Computational Methods for Large Algebraic Eigenproblems with Special Structures

具有特殊结构的大型代数本征问题的计算方法

• 批准号: 2111496
• 负责人: Fei Xue
• 金额: $ 25.23万
• 依托单位: Clemson University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111496&HistoricalAwards=false
• 关键词: Computational; Methods; Large; Algebraic; Eigenproblems

This project concerns development and analysis of new numerical methods for solving several important classes of large-scale and complex algebraic eigenvalue problems with special structures. Eigenvalues play an important role in many areas of applied mathematics and scientific computing. Fast and robust computations of physically relevant eigenvalues are essential to mathematical modeling and simulations for applications throughout computational sciences and engineering. This research will enhance the development and understanding of new solvers for large eigenproblems arising from condensed matter physics, quantum field theoretical systems, or dynamical systems with a need for reliable stability analysis. The new algorithms will help enable more efficient and robust large-scale modeling and simulations involving eigenvalues in many areas, including condensed matter physics, optical properties of materials, stabilities of dynamical systems arising from control problems, and many more. The project will also provide support for graduate students that will enhance their understanding of the essential techniques needed to analyze and solve these computational problems.Structure-preserving methods play a crucial role in solving eigenvalue problems arising from physics and mechanics, in both linear and nonlinear cases. Researchers need to take advantage of the special structures to design efficient problem-dependent methods that preserve the underlying physical properties of these problems. For eigenproblems with nonlinearity in eigenvalues, nontraditional problems such as computing the rightmost eigenvalues are relevant for understanding the stability of the associated dynamical systems. The project will investigate three classes of problems: (1) Computing ground states of Bose-Einstein condensation (BEC). Ground states of BEC are described by the solutions to the static Gross-Pitaevskii equation (GPE), a nonlinear eigenproblem with nonlinearity in eigenvectors, with the lowest total energy. Preconditioned optimization methods based on the structure of the energy functional will be studied. (2) Iterative methods for the complex Bethe-Salpeter Eigenvalue problem (BSE). BSE is a Hamiltonian eigenvalue problem, which can be transformed to a Hermitian problem with symmetric spectrum. The linear response eigenvalue problem is a subclass of BSE. Structure-preserving iterative methods will be investigated for computing a few smallest eigenvalues. (3) Reliable detection of instability of nonlinear eigenproblems. Evaluation of the distance of a nonlinear eigenvalue problem to instability largely depends on robust computation of the rightmost eigenvalues of a sequence of perturbed problems. Algorithms based on functions of matrices approximated by rational Krylov subspace methods will be explored.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目涉及新数值方法的开发和分析，用于解决具有特殊结构的几类重要的大规模复杂代数特征值问题。特征值在应用数学和科学计算的许多领域中发挥着重要作用。物理相关特征值的快速而稳健的计算对于整个计算科学和工程应用的数学建模和模拟至关重要。这项研究将加强对凝聚态物理、量子场理论系统或需要可靠稳定性分析的动力系统引起的大型本征问题的新求解器的开发和理解。新算法将有助于实现更高效、更稳健的大规模建模和模拟，涉及许多领域的特征值，包括凝聚态物理、材料的光学特性、控制问题引起的动力系统的稳定性等等。该项目还将为研究生提供支持，以增强他们对分析和解决这些计算问题所需的基本技术的理解。结构保持方法在解决物理和力学产生的线性和非线性特征值问题中发挥着至关重要的作用案例。研究人员需要利用特殊结构来设计有效的问题相关方法，以保留这些问题的潜在物理特性。对于特征值具有非线性的特征问题，非传统问题（例如计算最右边的特征值）对于理解相关动力系统的稳定性相关。该项目将研究三类问题：（1）计算玻色-爱因斯坦凝聚（BEC）的基态。 BEC 的基态由静态 Gross-Pitaevskii 方程 (GPE) 的解来描述，GPE 是一个具有特征向量非线性且总能量最低的非线性特征问题。研究基于能量泛函结构的预处理优化方法。 (2)复杂Bethe-Salpeter特征值问题(BSE)的迭代方法。 BSE是哈密顿特征值问题，可以转化为具有对称谱的埃尔米特问题。线性响应特征值问题是 BSE 的子类。将研究结构保持迭代方法来计算一些最小特征值。 (3)可靠地检测非线性特征值问题的不稳定性。非线性特征值问题与不稳定性的距离的评估很大程度上取决于对一系列扰动问题的最右边特征值的鲁棒计算。将探索基于理性 Krylov 子空间方法近似的矩阵函数的算法。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Inexact rational Krylov subspace method for eigenvalue problems

求解特征值问题的非精确有理 Krylov 子空间方法

• DOI: 10.1002/nla.2437
• 发表时间: 2022-03
• 期刊: Numerical Linear Algebra with Applications
• 影响因子: 4.3
• 作者: Xu, Shengjie;Xue, Fei
• 通讯作者: Xue, Fei