An efficient algorithm for bifurcation problems of variational inequalities

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

For a class of variational inequalities on a Hubert space H bifurcating solutions exist and may be characterized as critical points of a functional with respect to the intersection of the level surfaces of another functional and a closed convex subset K of H. In a recent paper [13] we have used a gradient-projection type algorithm to obtain the solutions for discretizations of the variational inequalities. A related but Newton-based method is given here. Global and asymptotically quadratic convergence is proved. Numerical results show that it may be used very efficiently in following the bifurcating branches and that it compares favorably with several other algorithms. The method is also attractive for a class of nonlinear eigenvalue problems (K = H) for which it reduces to a generalized Rayleigh-quotient iteration. So some results are included for the path following in turning-point problems.

Original languageEnglish (US)
Pages (from-to)473-485
Number of pages13
JournalMathematics of Computation
Volume41
Issue number164
DOIs
StatePublished - 1983
Externally publishedYes

Fingerprint

Bifurcation (mathematics)
Variational Inequalities
Efficient Algorithms
Bifurcation
Rayleigh Quotient Iteration
Gradient Projection
Path Following
Nonlinear Eigenvalue Problem
Quadratic Convergence
Hubert Space
Turning Point
Critical point
Branch
Intersection
Discretization
Closed
Numerical Results
Subset
Class

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

Cite this

An efficient algorithm for bifurcation problems of variational inequalities. / Mittelmann, Hans.

In: Mathematics of Computation, Vol. 41, No. 164, 1983, p. 473-485.

Research output: Contribution to journalArticle

@article{0d4119150b7d4c08bd9d4ce00c823314,
title = "An efficient algorithm for bifurcation problems of variational inequalities",
abstract = "For a class of variational inequalities on a Hubert space H bifurcating solutions exist and may be characterized as critical points of a functional with respect to the intersection of the level surfaces of another functional and a closed convex subset K of H. In a recent paper [13] we have used a gradient-projection type algorithm to obtain the solutions for discretizations of the variational inequalities. A related but Newton-based method is given here. Global and asymptotically quadratic convergence is proved. Numerical results show that it may be used very efficiently in following the bifurcating branches and that it compares favorably with several other algorithms. The method is also attractive for a class of nonlinear eigenvalue problems (K = H) for which it reduces to a generalized Rayleigh-quotient iteration. So some results are included for the path following in turning-point problems.",
author = "Hans Mittelmann",
year = "1983",
doi = "10.1090/S0025-5718-1983-0717697-7",
language = "English (US)",
volume = "41",
pages = "473--485",
journal = "Mathematics of Computation",
issn = "0025-5718",
publisher = "American Mathematical Society",
number = "164",

}

TY - JOUR

T1 - An efficient algorithm for bifurcation problems of variational inequalities

AU - Mittelmann, Hans

PY - 1983

Y1 - 1983

N2 - For a class of variational inequalities on a Hubert space H bifurcating solutions exist and may be characterized as critical points of a functional with respect to the intersection of the level surfaces of another functional and a closed convex subset K of H. In a recent paper [13] we have used a gradient-projection type algorithm to obtain the solutions for discretizations of the variational inequalities. A related but Newton-based method is given here. Global and asymptotically quadratic convergence is proved. Numerical results show that it may be used very efficiently in following the bifurcating branches and that it compares favorably with several other algorithms. The method is also attractive for a class of nonlinear eigenvalue problems (K = H) for which it reduces to a generalized Rayleigh-quotient iteration. So some results are included for the path following in turning-point problems.

AB - For a class of variational inequalities on a Hubert space H bifurcating solutions exist and may be characterized as critical points of a functional with respect to the intersection of the level surfaces of another functional and a closed convex subset K of H. In a recent paper [13] we have used a gradient-projection type algorithm to obtain the solutions for discretizations of the variational inequalities. A related but Newton-based method is given here. Global and asymptotically quadratic convergence is proved. Numerical results show that it may be used very efficiently in following the bifurcating branches and that it compares favorably with several other algorithms. The method is also attractive for a class of nonlinear eigenvalue problems (K = H) for which it reduces to a generalized Rayleigh-quotient iteration. So some results are included for the path following in turning-point problems.

UR - http://www.scopus.com/inward/record.url?scp=84966251938&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966251938&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1983-0717697-7

DO - 10.1090/S0025-5718-1983-0717697-7

M3 - Article

AN - SCOPUS:84966251938

VL - 41

SP - 473

EP - 485

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 164

ER -