ETNA - Electronic Transactions on Numerical Analysis
|
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: verlag@oeaw.ac.at |
|
DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
|
ETNA - Electronic Transactions on Numerical Analysis ISBN 978-3-7001-8258-0 Online Edition Research Article
Andreas Frommer,
Brigit Jacob,
Kartsen Kahl,
Christian Wyss,
Ian Zwaan
S. 541 - 561 doi:10.1553/etna_vol53s541 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol53s541
Abstract: The quadratic numerical range $W^2(A)$ is a subset of thestandard numerical range of a linear operator, which still contains itsspectrum. It arises naturally in operators that have a $2 \times 2$ blockstructure, and it consists of at most two connected components, none of whichnecessarily convex. The quadratic numerical range can thus reveal spectralgaps, and it can in particular indicate that the spectrum of an operator isbounded away from $0$.We exploit this property in the finite-dimensional setting to derive Krylovsubspace-type methods to solve the system $Ax = b$, in which the iteratesarise as solutions of low-dimensional models of the operator whosequadratic numerical range is contained in $W^2(A)$. This implies that theiterates are always well-defined and that, as opposed to standard FOM, largevariations in the approximation quality of consecutive iterates are avoided,although $0$ lies within the convex hull of the spectrum. We also considerGMRES variants that are obtained in asimilar spirit. We derive theoretical results on basic properties ofthese methods, review methods on how to compute the required bases in a stablemanner, and present results of several numerical experiments illustratingimprovements over standard FOM and GMRES. Keywords: quadratic numerical range, full orthogonalization method (FOM), generalized minimal residual method (GMRES), linear systems, projection methods, two-level orthogonal Arnoldi method Published Online: 2020/11/30 09:42:41 Object Identifier: 0xc1aa5576 0x003c08c5 Rights: . 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. …
|
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: verlag@oeaw.ac.at |