Fast algorithms for large-scale nonlinear algebraic eigenproblems

该项目涉及针对具有特征值、特征向量和参数非线性的大规模代数特征问题开发和分析新的数值算法。这些特征问题出现在电子结构计算、加速器腔设计、延迟微分方程、复杂结构的振动分析等中。最近开发的结构保持线性化技术对于中小多项式和有理本征问题具有竞争力，但由于线性化问题的维数显着扩大，它们需要大规模模拟的高计算成本。此外，线性化给预处理器的开发带来了相当大的复杂性，并且不适用于完全非线性的特征问题。 PI 应开发准确、鲁棒且高效的新型迭代投影方法，用于解决大规模真正非线性特征问题。这一目标可以通过探索不同类型的非线性特征问题的特殊性质来部分实现，这些特征可以实现与线性特征问题类似的解决策略。 本研究的重点是（1）新的预条件特征求解器，包括类共轭梯度和类最小残差方法，用于在使用或不使用变分原理的情况下有效求解特征值非线性问题的大量极端和内部特征值； (2) 快速不精确的类牛顿方法，用于解决参数相关的简并本征问题，以研究动力系统的稳定性； (3)用于解决凝聚态物理和电子结构计算中产生的特征向量非线性特征问题的有效算法。该研究将对数学理论和数值软件的开发进行系统和统一的处理。

This project concerns development and analysis of new numerical algorithms for large-scale algebraic eigenproblems with nonlinearity in eigenvalues, eigenvectors, and parameters. These eigenproblems arise in electronic structure calculation, design of accelerator cavities, delay differential equations, vibration analysis of complex structures, and many more. Structure-preserving linearization techniques that have been developed recently are competitive for small or medium polynomial and rational eigenproblems, but they entail high computational costs for large-scale simulations due to the significantly enlarged dimension of linearized problems. In addition, linearization introduces considerable complications for the development of preconditioners, and it is not applicable to eigenproblems with full nonlinearity. The PI shall develop novel iterative projection methods that are accurate, robust and efficient, for the solution of large-scale truly nonlinear eigenproblems. This goal can be achieved in part by exploration of special properties of different types of nonlinear eigenproblems that enable solution strategies similar to those for linear eigenproblems. This investigation is focused on ( 1) new preconditioned eigensolvers, including conjugate-gradient-like and minimal-residual-like methods, for efficient solution of a large number of extreme and interior eigenvalues of problems with nonlinearity in eigenvalues, with and without the variational principle; (2) fast inexact Newton-like methods to solve parameter-dependent degenerate eigenproblem for the study of (in)stabilities of dynamical systems; (3) efficient algorithms for solving eigenproblems with nonlinearity in eigenvectors arising from condensed matter physics and electronic structure calculation. The research will develop a systematic and unified treatment of mathematical theory and development of numerical software.