A NOTE ON INEXACT INNER PRODUCTS IN GMRES

Serge Gratton; Ehouarn Simon; David Titley-Peloquin; Philippe L. Toint

doi:10.1137/20m1320018

A NOTE ON INEXACT INNER PRODUCTS IN GMRES

Serge Gratton, Ehouarn Simon, David Titley-Peloquin, Philippe L. Toint

Résultats de recherche: Contribution à un journal/une revue › Article › Revue par des pairs

Résumé

We show to what extent the accuracy of the inner products computed in the GMRES iterative solver can be reduced as the iterations proceed without affecting the convergence rate or final accuracy achieved by the iterates. We bound the loss of orthogonality in GMRES with inexact inner products. We use this result to bound the ratio of the residual norm in inexact GMRES to the residual norm in exact GMRES and give a condition under which this ratio remains close to 1. We illustrate our results with examples in variable floating-point arithmetic.

langue originale	Anglais
Pages (de - à)	1406-1422
Nombre de pages	17
journal	SIAM Journal on Matrix Analysis and Applications
Volume	43
Numéro de publication	3
Les DOIs	https://doi.org/10.1137/20m1320018
Etat de la publication	Publié - 2022

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10.1137/20m1320018

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20m1320018
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Multigrilles algébriques en optimisation non linéaire
Toint, P. (Responsable du Projet) & Weber Mendonca, M. (Chercheur)
1/10/05 → 3/09/09
Projet: Projet de thèse

Contient cette citation

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title = "A NOTE ON INEXACT INNER PRODUCTS IN GMRES",

abstract = "We show to what extent the accuracy of the inner products computed in the GMRES iterative solver can be reduced as the iterations proceed without affecting the convergence rate or final accuracy achieved by the iterates. We bound the loss of orthogonality in GMRES with inexact inner products. We use this result to bound the ratio of the residual norm in inexact GMRES to the residual norm in exact GMRES and give a condition under which this ratio remains close to 1. We illustrate our results with examples in variable floating-point arithmetic.",

keywords = "Arnoldi algorithm, GMRES algorithm, inexact inner products, variable precision arithmetic",

author = "Serge Gratton and Ehouarn Simon and David Titley-Peloquin and Toint, {Philippe L.}",

note = "Funding Information: The work of the first and fourth authors was partially supported by the 3IA Artificial and Natural Intelligence Toulouse Institute, French {"}Investing for the Future - PIA3{"} program under grant ANR-19-PI3A-0004. The work of the second author was partially supported by the French National program LEFE (Les Enveloppes Fluides et l'Environnement). Publisher Copyright: {\textcopyright} 2022 Society for Industrial and Applied Mathematics.",

year = "2022",

doi = "10.1137/20m1320018",

language = "English",

volume = "43",

pages = "1406--1422",

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AU - Simon, Ehouarn

AU - Titley-Peloquin, David

AU - Toint, Philippe L.

N1 - Funding Information: The work of the first and fourth authors was partially supported by the 3IA Artificial and Natural Intelligence Toulouse Institute, French "Investing for the Future - PIA3" program under grant ANR-19-PI3A-0004. The work of the second author was partially supported by the French National program LEFE (Les Enveloppes Fluides et l'Environnement). Publisher Copyright: © 2022 Society for Industrial and Applied Mathematics.

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N2 - We show to what extent the accuracy of the inner products computed in the GMRES iterative solver can be reduced as the iterations proceed without affecting the convergence rate or final accuracy achieved by the iterates. We bound the loss of orthogonality in GMRES with inexact inner products. We use this result to bound the ratio of the residual norm in inexact GMRES to the residual norm in exact GMRES and give a condition under which this ratio remains close to 1. We illustrate our results with examples in variable floating-point arithmetic.

AB - We show to what extent the accuracy of the inner products computed in the GMRES iterative solver can be reduced as the iterations proceed without affecting the convergence rate or final accuracy achieved by the iterates. We bound the loss of orthogonality in GMRES with inexact inner products. We use this result to bound the ratio of the residual norm in inexact GMRES to the residual norm in exact GMRES and give a condition under which this ratio remains close to 1. We illustrate our results with examples in variable floating-point arithmetic.

KW - Arnoldi algorithm

KW - GMRES algorithm

KW - inexact inner products

KW - variable precision arithmetic

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A NOTE ON INEXACT INNER PRODUCTS IN GMRES

Résumé

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Projets

Multigrilles algébriques en optimisation non linéaire

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