Analysis of GMRES for Low‐Rank and Small‐Norm Perturbations of the Identity Matrix
dc.contributor.author | Carr, Arielle K. | en |
dc.contributor.author | de Sturler, Eric | en |
dc.contributor.author | Embree, Mark P. | en |
dc.date.accessioned | 2024-01-23T18:08:00Z | en |
dc.date.available | 2024-01-23T18:08:00Z | en |
dc.date.issued | 2023-03-24 | en |
dc.description.abstract | In 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.version | Accepted version | en |
dc.format.mimetype | application/pdf | en |
dc.identifier.doi | https://doi.org/10.1002/pamm.202200267 | en |
dc.identifier.issn | 1617-7061 | en |
dc.identifier.issue | 1 | en |
dc.identifier.orcid | De Sturler, Eric [0000-0002-9412-9360] | en |
dc.identifier.uri | https://hdl.handle.net/10919/117614 | en |
dc.identifier.volume | 22 | en |
dc.language.iso | en | en |
dc.publisher | Wiley | en |
dc.rights | In Copyright | en |
dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
dc.title | Analysis of GMRES for Low‐Rank and Small‐Norm Perturbations of the Identity Matrix | en |
dc.title.serial | Proceedings in Applied Mathematics and Mechanics | en |
dc.type | Article - Refereed | en |
dc.type.dcmitype | Text | en |
pubs.organisational-group | /Virginia Tech | en |
pubs.organisational-group | /Virginia Tech/Science | en |
pubs.organisational-group | /Virginia Tech/Science/Mathematics | en |
pubs.organisational-group | /Virginia Tech/All T&R Faculty | en |
pubs.organisational-group | /Virginia Tech/Science/COS T&R Faculty | en |