The dissertation concerns numerical methods for approximately solving certain linear partial differential equations. The foundation is a solution methodology for linear elliptic boundary value problems that we call the ``Hierarchical Poincare-Steklov (HPS)'' method. This method is based on a high-order multidomain spectral discretization that is designed to work particularly well...

This thesis describes a set of randomized algorithms for computing rank revealing factorizations of matrices. These algorithms are designed specifically to minimize the amount of data movement required, which is essential to high practical performance on modern computing hardware. The work presented builds on existing randomized algorithms for computing low-rank...

Bilinear programs and Phase Retrieval are two instances of nonconvex problems that arise in engineering and physical applications, and both occur with their fundamental difficulties. In this thesis, we consider various methods and algorithms for tackling these challenging problems and discuss their effectiveness. Bilinear programs (BLPs) are ubiquitous in engineering...

This is a thesis about how to characterize the statistical structure of the tails of degree distributions of real-world networks. The primary contribution is a statistical test of the prevalence of scale-free structure in real-world networks. A central claim in modern network science is that real-world networks are typically "scale...

This thesis consists of three distinct projects. The first is a study of microbial aggregate fragmentation, in which we develop a dynamical model of aggregate deformation and breakage and use it to obtain a post-fragmentation density function. The second and third projects deal with dimensionality reduction in machine learning problems....

Viscous fluid conduits provide an ideal system for the study of dissipationless, dispersive hydrodynamics. A dense, viscous fluid serves as the background medium through which a lighter, less viscous fluid buoyantly rises. If the interior fluid is continuously injected, a deformable pipe forms. The long wave interfacial dynamics are well-described...

This thesis consists of three distinct projects. The first is a study of microbial aggregate fragmentation, in which we develop a dynamical model of aggregate deformation and breakage and use it to obtain a post-fragmentation density function. The second and third projects deal with dimensionality reduction in machine learning problems....

This is a thesis about how to characterize the statistical structure of the tails of degree distributions of real-world networks. The primary contribution is a statistical test of the prevalence of scale-free structure in real-world networks. A central claim in modern network science is that real-world networks are typically "scale...

In this dissertation we consider numerical methods for a problem in each of numerical linear algebra, digital signal processing, and image processing for super-resolution fluorescence microscopy. We consider first a fast, randomized mixing operation applied to the unpivoted Householder QR factorization. The method is an adaptation of a slower randomized...

When solving elliptic partial differential equations (PDE's) multigrid algorithms often provide optimal solvers and preconditioners capable of providing solutions with O(N) computational cost, where N is the number of unknowns. As parallelism of modern super computers continues to grow towards exascale, however, the cost of communication has overshadowed the cost...