CAREER: Numerical Linear Algebra, Random Matrix Theory and Applications

数值算法在我们今天的生活中无处不在。例如，这些算法被用于向移动设备发送数据，以及在电影中模拟水火。许多最经典的算法，特别是那些在线性代数中的应用，已经非常有用了几十年，如果不是几个世纪的话。然而，其中一些算法的理解很差——它们可能会发生灾难性的失败，但很少发生。换句话说，最坏情况下的行为是糟糕的，但平均情况下的行为是好的。本研究将利用随机矩阵理论（RMT）不断扩大的领域中的技术来促进对这一现象的理解。反过来，这项研究在RMT中提出了新的问题。这个项目的一个重要特点是广泛的教育组成部分，将研究和教育相结合。这种整合是通过三管齐下的方法实现的，包括一个关于随机矩阵的夏季研讨会，高中参与，以及本科生/研究生指导。在数值线性代数中使用RMT的两种自然方法也与数值线性代数中两个最常见的主题相吻合：算法分析和算法开发。例如，RMT的非常详细的估计，如刚性和边缘通用性，已经在数值线性代数中找到了具体的应用，为经典幂方法提供了平均情况的性能。随机算法在大数据时代发挥了巨大的作用。这项研究将采用这两个主题，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作用为研究生，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。

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.