• 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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Fully algebraic domain decomposition preconditioners with adaptive spectral bounds

    Loïc Gouarin, Nicole Spillane

ETNA - Electronic Transactions on Numerical Analysis, pp. 169-196, 2024/04/05

doi: 10.1553/etna_vol60s169

doi: 10.1553/etna_vol60s169


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doi:10.1553/etna_vol60s169



doi:10.1553/etna_vol60s169

Abstract

In this article a new family of preconditioners is introduced for symmetricpositive definite linear systems. The new preconditioners, called the AWGpreconditioners (for Algebraic-Woodbury-GenEO), are constructedalgebraically. By this we mean that only the knowledge of the matrix $\mathbf{A}$for which the linear system is being solved is required. Thanks to the GenEOspectral coarse space technique, the condition number of the preconditionedoperator is bounded theoretically from above. This upper bound can be madesmaller by enriching the coarse space with more spectral modes.The novelty is that, unlike in previous work on the GenEO coarse spaces, noknowledge of a partially non-assembled form of $\mathbf{A}$ is required. Indeed, thespectral coarse space technique is not applied directly to $\mathbf{A}$ but to alow-rank modification of $\mathbf{A}$ of which a suitable non-assembled form is knownby construction. The extra cost is a second (and to this day rather expensive)coarse solve in the preconditioner. One of the AWG preconditioners has alreadybeen presented in a short preprint by Spillane [DomainDecomposition Methods in Science and EngineeringXXVI, Springer, Cham, 2022, pp. 745–752].This article is the first full presentation of the larger family of AWGpreconditioners. It includes proofs of the spectral bounds as well asnumerical illustrations.

Keywords: preconditioner, domain decomposition, coarse space, algebraic, linear system, Woodbury matrix identity