### Abstract

Read More: http://epubs.siam.org/doi/abs/10.1137/110827600

Language | English |
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Pages | A1584-A1606 |

Number of pages | 23 |

Journal | SIAM Journal on Scientific Computing |

Volume | 34 |

Issue number | 3 |

Early online date | 11 Jun 2012 |

DOIs | |

Status | Published - 2012 |

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*SIAM Journal on Scientific Computing*,

*34*(3), A1584-A1606. DOI: 10.1137/110827600

**Lyapunov inverse iteration for identifying Hopf bifurcations in models of incompressible flow.** / Elman, Howard C.; Meerbergen, Karl; Spence, Alastair; Wu, Minghao.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol 34, no. 3, pp. A1584-A1606. DOI: 10.1137/110827600

}

TY - JOUR

T1 - Lyapunov inverse iteration for identifying Hopf bifurcations in models of incompressible flow

AU - Elman,Howard C.

AU - Meerbergen,Karl

AU - Spence,Alastair

AU - Wu,Minghao

PY - 2012

Y1 - 2012

N2 - The identification of instability in large-scale dynamical systems caused by Hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems. A new method developed in [K. Meerbergen and A. Spence, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1982--1999] avoids this computation, instead performing an inverse iteration for a certain set of real eigenvalues that requires the solution of a large-scale Lyapunov equation at each iteration. In this study, we refine the Lyapunov inverse iteration method to make it more robust and efficient, and we examine its performance on challenging test problems arising from fluid dynamics. Various implementation issues are discussed, including the use of inexact inner iterations and the impact of the choice of iterative solution for the Lyapunov equations, and the effect of eigenvalue distribution on performance. Numerical experiments demonstrate the robustness of the algorithm.Read More: http://epubs.siam.org/doi/abs/10.1137/110827600

AB - The identification of instability in large-scale dynamical systems caused by Hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems. A new method developed in [K. Meerbergen and A. Spence, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1982--1999] avoids this computation, instead performing an inverse iteration for a certain set of real eigenvalues that requires the solution of a large-scale Lyapunov equation at each iteration. In this study, we refine the Lyapunov inverse iteration method to make it more robust and efficient, and we examine its performance on challenging test problems arising from fluid dynamics. Various implementation issues are discussed, including the use of inexact inner iterations and the impact of the choice of iterative solution for the Lyapunov equations, and the effect of eigenvalue distribution on performance. Numerical experiments demonstrate the robustness of the algorithm.Read More: http://epubs.siam.org/doi/abs/10.1137/110827600

UR - http://dx.doi.org/10.1137/110827600

U2 - 10.1137/110827600

DO - 10.1137/110827600

M3 - Article

VL - 34

SP - A1584-A1606

JO - SIAM Journal on Scientific Computing

T2 - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 3

ER -