Large sparse continuation problems
type: Item
instance of: scholarly article
title: Large sparse continuation problems
zbMATH Open document ID: 0688.65035
DOI: 10.1016/0377-0427(89)90144-1
author: Kurt Georg
author: Eugene L. Allgower
author: C.-S. Chien
published in: Journal of Computational and Applied Mathematics
publication date: 1989-01-01T00:00:00Z
review text: This article begins by reviewing continuation methods in general, using clear and careful notation; this includes a brief review of literature on exploitation of matrix structure in large continuation problems. The next section deals with incorporation of a nonlinear conjugate gradient method for the corrector phase (as opposed to using a linear conjugate gradient method for a Newton's method-based corrector, as other authors have done). In this method, the line searches are taylored to the fact that we can adjust the predictor steplength so that higher-order terms in the objective function are negligible. Despite the context of a solution manifold instead of isolated solutions, several convergence results are presented, and it is conjectured that the scheme is locally superlinearly convergent. Practicalities, such as preconditioners and use of higher-order predictors to offset the somewhat slower convergence of the conjugate gradient method, are discussed. Handling bifurcation in this context is discussed; local perturbations are used, since determinant sign changes are not readily available from the nonlinear conjugate gradient method. Numerical experiments dealing with nonlinear eigenvalue problems corresponding to buckled plates of various shapes are given; solutions are presented graphically. The reference list is sufficiently complete to guide a beginner through a thorough self-study of the underlying concepts.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 74K20
zbMATH DE Number: 4128307
zbMATH Keywords: large sparse systems
zbMATH Keywords: continuation methods
zbMATH Keywords: nonlinear conjugate gradient method
zbMATH Keywords: corrector phase
zbMATH Keywords: solution manifold
zbMATH Keywords: convergence
zbMATH Keywords: preconditioners
zbMATH Keywords: higher-order predictors
zbMATH Keywords: bifurcation
zbMATH Keywords: local perturbations
zbMATH Keywords: Numerical experiments
zbMATH Keywords: nonlinear eigenvalue problems
zbMATH Keywords: buckled plates
reviewed by: Ralph Baker Kearfott
describes a project that uses: PITCON
describes a project that uses: PLTMGC
MaRDI profile type: Publication
full work available at URL: 0377-0427(89)90144-1
OpenAlex ID: W2064386875
cites work: Descent Property and Global Convergence of the Fletcher—Reeves Method with Inexact Line Search
cites work: Continuation and Local Perturbation for Multiple Bifurcations
cites work: Q3327633
cites work: PLTMGC: A Multigrid Continuation Program for Parameterized Nonlinear Elliptic Systems
cites work: Q4146134
cites work: Q3883395
cites work: Q3219696
cites work: A Multigrid Continuation Method for Elliptic Problems with Folds
cites work: Finite dimensional approximation of nonlinear problems. III: Simple bifurcation points
cites work: Q3330399
cites work: Arc-Length Continuation and Multigrid Techniques for Nonlinear Elliptic Eigenvalue Problems
cites work: Q3948194
cites work: Rate of Convergence of Several Conjugate Gradient Algorithms
cites work: Bifurcation from simple eigenvalues
cites work: On Steplength Algorithms for a Class of Continuation Methods
cites work: Q3702408
cites work: Generalized conjugate directions
cites work: Function minimization by conjugate gradients
cites work: On Tracing an Implicitly Defined Curve by Quasi-Newton Steps and Calculating Bifurcation by Local Perturbations
cites work: Q4746734
cites work: Q4739659
cites work: Continuation-Conjugate Gradient Methods for the Least Squares Solution of Nonlinear Boundary Value Problems
cites work: Q5185900
cites work: Q3939124
cites work: Perturbed bifurcation theory
cites work: Nonlinear bifurcation
cites work: Q3703206
cites work: Q1110275
cites work: Numerical methods for bifurcation problems. Proceedings of the Conference at the University of Dortmund, August 22-26, 1983
cites work: Bifurcation: analysis, algorithms, applications. Proceedings of the Conference at the University of Dortmund, August 18-22, 1986
cites work: Alternative proofs of the convergence properties of the conjugate- gradient method
cites work: Q3326747
cites work: Q4774620
cites work: Q5563083
cites work: Restart procedures for the conjugate gradient method
cites work: Q3313210
cites work: Some global results for nonlinear eigenvalue problems
cites work: On the numerical analysis of the Von Kármán equations: Mixed finite element approximation and continuation techniques
cites work: Numerical Methods for a Class of Finite Dimensional Bifurcation Problems
cites work: On the monotone convergence of Newton's method
cites work: GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
cites work: On the Choice of Steplength in Path Following Methods
cites work: Q4117285
cites work: Q3347190
cites work: Some Superlinear Convergence Results for the Conjugate Gradient Method
cites work: Implementation of the GMRES Method Using Householder Transformations