Stability analysis of the two-level orthogonal Arnoldi procedure

Ding Lu; Yangfeng Su; Zhaojun Bai

doi:10.1137/151005142

Stability analysis of the two-level orthogonal Arnoldi procedure

Ding Lu
, Yangfeng Su
, Zhaojun Bai

Mathematics

Producción científica: Article › revisión exhaustiva

46 Citas (Scopus)

Resumen

The second-order Arnoldi (SOAR) procedure is an algorithm for computing an orthonormal basis of the second-order Krylov subspace. It has found applications in solving quadratic eigenvalue problems and model order reduction of second-order dynamical systems among others. Unfortunately, the SOAR procedure can be numerically unstable. The two-level orthogonal Arnoldi (TOAR) procedure has been proposed as an alternative to SOAR to cure the numerical instability. In this paper, we provide a rigorous stability analysis of the TOAR procedure. We prove that under mild assumptions, the TOAR procedure is backward stable in computing an orthonormal basis of the associated linear Krylov subspace. The benefit of the backward stability of TOAR is demonstrated by its high accuracy in structure-preserving model order reduction of second-order dynamical systems.

Idioma original	English
Páginas (desde-hasta)	195-214
Número de páginas	20
Publicación	SIAM Journal on Matrix Analysis and Applications
Volumen	37
N.º	1
DOI	https://doi.org/10.1137/151005142
Estado	Published - 2016

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Financiación

School of Mathematical Sciences, Fudan University, Shanghai 200433, China ([email protected], [email protected]). Part of this work was done while the first author was visiting the University of California, Davis, supported by China Scholarship Council. The research of the second author was supported in part by the Innovation Program of Shanghai Municipal Education Commission 13zz007, E-Institutes of Shanghai Municipal Education Commission N.E303004, and NSFC key project 91330201. ‡Department of Computer Science and Department of Mathematics, University of California, Davis, CA 95616, USA ([email protected]). The research of the this author was supported in part by the NSF grants DMS-1522697 and CCF-1527091.

Financiadores	Número del financiador
National Science Foundation (NSF)	CCF-1527091, DMS-1522697
National Natural Science Foundation of China (NSFC)	CA 95616, 91330201
Fudan University
Shanghai Municipal Education Commission	13zz007
China Scholarship Council

ASJC Scopus subject areas

Analysis

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AB - The second-order Arnoldi (SOAR) procedure is an algorithm for computing an orthonormal basis of the second-order Krylov subspace. It has found applications in solving quadratic eigenvalue problems and model order reduction of second-order dynamical systems among others. Unfortunately, the SOAR procedure can be numerically unstable. The two-level orthogonal Arnoldi (TOAR) procedure has been proposed as an alternative to SOAR to cure the numerical instability. In this paper, we provide a rigorous stability analysis of the TOAR procedure. We prove that under mild assumptions, the TOAR procedure is backward stable in computing an orthonormal basis of the associated linear Krylov subspace. The benefit of the backward stability of TOAR is demonstrated by its high accuracy in structure-preserving model order reduction of second-order dynamical systems.

KW - Backward stability

KW - Dynamical systems

KW - Model order reduction

KW - Second-order Arnoldi procedure

KW - Second-order Krylov subspace

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Stability analysis of the two-level orthogonal Arnoldi procedure

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