Theory and Algorithms for Eigenvector-Dependent Nonlinear Eigenvalue Problems

这个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、神经科学中的脑-计算机接口的信号处理和生物医学工程等应用中。这些问题的形式多种多样，对数学分析和数值解提出了有趣的挑战。该项目致力于推动分析和计算的最先进水平。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者提供工具，以便使用以前无法获得的先进模型在更短的时间内进行模拟。这个项目将把研究融入到教学和教育中，并让不同水平的学生参与计算数学和跨学科的研究。这个项目将在三年的拨款中每年资助一名毕业生。从技术上讲，本课题将重点研究一类重要的依赖于特征向量的非线性特征值问题NEPv，称为仿射线性NEPv(al-Nnev)。在al-nev中，nev的系数矩阵呈仿射线性结构。Al-Nnev的起源包括与跟踪相关的优化，如用于降维的跟踪比优化和用于处理数据不确定性的稳健瑞利商优化等。PI计划对al-Nnev进行系统分析和算法开发。拟议研究的主要组成部分包括三个方面：al-nev的分析，例如新的几何描述和变分表征；用于求解al-nev的自洽场(SCF)迭代的新的几何解释，以及用于处理局部最优问题和加速SCF收敛的SCF的变体；可用于从现实应用中收集Nnev的公共领域储存库。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.