Fast algorithms for large-scale nonlinear algebraic eigenproblems

该项目涉及大规模代数特征问题的新的数值算法的开发和分析，该问题具有特征值、特征向量和参数的非线性。这些本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方面。最近发展起来的结构保持线性化技术对于中小型多项式和有理特征问题是有竞争力的，但由于线性化问题的规模显著扩大，它们在大规模模拟中需要很高的计算代价。此外，线性化给预条件函数的发展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应开发准确、稳健和高效的新的迭代投影方法，用于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部分实现，这些特征问题使得解决策略类似于线性特征问题的策略。这项研究的重点是：(1)新的预条件本征解方法，包括共轭梯度类方法和最小余量类方法，用于有效地求解具有特征值非线性的问题的大量极值和内特征值，包括变分原理；(2)用于研究动力学系统稳定性的快速非精确牛顿类方法；(3)用于求解凝聚态物理和电子结构计算中特征向量非线性的特征问题的高效算法。本研究将为数学理论的系统统一处理和数值软件的开发奠定基础。

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.