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的脑机接口信号处理以及其他应用中，这些问题有多种形式，并对数学分析和数值提出了有趣的挑战。该项目旨在提高最先进的分析和计算水平。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使用以前的先进模型在更短的时间内进行模拟的工具。该项目将把研究整合到不可用中。教学和教育，并将吸引各个级别的学生参与计算数学和跨学科研究。该项目将在资助的三年中每年支持一名毕业生。从技术上讲，该项目将重点关注特征向量相关的非线性的重要类别。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。与迹线相关的优化，例如用于降维的迹线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。包括三个方面：对 al-NEPv 的分析，例如用于求解 al-NEPv 的自洽场 (SCF) 迭代的新颖几何描述和变分表征，以及用于处理局部最优的 SCF 变体；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进行评估，认为值得支持审查标准。

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