Natural preconditioning and iterative methods for saddle point systems

Jennifer Pestana, Andrew J. Wathen

Research output: Contribution to journalArticle

34 Citations (Scopus)
144 Downloads (Pure)

Abstract

The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or the discrete setting, so saddle point systems arising from the discretization of partial differential equation problems, such as those describing electromagnetic problems or incompressible flow, lead to equations with this structure, as do, for example, interior point methods and the sequential quadratic programming approach to nonlinear optimization. This survey concerns iterative solution methods for these problems and, in particular, shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness---in terms of rapidity of convergence---is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends.
Original languageEnglish
Pages (from-to)71–91
Number of pages21
JournalSIAM Review
Volume57
Issue number1
DOIs
Publication statusPublished - 2015

Keywords

  • inf-sup constant
  • iterative solvers
  • preconditioning
  • saddle point problems

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