Towards understanding CG and GMRES through examples

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10492943" target="_blank" >RIV/00216208:11320/24:10492943 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=Z1qIpG1m_g" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=Z1qIpG1m_g</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.laa.2024.04.003" target="_blank" >10.1016/j.laa.2024.04.003</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Towards understanding CG and GMRES through examples

Popis výsledku v původním jazyce

When the conjugate gradient (CG) method for solving linear algebraic systems was formulated about 70 years ago by Lanczos, Hestenes, and Stiefel, it was considered an iterative process possessing a mathematical finite termination property. With the deep insight of the original authors, CG was placed into a very rich mathematical context, including links with Gauss quadrature and continued fractions. The optimality property of CG was described via a normalized weighted polynomial least squares approximation to zero. This highly nonlinear problem explains the adaptation of CG iterates to the given data. Karush and Hayes immediately considered CG in infinite dimensional Hilbert spaces and investigated its superlinear convergence. Since then, the view of CG, as well as other Krylov subspace methods developed in the meantime, has changed. Today these methods are considered primarily as computational tools, and their behavior is typically characterized using linear upper bounds, or heuristics based on clustering of eigenvalues. Such simplifications limit the mathematical understanding of Krylov subspace methods, and also negatively affect their practical application. This paper offers a different perspective. Focusing on CG and the generalized minimal residual (GMRES) method, it presents mathematically important as well as practically relevant phenomena that uncover their behavior through a discussion of computed examples. These examples provide an easily accessible approach that enables understanding of the methods, while pointers to more detailed analyses in the literature are given. This approach allows readers to choose the level of depth and thoroughness appropriate for their intentions. Some of the points made in this paper illustrate well known facts. Others challenge mainstream views and explain existing misunderstandings. Several points refer to recent results leading to open problems. We consider CG and GMRES crucially important for the mathematical understanding, further development, and practical applications also of other Krylov subspace methods. The paper additionally addresses the motivation of preconditioning. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY -NC license (http:// creativecommons .org /licenses /by -nc /4 .0/).

Název v anglickém jazyce

Towards understanding CG and GMRES through examples

Popis výsledku anglicky

When the conjugate gradient (CG) method for solving linear algebraic systems was formulated about 70 years ago by Lanczos, Hestenes, and Stiefel, it was considered an iterative process possessing a mathematical finite termination property. With the deep insight of the original authors, CG was placed into a very rich mathematical context, including links with Gauss quadrature and continued fractions. The optimality property of CG was described via a normalized weighted polynomial least squares approximation to zero. This highly nonlinear problem explains the adaptation of CG iterates to the given data. Karush and Hayes immediately considered CG in infinite dimensional Hilbert spaces and investigated its superlinear convergence. Since then, the view of CG, as well as other Krylov subspace methods developed in the meantime, has changed. Today these methods are considered primarily as computational tools, and their behavior is typically characterized using linear upper bounds, or heuristics based on clustering of eigenvalues. Such simplifications limit the mathematical understanding of Krylov subspace methods, and also negatively affect their practical application. This paper offers a different perspective. Focusing on CG and the generalized minimal residual (GMRES) method, it presents mathematically important as well as practically relevant phenomena that uncover their behavior through a discussion of computed examples. These examples provide an easily accessible approach that enables understanding of the methods, while pointers to more detailed analyses in the literature are given. This approach allows readers to choose the level of depth and thoroughness appropriate for their intentions. Some of the points made in this paper illustrate well known facts. Others challenge mainstream views and explain existing misunderstandings. Several points refer to recent results leading to open problems. We consider CG and GMRES crucially important for the mathematical understanding, further development, and practical applications also of other Krylov subspace methods. The paper additionally addresses the motivation of preconditioning. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY -NC license (http:// creativecommons .org /licenses /by -nc /4 .0/).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Linear Algebra and Its Applications

ISSN

0024-3795

e-ISSN

1873-1856

Svazek periodika

692

Číslo periodika v rámci svazku

April

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

51

Strana od-do

241-291

Kód UT WoS článku

001230382800001

EID výsledku v databázi Scopus

2-s2.0-85190467299