CAREER: Numerical Linear Algebra, Random Matrix Theory and Applications

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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.