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.

该项目涉及对新的数值方法的开发和分析，以解决与特殊结构的几种重要类别的大规模和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都起着重要作用。对物理相关特征值的快速和强大的计算对于整个计算科学和工程的应用是数学建模和模拟至关重要的。这项研究将增强对新求解器的开发和理解，用于由凝结物理学，量子场理论系统或需要可靠稳定性分析的动力学系统引起的大型本征局问题。新算法将有助于在许多领域（包括冷凝物质物理学，材料的光学特性，由控制问题引起的动力学系统的稳定性等）在许多领域中涉及特征值的更有效，更强大的大规模建模和模拟。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。在线性和非线性案例中，结构性推动的方法在解决物理学和力学引起的特征值问题方面起着至关重要的作用。研究人员需要利用特殊的结构来设计有效的问题依赖性方法，以保留这些问题的潜在物理特性。对于特征值中非线性的本本特征，非传统问题（例如计算最右边的特征值）与理解相关动力学系统的稳定性有关。该项目将研究三类问题：（1）计算Bose-Einstein凝结（BEC）的基础状态。 BEC的基态由静态GROSS-PITAEVSKII方程（GPE）的解决方案描述，这是一种非线性本本特征，具有最低的总能量，具有非线性。将研究基于能量功能的结构的预处理优化方法。 （2）复杂的伯特盐特征值问题（BSE）的迭代方法。 BSE是一个汉密尔顿特征值问题，可以通过对称频谱转化为冬宫问题。线性响应特征值问题是BSE的子类。将研究具有结构的迭代方法，以计算一些最小的特征值。 （3）可靠检测非线性本本特征问题的不稳定性。评估非线性特征值问题与不稳定性的距离很大程度上取决于对一系列扰动问题的最右边特征值的强大计算。将探讨基于由理性Krylov子空间方法近似的矩阵功能的算法。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估标准通过评估来支持的。

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
• 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