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

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	1406-1422
Number of pages	17
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	43
Issue number	3
DOIs	https://doi.org/10.1137/20m1320018
Publication status	Published - 2022

Keywords

Arnoldi algorithm
GMRES algorithm
inexact inner products
variable precision arithmetic

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10.1137/20m1320018

Cite this

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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",

journal = "SIAM Journal on Matrix Analysis and Applications",

issn = "0895-4798",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "3",

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TY - JOUR

T1 - A NOTE ON INEXACT INNER PRODUCTS IN GMRES

AU - Gratton, Serge

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.

PY - 2022

Y1 - 2022

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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U2 - 10.1137/20m1320018

DO - 10.1137/20m1320018

M3 - Article

AN - SCOPUS:85138488898

SN - 0895-4798

VL - 43

SP - 1406

EP - 1422

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 3

ER -

A NOTE ON INEXACT INNER PRODUCTS IN GMRES

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Use of algebraic multigrid techniques in constrained nonconvex optimization

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A NOTE ON INEXACT INNER PRODUCTS IN GMRES

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Use of algebraic multigrid techniques in constrained nonconvex optimization

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