CAREER: Numerical Linear Algebra, Random Matrix Theory and Applications

职业：数值线性代数、随机矩阵理论及应用

基本信息

批准号：
1945652
负责人：
Thomas Trogdon
金额：
$ 43.24万
依托单位：
University of Washington
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945652&HistoricalAwards=false
关键词：
CAREER Numerical Linear Algebra Random

项目摘要

Numerical algorithms are pervasive in our lives today. These algorithms are used, for example, to send data to mobile devices and to simulate fire and water in movies. Many of the most classical algorithms, and particularly those with applications in linear algebra, have been extremely useful for decades, if not centuries. Yet, some of these algorithms are poorly understood -- they can fail catastrophically, but rarely do. In other words, the worst-case behavior is poor, but the average-case behavior is good. This research will advance the understanding of this phenomenon by employing techniques from the ever-expanding field of random matrix theory (RMT). In turn, this research gives rise to new questions within RMT. A substantial feature of this project is the extensive educational component that integrates research and education. This integration is achieved via a three-pronged approach including a summer workshop on random matrices, high school engagement, and undergraduate/graduate student mentoring.Two natural ways to employ RMT within numerical linear algebra also coincide with the two most common themes in numerical linear algebra: Algorithm analysis and algorithm development. For example, remarkably detailed estimates from RMT such as rigidity and edge universality have found concrete applications within numerical linear algebra giving average-case performance for the classical power method. Randomized algorithms have found great utility in the big-data age. This research will employ both of these themes with applications to data science, psychology and beyond. The synergy of the fields of RMT and numerical linear algebra provides a unique educational opportunity for graduate, undergraduate and high school students.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.

当今我们的生活中，数值算法普遍存在。例如，使用这些算法将数据发送到移动设备并模拟电影中的火和水。数十年来，许多最古典的算法，尤其是在线性代数中应用的算法，即使不是几个世纪，都非常有用。然而，其中一些算法的理解很少 - 它们可能会灾难性地失败，但很少这样做。换句话说，最坏的案例行为很差，但平均案例行为良好。这项研究将通过采用不断扩展的随机矩阵理论（RMT）领域的技术来提高对这一现象的理解。反过来，这项研究引起了RMT内的新问题。该项目的一个实质特征是整合研究和教育的广泛教育组成部分。这种整合是通过三管齐下的方法来实现的，包括关于随机矩阵，高中参与度和本科/研究生心理的夏季研讨会。在数值线性代数中使用RMT的两种自然方法也与数值线性代数中的两个最常见的主题相吻合。例如，诸如刚性和边缘宇宙之类的RMT的详细估计发现了数值线性代数中的具体应用，为经典功率方法提供了平均案例性能。随机算法在大数据时代发现了极大的效用。这项研究将采用这两个主题，并应用于数据科学，心理学及其他方面。 RMT和数值线性代数的协同作用为研究生，本科和高中生提供了独特的教育机会。该奖项反映了NSF的法定任务，并被认为值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来进行评估。