Numerical methods for large-scale parameter-dependent systems

Duration: | June 2009 - May 2014 |

Project Heads: | |

Members: | A. Gaul, P. Losse |

Associated member: | C. SchrÃ¶der |

Cooperation: |
There are connections to MATHEON projects
C22,
D9 and
D21. Z. Bai and R. Freund (UC Davis) D. Chu and X. Liu (National U of Singapore) E. de Sturler (Virginia Tech) Y. Erlangga (UBC Vancouver) N. Higham and F. Tisseur (U of Manchester) D. Kressner (ETH Zurich) W.-W. Lin (National Tsinghua U. Hsinchu, Taiwan) D. S. Mackey and N. Mackey (Western Michigan U) A. Ran (VU Amsterdam) L. Rodman (Williamsburg) O. Schenk (U of Basel) H. Schwetlick (TU Dresden) V. Simoncini (U Bologna) D. Sorensen (Rice U, Texas) Z. Strakos, P. Tichy and M. Tuma (Czech Academy of Sciences) H. Voss (TU Hamburg-Harburg) K. Vuik (TU Delft) D. Watkins (UW Pullman) H. Xu (U of Kansas) |

Activities: |
Diplomanden- und Doktorandenseminar Numerische Mathematik SS 2010 |

Support: | DFG Research Center MATHEON "Mathematics for Key Technologies" |

MATHEON project website: | Project C29 |

Description: |
This project is a continuation and extension of the previous MATHEON project C4, Numerical solution of large nonlinear eigenvalue problems. For previous activities and results see the web site of C4. The numerical solution of large nonlinear eigenvalue problems (particularly polynomial eigenvalue problems) is a crucial component in numerous technologies. In particular, the trend to extreme designs (high speed trains, larger aircraft, higher and higher buildings in earthquake zones, optoelectronic devices, micro-electromechanical systems) presents a challenge for the numerical computation of the important eigenvalues, especially when the problem occurs inside an optimization loop. These extreme designs often lead to very large and poorly conditioned eigenvalue problems. On the other hand, the physics of the underlying system lead to specific algebraic structures of the nonlinear eigenvalue problem that result in symmetries in the spectrum and also in specific properties of the corresponding eigenvectors or invariant subspaces. In numerical computations, for accuracy and complexity reasons, it is essential to use algorithms that preserve these structures in order to obtain physically meaningful results. In many applications, the structured eigenvalue problems depend on parameters that describe topology, geometry and material properties of a system. Examples include the optimization of acoustic fields inside cars, or the modal analysis of rotor dynamics. Parameters have to be chosen to optimize the system behavior. Within the iterative scheme for the computation of the desired eigenvalues, large, sparse, and parameter-dependent linear algebraic systems have to be solved very frequently and for large parameter variations. An essential question in this context is how to reduce the cost of solving a sequence of linear systems by exploiting useful information that has already been computed, and how to update this information from one step to the next. This concerns the solver and also the construction and updating of preconditioners. Due to the continuous nature of the change in the systems, none of the well known low-rank updating techniques that are widely used in a discrete setting (e.g. in the Simplex algorithm) can be applied here, and direct factorization methods alone are not sufficient to solve problems in industrial applications. |