Fast algorithms for large-scale nonlinear algebraic eigenproblems

大规模非线性代数本征问题的快速算法

• 批准号: 1419100
• 负责人: Fei Xue
• 金额: $ 18万
• 依托单位: University of Louisiana at Lafayette
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2017-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1419100&HistoricalAwards=false
• 关键词: Fast; algorithms; large; scale; nonlinear; Fast; algorithms; large; scale; nonlinear

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.

该项目涉及针对特征值，特征向量和参数具有非线性的大规模代数本质问题的新数值算法的开发和分析。这些本本特征出现在电子结构计算，加速器腔的设计，延迟微分方程，复杂结构的振动分析等。最近开发的具有结构的线性化技术对于中小型多项式和理性本本特征问题具有竞争力，但由于线性化问题的尺寸显着扩大，它们需要大量模拟的高计算成本。此外，线性化为预处理的开发带来了相当大的并发症，并且不适用于具有完全非线性的本本特征。 PI应开发出新颖的迭代投影方法，这些方法是准确，健壮和有效的，以解决大规模的真正非线性本本特征问题。可以部分通过探索不同类型的非线性本本特征问题的特殊特性来实现此目标，从而实现与线性本本特征问题相似的解决方案策略。 这项研究的重点是（1）新的预处理特征体，包括类似结合级别的类似于和最小的剩余方法，用于有效解决大量的极端和内部特征值，这些特征性在特征值，具有各种原理的特征性问题中存在非线性问题； （2）求解参数依赖性的变性本征征的快速不进行牛顿的方法，用于研究动态系统的（IN）稳定性； （3）在凝结物理学和电子结构计算引起的特征向量中求解具有非线性本本特征的有效算法。这项研究将开发出数值软件的数学理论和开发的系统和统一处理。