Computational Methods for Large Algebraic Eigenproblems with Special Structures

本计画系关于发展与分析新的数值方法，以解决几类重要的大型与特殊结构的复代数特征值问题。特征值在应用数学和科学计算的许多领域中起着重要的作用。物理相关特征值的快速和鲁棒计算对于整个计算科学和工程应用的数学建模和仿真是必不可少的。这项研究将加强对凝聚态物理学，量子场论系统或需要可靠稳定性分析的动力系统所产生的大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有效和更强大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，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，这将提高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线性和非线性情况下的物理和力学特征值问题中起着至关重要的作用。研究人员需要利用这些特殊的结构来设计有效的问题相关方法，以保持这些问题的基本物理性质。对于非线性特征值的特征值问题，非传统的问题，如计算最右边的特征值是相关的了解相关的动力系统的稳定性。本项目将研究三类问题：（1）玻色-爱因斯坦凝聚（BEC）基态的计算。BEC的基态由静态Gross-Pitaevskii方程（GPE）的解描述，GPE是一个具有非线性特征向量的非线性特征问题，具有最低的总能量。基于能量泛函结构的预条件优化方法将被研究。(2)复杂Bethe-Salpeter特征值问题（BSE）的迭代方法。BSE是一个Hamilton特征值问题，它可以转化为具有对称谱的Hermite问题。线性响应特征值问题是BSE的一个子类。结构保持迭代方法将研究计算一些最小的特征值。(3)非线性特征值问题不稳定性的可靠检测。一个非线性特征值问题的不稳定性的距离的评估在很大程度上取决于一系列扰动问题的最右边的特征值的鲁棒计算。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。

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.