Fast algorithms for large-scale nonlinear algebraic eigenproblems

本计画是关于大规模代数特征值、特征向量与参数之非线性问题之新数值演算法之发展与分析。这些本征值问题出现在电子结构计算、加速器腔体设计、延迟微分方程、复杂结构振动分析等许多领域。最近发展起来的保结构线性化技术对于中小多项式和有理特征值问题是有竞争力的，但是由于线性化问题的维数显著增大，它们对于大规模模拟需要很高的计算成本。此外，线性化引入了相当大的复杂性的发展的预条件，它是不适用于本征问题与充分的非线性。PI应开发新的迭代投影方法，这些方法是准确的，鲁棒的和有效的，用于解决大规模的真正非线性特征值问题。这一目标可以通过探索不同类型的非线性特征值问题的特殊性质来实现，这些特征值问题的求解策略与线性特征值问题的求解策略相似。 本文主要研究（1）新的预条件特征值求解器，包括共轭梯度类方法和最小残差类方法，用于有效地求解非线性特征值问题的大量极值和内特征值，包括变分原理和非变分原理;（2）求解参数依赖的退化特征值问题的快速不精确Newton类方法，用于动力系统稳定性的研究;（3）求解凝聚态物理和电子结构计算中本征向量具有非线性的本征问题的有效算法。该研究将发展一个系统和统一的数学理论和数值软件的开发处理。

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.