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: verlag@oeaw.ac.at |
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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)
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ETNA - Electronic Transactions on Numerical Analysis ISBN 978-3-7001-8258-0 Online Edition Research Article
Fei Xue
One-step convergence of inexact Anderson acceleration for contractive and non-contractive mappings ()
S. 285 - 309doi:10.1553/etna_vol55s285 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol55s285
Abstract: We give a one-step convergence analysis of inexact Anderson acceleration for the fixed point iteration $x_{k+1}=g(x_k)$ with a potentially non-contractive mapping $g$, where $g(x_k)$ is evaluated approximately and the minimization of the nonlinear residual norms is performed in the vector 2-norm by the linear least-squares method. If $g$ is non-contractive, then the original fixed point iteration does not converge, but a recent analysis by S. Pollock and L. Rebholz [IMA J. Numer. Anal., 41 (2021),pp. 2841–2872] shows that Anderson acceleration may still converge provided that the minimization at each step has a sufficient gain. In this paper, we show that inexact Anderson acceleration exhibits essentially the same convergence behavior as the exact algorithm if each $g(x_k)$ is evaluated with an error proportional to the nonlinear residual norm $\|w_k\|=\|g(x_k)-x_k\|$, regardless of whether $g$ is contractive or not. This means that the existing relationship between exact and inexact Anderson acceleration can be generalized in a unified framework for both contractive and non-contractive mappings. Numerical experiments show that the inexact algorithm can converge as rapidly as the exact counterpart while it can lower the computational cost. Keywords: fixed point iteration, inexact Anderson acceleration, non-contractive mapping, one-step convergence Published Online: 2022/01/25 10:29:54 Object Identifier: 0xc1aa5572 0x003d32c8 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. …
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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: verlag@oeaw.ac.at |