Theory and Algorithms for Eigenvector-Dependent Nonlinear Eigenvalue Problems

特征向量相关的非线性特征值问题的理论和算法

• 批准号: 2110731
• 负责人: Ding Lu
• 金额: $ 32.48万
• 依托单位: University of Kentucky Research Foundation
• 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: 2021-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2110731&HistoricalAwards=false
• 关键词: Theory; Algorithms; Eigenvector; Dependent; Nonlinear

The mathematical questions in this project arise in electronic structure calculations in computational materials science, signal processing of brain–computer interfaces in neuroscience and biomedical engineering, among other applications. The questions come in a variety of forms and pose intriguing challenges for mathematical analysis and numerical solutions. This project seeks to advance the state-of-the-art of the analysis and computation. The project's outcome will advance understanding of the mathematical problems and provide tools for researchers and practitioners to perform simulations in less time using advanced models that were previously unavailable. This project will integrate research into teaching and education and will engage students at various levels in computational mathematics and interdisciplinary research.This project will support one graduate per year in each of the three years of the grant. Technically, this project will focus on an important class of Eigenvector-dependent Nonlinear Eigenvalue Problems NEPv, called affine-linear NEPv (al-NEPv). In an al-NEPv, the coefficient matrix of NEPv poses an affine-linear structure. Origins of al-NEPv include trace-related optimizations such as the trace-ratio optimization for dimension reduction and robust Rayleigh-quotient optimization for handling data uncertainties, among others. The PI plans to conduct systematic analysis and algorithmic development for al-NEPv. The main components of the proposed research are threefold: analysis of al-NEPv, such as a novel geometric description and a variational characterization; new geometric interpretation of the self-consistent field (SCF) iteration for solving al-NEPv, and variants of SCF for handling the local optimal issue and for accelerating the convergence of SCF; availability of a public-domain repository for the collection of NEPv from real-life applications.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.

该项目中的数学问题是在计算材料科学中的电子结构计算中出现的，神经科学和生物医学工程中大脑 - 计算机接口的信号处理以及其他应用。这些问题以多种形式出现，并为数学分析和数值解决方案带来了有趣的挑战。该项目旨在推进分析和计算的最新技术。该项目的结果将提高对数学问题的理解，并为研究人员和实践者提供工具，以减少使用以前无法使用的高级模型在更少的时间内进行仿真。该项目将将研究整合到教学和教育中，并将吸引各个级别的学生参与计算数学和跨学科研究。该项目将在赠款的每三年中每年支持一名毕业生。从技术上讲，该项目将集中于一类重要的特征向量依赖性非线性特征值问题NEPV，称为仿射线性NEPV（al-NEPV）。在AL-NEPV中，NEPV的核心基质具有仿射线性结构。 AL-NEPV的起源包括与痕量相关的优化，例如用于降低尺寸的痕量比率优化和用于处理数据不确定性的强大瑞利（Rayleigh）优化优化。 PI计划为Al-NEPV进行系统分析和算法开发。拟议研究的主要组成部分是三重：Al-NEPV的分析，例如新颖的几何描述和多样性的表征；用于解决al-NEPV的自洽场（SCF）迭代的新几何解释，以及用于处理局部最佳问题并加速SCF收敛性的SCF变体；该奖项反映了NSF的法定任务，可用于收集NEPV的公共域存储库，并使用基金会的知识分子优点和更广泛的影响评估标准来评估我们的法定任务。

项目成果

Sharp Estimation of Convergence Rate for Self-Consistent Field Iteration to Solve Eigenvector-Dependent Nonlinear Eigenvalue Problems

• DOI: 10.1137/20m136606x
• 发表时间: 2022-02
• 期刊: SIAM J. Matrix Anal. Appl.
• 作者: Z. Bai;Ren-Cang Li;Ding Lu