Analysis of GMRES for Low‐Rank and Small‐Norm Perturbations of the Identity Matrix

dc.contributor.authorCarr, Arielle K.en
dc.contributor.authorde Sturler, Ericen
dc.contributor.authorEmbree, Mark P.en
dc.date.accessioned2024-01-23T18:08:00Zen
dc.date.available2024-01-23T18:08:00Zen
dc.date.issued2023-03-24en
dc.description.abstractIn many applications, linear systems arise where the coefficient matrix takes the special form I + K + E, where I is the identity matrix of dimension n, rank(K) = p ≪ n, and ∥E∥ ≤ ϵ < 1. GMRES convergence rates for linear systems with coefficient matrices of the forms I+K and I+E are guaranteed by well-known theory, but only relatively weak convergence bounds specific to matrices of the form I + K + E currently exist. In this paper, we explore the convergence properties of linear systems with such coefficient matrices by considering the pseudospectrum of I + K. We derive a bound for the GMRES residual in terms of ϵ when approximately solving the linear system (I+K+E)x = b and identify the eigenvalues of I + K that are sensitive to perturbation. In particular, while a clustered spectrum away from the origin is often a good indicator of fast GMRES convergence, that convergence may be slow when some of those eigenvalues are ill-conditioned. We show there can be at most 2p eigenvalues of I + K that are sensitive to small perturbations. We present numerical results when using GMRES to solve a sequence of linear systems of the form (I + K<sub>j</sub> + E<sub>j</sub> )x<sub>j</sub> = b<sub>j</sub> that arise from the application of Broyden’s method to solve a nonlinear partial differential equation.en
dc.description.versionAccepted versionen
dc.format.mimetypeapplication/pdfen
dc.identifier.doihttps://doi.org/10.1002/pamm.202200267en
dc.identifier.issn1617-7061en
dc.identifier.issue1en
dc.identifier.orcidDe Sturler, Eric [0000-0002-9412-9360]en
dc.identifier.urihttps://hdl.handle.net/10919/117614en
dc.identifier.volume22en
dc.language.isoenen
dc.publisherWileyen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.titleAnalysis of GMRES for Low‐Rank and Small‐Norm Perturbations of the Identity Matrixen
dc.title.serialProceedings in Applied Mathematics and Mechanicsen
dc.typeArticle - Refereeden
dc.type.dcmitypeTexten
pubs.organisational-group/Virginia Techen
pubs.organisational-group/Virginia Tech/Scienceen
pubs.organisational-group/Virginia Tech/Science/Mathematicsen
pubs.organisational-group/Virginia Tech/All T&R Facultyen
pubs.organisational-group/Virginia Tech/Science/COS T&R Facultyen

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