Fast algorithms for large-scale nonlinear algebraic eigenproblems

本项目涉及在特征值、特征向量和参数上具有非线性的大规模代数特征问题的新数值算法的开发和分析。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许多领域。近年来发展起来的保持结构的线性化技术对中小型多项式和有理特征问题具有一定的竞争力，但由于线性化问题的维数大大增加，它们在大规模模拟中需要很高的计算成本。此外，线性化给预调节器的开发带来了相当大的复杂性，并且不适用于具有完全非线性的特征问题。PI应该开发新的迭代投影方法，准确，鲁棒和高效，解决大规模真正的非线性特征问题。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实现，这些特征问题的解决策略与线性特征问题的解决策略相似。本研究的重点是：(1)新的预条件特征解，包括共轭梯度法和最小残差法，用于有效解具有或不具有变分原理的特征值非线性问题的大量极值和内特征值；(2)求解动力系统稳定性研究中参数相关退化特征问题的快速非精确类牛顿方法；(3)凝聚态物理和电子结构计算中特征向量非线性特征问题的有效求解算法。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.