An analysis of low-rank modifications of preconditioners for saddle point systems

Chen Greif, Michael L. Overton

Research output: Contribution to journalArticle

Abstract

We characterize the spectral behavior of a primal Schur-complement-based block diagonal preconditioner for saddle point systems, subject to low-rank modifications. This is motivated by a desire to reduce as much as possible the computational cost of matrix-vector products with the (1,1) block, while keeping the eigenvalues of the preconditioned matrix reasonably clustered. The formulation leads to a perturbed hyperbolic quadratic eigenvalue problem. We derive interlacing results, highlighting the differences between this problem and perturbed linear eigenvalue problems. As an example, we consider primal-dual interior point methods for semidefinite programs, and express the eigenvalues of the preconditioned matrix in terms of the centering parameter.

Original languageEnglish (US)
Pages (from-to)307-320
Number of pages14
JournalElectronic Transactions on Numerical Analysis
Volume37
StatePublished - 2010

Fingerprint

Saddle Point Systems
Preconditioner
Quadratic Eigenvalue Problem
Eigenvalue
Primal-dual Interior Point Method
Interlacing
Semidefinite Program
Schur Complement
Cross product
Matrix Product
Eigenvalue Problem
Computational Cost
Express
Formulation

Keywords

  • Preconditioners
  • Saddle point systems
  • Schur complement
  • Semidefinite programming

ASJC Scopus subject areas

  • Analysis

Cite this

An analysis of low-rank modifications of preconditioners for saddle point systems. / Greif, Chen; Overton, Michael L.

In: Electronic Transactions on Numerical Analysis, Vol. 37, 2010, p. 307-320.

Research output: Contribution to journalArticle

@article{713b6aef4d9d425f93bf515328bf7bb9,
title = "An analysis of low-rank modifications of preconditioners for saddle point systems",
abstract = "We characterize the spectral behavior of a primal Schur-complement-based block diagonal preconditioner for saddle point systems, subject to low-rank modifications. This is motivated by a desire to reduce as much as possible the computational cost of matrix-vector products with the (1,1) block, while keeping the eigenvalues of the preconditioned matrix reasonably clustered. The formulation leads to a perturbed hyperbolic quadratic eigenvalue problem. We derive interlacing results, highlighting the differences between this problem and perturbed linear eigenvalue problems. As an example, we consider primal-dual interior point methods for semidefinite programs, and express the eigenvalues of the preconditioned matrix in terms of the centering parameter.",
keywords = "Preconditioners, Saddle point systems, Schur complement, Semidefinite programming",
author = "Chen Greif and Overton, {Michael L.}",
year = "2010",
language = "English (US)",
volume = "37",
pages = "307--320",
journal = "Electronic Transactions on Numerical Analysis",
issn = "1068-9613",
publisher = "Kent State University",

}

TY - JOUR

T1 - An analysis of low-rank modifications of preconditioners for saddle point systems

AU - Greif, Chen

AU - Overton, Michael L.

PY - 2010

Y1 - 2010

N2 - We characterize the spectral behavior of a primal Schur-complement-based block diagonal preconditioner for saddle point systems, subject to low-rank modifications. This is motivated by a desire to reduce as much as possible the computational cost of matrix-vector products with the (1,1) block, while keeping the eigenvalues of the preconditioned matrix reasonably clustered. The formulation leads to a perturbed hyperbolic quadratic eigenvalue problem. We derive interlacing results, highlighting the differences between this problem and perturbed linear eigenvalue problems. As an example, we consider primal-dual interior point methods for semidefinite programs, and express the eigenvalues of the preconditioned matrix in terms of the centering parameter.

AB - We characterize the spectral behavior of a primal Schur-complement-based block diagonal preconditioner for saddle point systems, subject to low-rank modifications. This is motivated by a desire to reduce as much as possible the computational cost of matrix-vector products with the (1,1) block, while keeping the eigenvalues of the preconditioned matrix reasonably clustered. The formulation leads to a perturbed hyperbolic quadratic eigenvalue problem. We derive interlacing results, highlighting the differences between this problem and perturbed linear eigenvalue problems. As an example, we consider primal-dual interior point methods for semidefinite programs, and express the eigenvalues of the preconditioned matrix in terms of the centering parameter.

KW - Preconditioners

KW - Saddle point systems

KW - Schur complement

KW - Semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=78651483682&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78651483682&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:78651483682

VL - 37

SP - 307

EP - 320

JO - Electronic Transactions on Numerical Analysis

JF - Electronic Transactions on Numerical Analysis

SN - 1068-9613

ER -