Other Applied Mathematics Commons^™

Models Of Functional Redundancy In Ecological Communities, Sandra Annie Tsiorintsoa

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Functional redundancy is the number of taxa that perform a given function within a given community. In most systems, high levels of functional redundancy are important, because they contribute to ecosystem stability. However, we currently have very little understanding of why functional redundancy varies among communities. One possible factor that could affect functional redundancy is environmental complexity. Many studies show that simplified ecosystems harbor communities with lower taxon diversity. What is less clear is if this simplicity and lower taxon diversity also affects functional redundancy. To answer this question, we use metacommunity models to explore the connection between environmental complexity …

Aspects Of Stochastic Geometric Mechanics In Molecular Biophysics, David Frost

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In confocal single-molecule FRET experiments, the joint distribution of FRET efficiency and donor lifetime distribution can reveal underlying molecular conformational dynamics via deviation from their theoretical Forster relationship. This shift is referred to as a dynamic shift. In this study, we investigate the influence of the free energy landscape in protein conformational dynamics on the dynamic shift by simulation of the associated continuum reaction coordinate Langevin dynamics, yielding a deeper understanding of the dynamic and structural information in the joint FRET efficiency and donor lifetime distribution. We develop novel Langevin models for the dye linker dynamics, including rotational dynamics, based …

Asymptotic Cones Of Quadratically Defined Sets And Their Applications To Qcqps, Alexander Joyce

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Quadratically constrained quadratic programs (QCQPs) are a set of optimization problems defined by a quadratic objective function and quadratic constraints. QCQPs cover a diverse set of problems, but the nonconvexity and unboundedness of quadratic constraints lead to difficulties in globally solving a QCQP. This thesis covers properties of unbounded quadratic constraints via a description of the asymptotic cone of a set defined by a single quadratic constraint. A description of the asymptotic cone is provided, including properties such as retractiveness and horizon directions.

Using the characterization of the asymptotic cone, we generalize existing results for bounded quadratically defined regions with …

Multi-Commodity Flow Models For Logistic Operations Within A Contested Environment, Isabel Strinsky

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Today's military logistics officers face a difficult challenge, generating route plans for mass deployments within contested environments. The current method of generating route plans is inefficient and does not assess the vulnerability within supply networks and chains. There are few models within the current literature that provide risk-averse solutions for multi-commodity flow models. In this thesis, we discuss two models that have the potential to aid military planners in creating route plans that account for risk and uncertainty. The first model we introduce is a continuous time model with chance constraints. The second model is a two-stage discrete time model …

Null Space Removal In Finite Element Discretizations, Pengfei Jia

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Partial differential equations are frequently utilized in the mathematical formulation of physical problems. Boundary conditions need to be applied in order to obtain the unique solution to such problems. However, some types of boundary conditions do not lead to unique solutions because the continuous problem has a null space. In this thesis, we will discuss how to solve such problems effectively. We first review the foundation of all three problems and prove that Laplace problem, linear elasticity problem and Stokes problem can be well posed if we restrict the test and trial space in the continuous and discrete finite element …

On Variants Of Sliding And Frank-Wolfe Type Methods And Their Applications In Video Co-Localization, Seyed Hamid Nazari

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In this dissertation, our main focus is to design and analyze first-order methods for computing approximate solutions to convex, smooth optimization problems over certain feasible sets. Specifically, our goal in this dissertation is to explore some variants of sliding and Frank-Wolfe (FW) type algorithms, analyze their convergence complexity, and examine their performance in numerical experiments. We achieve three accomplishments in our research results throughout this dissertation. First, we incorporate a linesearch technique to a well-known projection-free sliding algorithm, namely the conditional gradient sliding (CGS) method. Our proposed algorithm, called the conditional gradient sliding with linesearch (CGSls), does not require the …

Efficiency Of Homomorphic Encryption Schemes, Kyle Yates

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In 2009, Craig Gentry introduced the first fully homomorphic encryption scheme using bootstrapping. In the 13 years since, a large amount of research has gone into improving efficiency of homomorphic encryption schemes. This includes implementing leveled homomorphic encryption schemes for practical use, which are schemes that allow for some predetermined amount of additions and multiplications that can be performed on ciphertexts. These leveled schemes have been found to be very efficient in practice. In this thesis, we will discuss the efficiency of various homomorphic encryption schemes. In particular, we will see how to improve sizes of parameter choices in homomorphic …

Optimal First Order Methods For Reducing Gradient Norm In Unconstrained Convex Smooth Optimization, Yunheng Jiang

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In this thesis, we focus on convergence performance of first-order methods to compute an $\epsilon$-approximate solution of minimizing convex smooth function $f$ at the $N$-th iteration.

In our introduction of the above research question, we first introduce the gradient descent method with constant step size $h=1/L$. The gradient descent method has a $\mathcal{O}(L^2\|x_0-x^*\|^2/\epsilon)$ convergence with respect to $\|\nabla f(x_N)\|^2$. Next we introduce Nesterov’s accelerated gradient method, which has an $\mathcal{O}(L\|x_0-x^*\|\sqrt{1/\epsilon})$ complexity in terms of $\|\nabla f(x_N)\|^2$. The convergence performance of Nesterov’s accelerated gradient method is much better than that of the gradient descent method but still not optimal. We also …

Advancements In Gaussian Process Learning For Uncertainty Quantification, John C. Nicholson

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Gaussian processes are among the most useful tools in modeling continuous processes in machine learning and statistics. The research presented provides advancements in uncertainty quantification using Gaussian processes from two distinct perspectives. The first provides a more fundamental means of constructing Gaussian processes which take on arbitrary linear operator constraints in much more general framework than its predecessors, and the other from the perspective of calibration of state-aware parameters in computer models. If the value of a process is known at a finite collection of points, one may use Gaussian processes to construct a surface which interpolates these values to …

An Algorithm For Biobjective Mixed Integer Quadratic Programs, Pubudu Jayasekara Merenchige

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Multiobjective quadratic programs (MOQPs) are appealing since convex quadratic programs have elegant mathematical properties and model important applications. Adding mixed-integer variables extends their applicability while the resulting programs become global optimization problems. Thus, in this work, we develop a branch and bound (BB) algorithm for solving biobjective mixed-integer quadratic programs (BOMIQPs). An algorithm of this type does not exist in the literature.

The algorithm relies on five fundamental components of the BB scheme: calculating an initial set of efficient solutions with associated Pareto points, solving node problems, fathoming, branching, and set dominance. Considering the properties of the Pareto set of …