Betcke Marta M, Voss Heinrich
Department of Computer Science, University College London, Gower Street, London, WC1E 6BT UK.
Institute of Mathematics, Hamburg University of Technology, 21071 Hamburg, Germany.
Numer Math (Heidelb). 2017;135(2):397-430. doi: 10.1007/s00211-016-0804-3. Epub 2016 May 14.
In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. Our technique makes use of the minmax induced of the eigenvalues in the inner iteration. In contrast to global numbering which requires including all the previously computed eigenvectors in the search subspace, the proposed local numbering only requires a presence of one eigenvector in the search subspace. This effectively eliminates the search subspace growth and therewith the super-linear increase of the computational costs if a large number of eigenvalues or eigenvalues in the interior of the spectrum are to be computed. The new restart technique is integrated into nonlinear iterative projection methods like the Nonlinear Arnoldi and Jacobi-Davidson methods. The efficiency of our new restart framework is demonstrated on a range of nonlinear eigenvalue problems: quadratic, rational and exponential including an industrial real-life conservative gyroscopic eigenvalue problem modeling free vibrations of a rolling tire. We also present an extension of the method to problems without minmax property but with eigenvalues which have a dominant either real or imaginary part and test it on two quadratic eigenvalue problems.
在这项工作中，我们针对非线性特征值问题的迭代投影方法提出了一种新的重启技术，这些问题的特征值具有极小极大特征。我们的技术利用了内迭代中特征值的极小极大诱导。与需要在搜索子空间中包含所有先前计算的特征向量的全局编号不同，所提出的局部编号仅要求在搜索子空间中存在一个特征向量。如果要计算大量特征值或谱内部的特征值，这有效地消除了搜索子空间的增长，从而消除了计算成本的超线性增加。新的重启技术被集成到非线性迭代投影方法中，如非线性阿诺尔迪方法和雅可比 - 戴维森方法。我们在一系列非线性特征值问题上展示了新重启框架的效率：二次、有理和指数问题，包括一个模拟滚动轮胎自由振动的工业实际保守陀螺特征值问题。我们还将该方法扩展到不具有极小极大性质但特征值具有主导实部或虚部的问题，并在两个二次特征值问题上进行了测试。
