Optimal rank matrix algebras preconditioners

F. Tudisco, C. Di Fiore, E. E. Tyrtyshnikov

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

When a linear system Ax=y is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner P and move to the preconditioned system P-1Ax=P-1y. The use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate. The construction of optimal rank preconditioners is strongly related to the possibility of splitting A as A=P+R+E, where E is a small perturbation and R is of low rank (Tyrtyshnikov, 1996) [1]. In the present work we extend the black-dot algorithm for the computation of such splitting for P circulant (see Oseledets and Tyrtyshnikov, 2006 [2]), to the case where P is in A, for several known low-complexity matrix algebras A. The algorithm so obtained is particularly efficient when A is Toeplitz plus Hankel like. We finally discuss in detail the existence and the properties of the decomposition A=P+R+E when A is Toeplitz, also extending to the φ-circulant and Hartley-type cases some results previously known for P circulant.
LanguageEnglish
Pages405-427
Number of pages23
JournalLinear Algebra and its Applications
Volume438
Issue number1
Early online date19 Sep 2012
DOIs
Publication statusPublished - 1 Jan 2013

Fingerprint

Matrix Algebra
Preconditioner
Otto Toeplitz
Convergence Rate
Iterative methods
Linear systems
GMRES
Hankel
P Systems
Decomposition
Small Perturbations
Low Complexity
Linear Systems
Iteration
Decompose

Keywords

  • preconditioning
  • matrix algebras
  • Toeplitz
  • Hankel
  • clustering
  • fast discrete transforms

Cite this

Tudisco, F. ; Di Fiore, C. ; Tyrtyshnikov, E. E. / Optimal rank matrix algebras preconditioners. In: Linear Algebra and its Applications. 2013 ; Vol. 438, No. 1. pp. 405-427.
@article{980eacb6cba04ce78e089277a658b00b,
title = "Optimal rank matrix algebras preconditioners",
abstract = "When a linear system Ax=y is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner P and move to the preconditioned system P-1Ax=P-1y. The use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate. The construction of optimal rank preconditioners is strongly related to the possibility of splitting A as A=P+R+E, where E is a small perturbation and R is of low rank (Tyrtyshnikov, 1996) [1]. In the present work we extend the black-dot algorithm for the computation of such splitting for P circulant (see Oseledets and Tyrtyshnikov, 2006 [2]), to the case where P is in A, for several known low-complexity matrix algebras A. The algorithm so obtained is particularly efficient when A is Toeplitz plus Hankel like. We finally discuss in detail the existence and the properties of the decomposition A=P+R+E when A is Toeplitz, also extending to the φ-circulant and Hartley-type cases some results previously known for P circulant.",
keywords = "preconditioning, matrix algebras, Toeplitz, Hankel, clustering, fast discrete transforms",
author = "F. Tudisco and {Di Fiore}, C. and Tyrtyshnikov, {E. E.}",
year = "2013",
month = "1",
day = "1",
doi = "10.1016/j.laa.2012.07.042",
language = "English",
volume = "438",
pages = "405--427",
journal = "Linear Algebra and its Applications",
issn = "0024-3795",
number = "1",

}

Tudisco, F, Di Fiore, C & Tyrtyshnikov, EE 2013, 'Optimal rank matrix algebras preconditioners' Linear Algebra and its Applications, vol. 438, no. 1, pp. 405-427. https://doi.org/10.1016/j.laa.2012.07.042

Optimal rank matrix algebras preconditioners. / Tudisco, F.; Di Fiore, C.; Tyrtyshnikov, E. E.

In: Linear Algebra and its Applications, Vol. 438, No. 1, 01.01.2013, p. 405-427.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Optimal rank matrix algebras preconditioners

AU - Tudisco, F.

AU - Di Fiore, C.

AU - Tyrtyshnikov, E. E.

PY - 2013/1/1

Y1 - 2013/1/1

N2 - When a linear system Ax=y is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner P and move to the preconditioned system P-1Ax=P-1y. The use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate. The construction of optimal rank preconditioners is strongly related to the possibility of splitting A as A=P+R+E, where E is a small perturbation and R is of low rank (Tyrtyshnikov, 1996) [1]. In the present work we extend the black-dot algorithm for the computation of such splitting for P circulant (see Oseledets and Tyrtyshnikov, 2006 [2]), to the case where P is in A, for several known low-complexity matrix algebras A. The algorithm so obtained is particularly efficient when A is Toeplitz plus Hankel like. We finally discuss in detail the existence and the properties of the decomposition A=P+R+E when A is Toeplitz, also extending to the φ-circulant and Hartley-type cases some results previously known for P circulant.

AB - When a linear system Ax=y is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner P and move to the preconditioned system P-1Ax=P-1y. The use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate. The construction of optimal rank preconditioners is strongly related to the possibility of splitting A as A=P+R+E, where E is a small perturbation and R is of low rank (Tyrtyshnikov, 1996) [1]. In the present work we extend the black-dot algorithm for the computation of such splitting for P circulant (see Oseledets and Tyrtyshnikov, 2006 [2]), to the case where P is in A, for several known low-complexity matrix algebras A. The algorithm so obtained is particularly efficient when A is Toeplitz plus Hankel like. We finally discuss in detail the existence and the properties of the decomposition A=P+R+E when A is Toeplitz, also extending to the φ-circulant and Hartley-type cases some results previously known for P circulant.

KW - preconditioning

KW - matrix algebras

KW - Toeplitz

KW - Hankel

KW - clustering

KW - fast discrete transforms

UR - http://www.sciencedirect.com/science/journal/00243795

U2 - 10.1016/j.laa.2012.07.042

DO - 10.1016/j.laa.2012.07.042

M3 - Article

VL - 438

SP - 405

EP - 427

JO - Linear Algebra and its Applications

T2 - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

IS - 1

ER -