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A Posteriori Error Estimates For Maxwell's Equations Using Auxiliary Subspace Techniques, Ahmed El Sakori

Dissertations and Theses

The aim of our work is to construct provably efficient and reliable error estimates of discretization error for Nédélec (edge) element discretizations of Maxwell's equations on tetrahedral meshes. Our general approach for estimating the discretization error is to compute an approximate error function by solving an associated problem in an auxiliary space that is chosen so that:

-Efficiency and reliability results for the computed error estimates can be established under reasonable and verifiable assumptions.

-The linear system used to compute the approximate error function has condition number bounded independently of the discretization parameter.

In many applications, it is some functional …

Guided Reinvention As A Context For Investigating Students' Thinking About Mathematical Language And For Supporting Students In Gaining Fluency, Kristen Vroom

Dissertations and Theses

Fluency with mathematical language is important for students' engagement in many disciplinary practices such as defining, conjecturing, and proving; yet, there is growing evidence that mathematical language is challenging for undergraduate students. This dissertation study draws on two design experiments with pairs of students who were supported to encode their mathematical meanings with more formal language. I aimed to investigate the teaching and learning of mathematical language, and particularly the language in statements with multiple quantifiers, by engaging students in this type of activity. In the first paper, I investigated the complex ways in which the students in my study …

Convex And Nonconvex Optimization Techniques For Multifacility Location And Clustering, Tuyen Dang Thanh Tran

Dissertations and Theses

This thesis contains contributions in two main areas: calculus rules for generalized differentiation and optimization methods for solving nonsmooth nonconvex problems with applications to multifacility location and clustering. A variational geometric approach is used for developing calculus rules for subgradients and Fenchel conjugates of convex functions that are not necessarily differentiable in locally convex topological and Banach spaces. These calculus rules are useful for further applications to nonsmooth optimization from both theoretical and numerical aspects. Next, we consider optimization methods for solving nonsmooth optimization problems in which the objective functions are not necessarily convex. We particularly focus on the class …

On Dc And Local Dc Functions, Liam Jemison

University Honors Theses

In this project we investigate the class of functions which can be represented by a difference of convex functions, hereafter referred to simply as 'DC' functions. DC functions are of interest in optimization because they allow the use of convex optimization techniques in certain non-convex problems. We present known results about DC and locally DC functions, including detailed proofs of important theorems by Hartman and Vesely.

We also investigate the DCA algorithm for optimizing DC functions and implement it to solve the support vector machine problem.

Laurent Series Expansion And Its Applications, Anna Sobczyk

University Honors Theses

The Laurent expansion is a well-known topic in complex analysis for its application in obtaining residues of complex functions around their singularities. Computing the Laurent series of a function around its singularities turns out to be an efficient way to determine the residue of the function as well as to compute the integral of the function along any closed curves around its singularities. Based on the theory of the Laurent series, this paper provides several working examples where the Laurent series of a function is determined and then used to calculate the integral of the function along any closed curve …

Leveraging Model Flexibility And Deep Structure: Non-Parametric And Deep Models For Computer Vision Processes With Applications To Deep Model Compression, Anthony D. Rhodes

Dissertations and Theses

My dissertation presents several new algorithms incorporating non-parametric and deep learning approaches for computer vision and related tasks, including object localization, object tracking and model compression. With respect to object localization, I introduce a method to perform active localization by modeling spatial and other relationships between objects in a coherent "visual situation" using a set of probability distributions. I further refine this approach with the Multipole Density Estimation with Importance Clustering (MIC-Situate) algorithm. Next, I formulate active, "situation" object search as a Bayesian optimization problem using Gaussian Processes. Using my Gaussian Process Context Situation Learning (GP-CL) algorithm, I demonstrate improved …

Modeling And Visualizing Power Amplification In Fiber Optic Cables, Gil Parnon

University Honors Theses

Transverse mode instability in fiber optic cables causes power amplification to exhibit chaotic behavior. Due to this, numerical modeling of fiber optic power amplification is extremely computationally expensive. In this paper I work through modeling similar behavior in a simpler system. I also visualize the three-dimensional phase portrait of the system in order to better understand the behavior and hopefully relate it to more well-understood problems.

Dictionary Learning For Image Reconstruction Via Numerical Non-Convex Optimization Methods, Lewis M. Hicks

University Honors Theses

This thesis explores image dictionary learning via non-convex (difference of convex, DC) programming and its applications to image reconstruction. First, the image reconstruction problem is detailed and solutions are presented. Each such solution requires an image dictionary to be specified directly or to be learned via non-convex programming. The solutions explored are the DCA (DC algorithm) and the boosted DCA. These various forms of dictionary learning are then compared on the basis of both image reconstruction accuracy and number of iterations required to converge.