Computational Methods for Large Algebraic Eigenproblems with Special Structures

本项目涉及发展和分析新的数值方法来解决几类重要的具有特殊结构的大规模和复杂的代数特征值问题。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模拟至关重要。本研究将促进对新求解器的发展和理解，这些求解器来自凝聚态物理、量子场理论系统或需要可靠稳定性分析的动力系统。新的算法将有助于在许多领域实现更高效、更稳健的大规模建模和模拟，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。结构保持方法在求解线性和非线性物理和力学中的特征值问题中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方法，以保持这些问题的潜在物理性质。对于具有非线性特征值的特征问题，计算最右特征值等非传统问题与理解相关动力系统的稳定性有关。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是一个具有非线性特征向量的非线性特征问题，具有最低的总能量。本文将研究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（BSE）的迭代方法。BSE是一个哈密顿特征值问题，它可以转化为具有对称谱的厄米问题。线性响应特征值问题是BSE的一个子类。我们将研究保持结构的迭代方法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。非线性特征值问题到不稳定的距离的评估很大程度上取决于扰动问题序列最右特征值的鲁棒计算。我们将探讨基于由有理Krylov子空间方法近似的矩阵函数的算法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.