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.

本项目致力于发展和分析求解几类重要的具有特殊结构的大型复杂代数特征值问题的新的数值方法。本征值在应用数学和科学计算的许多领域中都扮演着重要的角色。对物理上相关的特征值的快速和健壮的计算对于整个计算科学和工程中的应用的数学建模和模拟是必不可少的。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中出现的大型本征问题的新求解器的开发和理解。新的算法将有助于实现更高效和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、材料的光学性质、由控制问题引起的动力系统的稳定性等。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。保结构方法在解决物理和力学中的本征问题方面发挥着至关重要的作用，无论是在线性还是非线性情况下。研究人员需要利用这种特殊的结构来设计有效的依赖于问题的方法，以保持这些问题的潜在物理性质。对于特征值具有非线性的特征问题，诸如计算最右特征值等非传统问题对于理解相关动力系统的稳定性是相关的。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的基态可以用总能量最低的静态Gross-Pitaevskii方程(GPE)的解来描述，GPE是一个本征向量具有非线性的非线性本征问题。研究了基于能量泛函结构的预条件优化方法。(2)复Bethe-Salpeter特征值问题的迭代方法。BSE是一个哈密顿本征值问题，可以转化为具有对称谱的厄米特问题。线性响应特征值问题是BSE的一个子类。保结构迭代方法将被用来计算几个最小特征值。(3)非线性特征问题不稳定性的可靠检测。估计一个非线性特征值问题到失稳的距离在很大程度上取决于一系列摄动问题最右边的特征值的稳健计算。将探索基于有理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优势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
• 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