rank alternating direction implicit iteration for large Sylvester equations

    Patrick Kürschner

ETNA - Electronic Transactions on Numerical Analysis, pp. 119-137, 2024/09/19

doi: 10.1553/etna_vol62s119

doi: 10.1553/etna_vol62s119


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doi:10.1553/etna_vol62s119



doi:10.1553/etna_vol62s119

Abstract

We consider iteration for approximately solving large-scale algebraic Sylvester equations. Inside every iteration step of this iterative process, a pair of linear systems of equations has to be solved. We investigate the situation when those inner linear systems are solved inexactly by an iterative method such as, for example, preconditioned Krylov subspace methods. The main contribution of this work are thresholds for the required accuracies regarding the inner linear systems, which dictate when the employed inner Krylov subspace methods can be safely terminated. The goal is to save computational effort by solving the inner linear system as inaccurately as possible without endangering the functionality of the low-rank Sylvester–ADI method. Ideally, the inexact ADI method mimics the convergence behavior of the more expensive exact ADI method, where the linear systems are solved directly. Alongside the theoretical results, strategies for an actual practical implementation of the stopping criteria are also developed. Numerical experiments confirm the effectiveness of the proposed strategies.

Keywords: Sylvester equation, alternating direction implicit, low-rank approximation, inner–outer methods

Published Online: 2024/09/19

Object Identifier: 0xc1aa5572 0x003f6020

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