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，神经科学和生物医学工程中脑机接口的信号处理以及其他应用中。这些问题以各种形式出现，对数学分析和数值解决方案提出了有趣的挑战。该项目旨在推进分析和计算的最新技术。该项目的成果将促进对数学问题的理解，并为研究人员和实践者提供工具，使用以前无法获得的先进模型在更短的时间内进行模拟。该项目将把研究与教学和教育相结合，并将吸引不同水平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，该项目将重点关注一类重要的特征向量相关非线性特征值问题NEPv，称为仿射线性NEPv （al-NEPv）。在al-NEPv中，NEPv的系数矩阵呈仿射线性结构。al-NEPv的起源包括与跟踪相关的优化，例如用于降维的跟踪比率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。本研究的主要内容包括三个方面：对al-NEPv的分析，如新颖的几何描述和变分表征；求解al-NEPv的自洽场迭代的新几何解释，以及用于处理局部最优问题和加速自洽场收敛的自洽场的变体；从实际应用程序中收集NEPv的公共领域存储库的可用性。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。

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.