Physical Sciences and Mathematics Commons^™

A Connectivity Framework To Explore The Role Of Anthropogenic Activity And Climate On The Propagation Of Water And Sediment At The Catchment Scale, Christos Giannopoulos

Doctoral Dissertations

Anthropogenic disturbance in intensively managed landscapes (IMLs) has dramatically altered critical zone processes, resulting in fundamental changes in material fluxes. Mitigating the negative effects of anthropogenic disturbance and making informed decisions for optimal placement and assessment of best management practices (BMPs) requires fundamental understanding of how different practices affect the connectivity or lack thereof of governing transport processes and resulting material fluxes across different landscape compartments within the hillslope-channel continuum of IMLs. However, there are no models operating at the event timescale that can accurately predict material flux transport from the hillslope to the catchment scale capturing the spatial and …

Coarse Proximity Spaces, Jeremy D. Siegert

Doctoral Dissertations

This work is meant to present the current general landscape of the theory of coarse proximity spaces. It is largely comprised of two parts that are heavily interrelated, the study of boundaries of coarse proximity spaces, and the dimension theory of coarse proximity spaces. Along the way a study of the relationships between coarse proximity spaces and other structures in coarse geometry are explored.

We begin in chapter 2 by going over the necessary preliminary definitions and concepts from the study of small scale proximity spaces as well as coarse geometry. We then quickly proceed to the introduction of coarse …

Preconditioned Nesterov’S Accelerated Gradient Descent Method And Its Applications To Nonlinear Pde, Jea Hyun Park

Doctoral Dissertations

We develop a theoretical foundation for the application of Nesterov’s accelerated gradient descent method (AGD) to the approximation of solutions of a wide class of partial differential equations (PDEs). This is achieved by proving the existence of an invariant set and exponential convergence rates when its preconditioned version (PAGD) is applied to minimize locally Lipschitz smooth, strongly convex objective functionals. We introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requires a natural time step restriction for energy stability. At the continuous time level, we show …