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 local enumeration 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.

中文翻译：

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重新启动具有 minmax 属性的 Hermitian 非线性特征值问题的迭代投影方法

在这项工作中，我们提出了一种新的重新启动技术，用于非线性特征值问题的迭代投影方法，允许其特征值的 minmax 表征。我们的技术利用了内部迭代中特征值的 minmax 引起的局部枚举。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相反，所提出的局部编号只需要在搜索子空间中存在一个特征向量。如果要计算大量的特征值或频谱内部的特征值，则这有效地消除了搜索子空间的增长以及计算成本的超线性增长。新的重新启动技术被集成到非线性迭代投影方法中，如非线性 Arnoldi 和 Jacobi-Davidson 方法。我们的新重启框架的效率在一系列非线性特征值问题上得到了证明：二次、有理和指数问题，包括对滚动轮胎的自由振动进行建模的工业现实保守陀螺特征值问题。我们还将该方法扩展到没有 minmax 属性但特征值具有主要实部或虚部的问题，并在两个二次特征值问题上对其进行测试。