CAREER: Synergistic interactions between Numerical Linear Algebra and Stochastic Eigenanalysis (Random Matrix Theory)

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目的目的有三个方面：开发和分析快速，随机，易于并行化的算法，研究大型随机矩阵的特征统计的平均行为和限制分布（这对于理解数值算法的性能至关重要），以及用于符号和数值计算随机矩阵特征统计的软件开发。为此，研究者将使用数值线性代数、正交多项式理论、概率论、摄动理论和组合学等工具。数值线性代数和随机特征分析（经典称为随机矩阵理论）是与科学和工程有着深刻和各种联系的数学领域，有着广泛而深远的应用。其中一个应用程序是随机高性能计算。当一个工业规模的问题被定义为6位或更多的数字时，计算机处理速度的天文数字般的增长并不能弥补存储器速度的缓慢增长；结果，处理器和内存之间的差距越来越大——这是目前提高计算机性能的主要障碍。随机化提供了一种成功解决这一问题的方法。该项目侧重于将数值线性代数的方法“移植”到随机特征分析的研究中，并将由此获得的理论结果用于高性能计算算法的开发，以及其他应用，如研究Internet网络，构建更可靠的手机网络等。该项目还包括一些教育方面的努力，包括在华盛顿大学发起一系列公开讲座，展示数学在科学、经济等领域的应用，以及增加女性在大学生数学竞赛中的参与度。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The purpose of this project is threefold: development and analysis of fast, randomized, easily parallelizable algorithms, study of average behavior and limiting distributions for eigenstatistics of large random matrices (which is crucial in understanding the performance of numerical algorithms), and software development for symbolically and numerically computing eigenstatistics of random matrices. To this extent, the investigator will use tools from numerical linear algebra, orthogonal polynomials theory, probability, perturbation theory, and combinatorics.Numerical linear algebra and stochastic eigenanalysis (classically known as random matrix theory) are mathematical fields with deep and various connections to sciences and engineering, as well as wide and far-reaching applications. One such application is randomized high-performance computing. At a time when an industrial-sized problem is being defined by six or more digits, the astronomical rate of increase in computer processing speed does not compensate for the far slower growth of memory speed; as a result, the processor-memory gap is increasing -- a fact which is now the main obstacle to improved computer performance. Randomization offers a way to successfully address this issue. The project focuses on "transplanting" methods of numerical linear algebra to the study of stochastic eigenanalysis, and using the theoretical results thus obtained for the development of high-performance computational algorithms, as well as in other applications such as the study of Internet networks, building more reliable cell phone networks, and so on. The project also includes several educational endeavors, including initiation of a series of public lectures held at University of Washington to showcase applications of mathematics to sciences, economy, etc. and increasing female participation in undergraduate mathematical competitions.