Condition estimates for pseudo-arclength continuation

K. I. Dickson, C. T. Kelley, I. C.F. Ipsent, I. G. Kevrekidis

Research output: Contribution to journalArticle

18 Scopus citations

Abstract

We bound the condition number of the Jacobian in pseudo-arclength continuation problems, and we quantify the effect of this condition number on the linear system solution in a Newton-GMRES solve. Pseudo-arclength continuation solves parameter dependent nonlinear equations G(u, λ) = 0 by introducing a new parameter s, which approximates arclength, and viewing the vector x = (u, λ) as a function of s. In this way simple fold singularities can be computed directly by solving a larger system F(x, s) = 0 by simple continuation in the new parameter s. It is known that the Jacobian Fx of F with respect to x = (u, λ) is nonsingular if the path contains only regular points and simple fold singularities. We introduce a new characterization of simple folds in terms of the singular value decomposition, and we use it to derive a new bound for the norm of Fx-1. We also show that the convergence rate of GMRES in a Newton step for F(x, s) = 0 is essentially the same as that of the original problem G(u, λ) = 0. In particular, we prove that the bounds on the degrees of the minimal polynomials of the Jacobians Fx and Gu differ by at most 2. We illustrate the effectiveness of our bounds with an example from radiative transfer theory.

Original languageEnglish (US)
Pages (from-to)263-276
Number of pages14
JournalSIAM Journal on Numerical Analysis
Volume45
Issue number1
DOIs
StatePublished - Dec 1 2007

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Eigenvalues
  • Fold point
  • GMRES
  • Limit point
  • Pseudo-arclength continuation
  • Rank-one update
  • Simple fold
  • Singular vectors
  • Singularity
  • Turning point

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    Dickson, K. I., Kelley, C. T., Ipsent, I. C. F., & Kevrekidis, I. G. (2007). Condition estimates for pseudo-arclength continuation. SIAM Journal on Numerical Analysis, 45(1), 263-276. https://doi.org/10.1137/060654384