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出现在计算材料科学中的电子结构计算、神经科学和生物医学工程中脑机接口的信号处理等应用中。这些问题以各种形式出现，并对数学分析和数值解提出了有趣的挑战。该项目旨在推进最先进的分析和计算。该项目的成果将促进对数学问题的理解，并为研究人员和从业人员提供工具，使用以前无法使用的先进模型在更短的时间内进行模拟。该项目将把研究融入教学和教育中，让不同层次的学生参与计算数学和跨学科研究。该项目将在三年的资助期内每年资助一名毕业生。 从技术上讲，这个项目将集中在一类重要的特征向量相关的非线性特征值问题NEPv，称为仿射线性NEPv（al-NEPv）。在al-NEPv中，NEPv的系数矩阵构成仿射线性结构。al-NEPv的起源包括迹相关的优化，例如用于降维的迹比优化和用于处理数据不确定性的鲁棒瑞利商优化等。PI计划对al-NEPv进行系统分析和算法开发。所提出的研究的主要组成部分有三个方面：分析al-NEPv，如一个新的几何描述和变分表征;新的几何解释的自洽场（SCF）迭代求解al-NEPv，和变体的SCF处理局部最优问题和加速收敛的SCF;从真实的收集NEPv的公共域存储库的可用性-该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的支持影响审查标准。

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