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
Alessandro Buccini,
Lucas Onisk,
Lothar Reichel
S. 348 - 377 doi:10.1553/etna_vol58s348 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol58s348
Abstract: Linear systems of equations with a matrix whose singular values decay to zero with increasing index number, and without a significant gap, are commonly referred to as linear discrete ill-posed problems. Such systems arise, e.g., when discretizing a Fredholm integral equation of the first kind. The right-hand side vectors of linear discrete ill-posed problems that arise in science and engineering often represent an experimental measurement that is contaminated by measurement error. The solution to these problems typically is very sensitive to this error. Previous works have shown that error propagation into the computed solution may be reduced by using specially designed iterative methods that allow the user to select the subspace in which the approximate solution is computed. Since the dimension of this subspace often is quite small, its choice is important for the quality of the computed solution. This work describes algorithms for three iterative methods that modify the GMRES, block GMRES, and global GMRES methods for the solution of appropriate linear systems of equations. We contribute to the work already available on this topic by introducing two block variants for the solution of linear systems of equations with multiple right-hand side vectors. The dominant computational aspects are discussed, and software for each method is provided. Additionally, we illustrate the utility of these iterative subspace methods through numerical examples focusing on image reconstruction. This paper is accompanied by software. Keywords: ill-posed problems, iterative method, Arnoldi process, block Arnoldi process, global Arnoldi process Published Online: 2023/04/03 14:53:57 Object Identifier: 0xc1aa5572 0x003e29ef 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 |