### Abstract

Language | English |
---|---|

Pages | 71–91 |

Number of pages | 21 |

Journal | SIAM Review |

Volume | 57 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

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

### Cite this

*SIAM Review*,

*57*(1), 71–91. https://doi.org/10.1137/130934921

}

*SIAM Review*, vol. 57, no. 1, pp. 71–91. https://doi.org/10.1137/130934921

**Natural preconditioning and iterative methods for saddle point systems.** / Pestana, Jennifer; Wathen, Andrew J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Natural preconditioning and iterative methods for saddle point systems

AU - Pestana, Jennifer

AU - Wathen, Andrew J.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - inf-sup constant

KW - iterative solvers

KW - preconditioning

KW - saddle point problems

UR - http://epubs.siam.org/doi/abs/10.1137/130934921

U2 - 10.1137/130934921

DO - 10.1137/130934921

M3 - Article

VL - 57

SP - 71

EP - 91

JO - SIAM Review

T2 - SIAM Review

JF - SIAM Review

SN - 0036-1445

IS - 1

ER -