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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：
1753185
负责人：
Thomas Trogdon
金额：
$ 41.8万
依托单位：
University of California-Irvine
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2019-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1753185&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.

数值算法在我们今天的生活中无处不在。例如，这些算法被用来向移动设备发送数据，以及模拟电影中的水和火。许多最经典的算法，特别是那些在线性代数中应用的算法，几十年来一直非常有用，如果不是几个世纪的话。到目前为止，人们对其中一些算法知之甚少--它们可能会灾难性地失败，但很少会失败。换句话说，最糟糕的情况是糟糕的行为。但平均情况下的行为是好的。本研究将通过使用不断扩大的随机矩阵理论(RMT)领域的技术来推进对这一现象的理解。反过来，本研究在RMT中引发了新的问题。这个项目的一个重要特点是广泛的教育组成部分，融合了研究和教育。这种整合是通过三管齐下的方法实现的，包括关于随机矩阵的夏季研讨会、高中参与和本科生/研究生指导。在数值线性代数中使用RMT的两种自然方式也与数值线性代数中最常见的两个主题相吻合：算法分析和算法开发。例如，RMT的非常详细的估计，如刚性和边缘普适性，在数值线性代数中找到了具体的应用，给出了经典幂方法的平均情况性能。随机化算法在大数据时代发现了巨大的实用价值。这项研究将使用这两个主题，并将其应用于数据科学、心理学和其他领域。RMT和数值线性代数领域的协同为研究生、本科生和高中生提供了独特的教育机会。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。