This paper explores some optimization methods such as the gradient descent method, Gauss-Newton method, and stochastic gradient method. Some examples of minimizing objective functions are given to validate the theories. Then we introduce a simple example of artificial neural networks, define its structure, and apply the optimization methods to it....
We now have broad consensus in the mathematics education research community that active, inquiry-based classrooms provide a wealth of learning benefits for students (Freeman et al., 2014; Laursen et al., 2014; Theobald et al., 2020). Classrooms that utilize inquiry throughout the entire structure of the course, as opposed to the...
Aggregation equations have been used to model phenomena such as insect swarming and chemotaxis. Previous work on aggregation equations in the area of analysis applied to PDE has proven well-posedness of certain classes of aggregation equations in Lebesgue spaces. We will prove local existence of solutions in H^1 to an...
The HyperLogLog (HLL) algorithm is used to estimate the cardinality of large sets. This thesis gives a novel analysis of the HyperLogLog algorithm by using techniques from statistics and probability. Initially, closed form bounds for the mean and variance of the max of n independent and identically distributed geometric random...
Many geophysical phenomena exhibit complicated dynamics that, due to a variety of factors, diverge quickly from physical models. The arrival of new observations allows researchers to combine the model estimate with measurements in a statistical process called data assimilation to produce a revised estimate of the phenomenon. This assimilation of...
In this dissertation we consider two application specific flow and transport models in porous media at multiple scales: 1) methane gas transport models for hydrate formation and dissociation in the subsurface under two-phase conditions, and 2) coupled flow and biomass-nutrient model for biofilm growth in complex geometries with biofilm, and...
The aim of this dissertation is to construct a virtual element method (VEM) for models in magneto-hydrodynamics (MHD), an area that studies the behavior and properties of electrically conducting fluids such as a plasma. MHD models are a coupling of the Maxwell’s equations for electromagnetics and models for fluid flow....
In this paper we explain at the theoretical level how discrete Morse theory can provide us a more efficient approach to compute persistent homologies. In achieving so we also provide a framework for discrete Morse theory to be applied to persistent homology for other purposes.
An important problem in computer graphics is to determine where contour lines and ridges appear in a surface constructed from a triangle mesh. In this presentation we will investigate a new answer to this problem – the horizon measure. The horizon measure determines the likelihood of contour lines to appear...
This dissertation explores mathematical theory of the 3 dimensional incompressible Navier-Stokes equations that consists a set of partial differential equations which govern the motion of Newtonian fluids and can be seen as Newton's second law of motion for fluids. The main interest of this work focuses on how local perturbation...