<p> Many problems in image and video processing may be formulated in the language of constrained optimization. Algorithms for solving general constrained optimization problems may not guarantee solutions or be computationally efficient, particularly if the problem is nonlinear or non-convex. Oftentimes these constrained optimization problems may be relaxed into the form of a convex problem. This allows for the use of convex solvers such as the Augmented Lagrangian method and the Split Bregman iteration. In this thesis, we will study the advantages of incorporating convexity into constrained optimization problems. These problems will be motivated from the standpoint of hyperspectral image processing, particularly the detection and identification of airborne chemicals in gas cloud releases. </p>

Under certain conditions, flow down an inclined plane destabilizes and a persistent series of interfacial waves develop. An interest in determining under what conditions a flow becomes unstable and how the interface develops has motivated researchers to derive several models for analyzing this problem. The first part of this thesis compares three models for flow down a wavy, inclined plane with the goal of determining which best predicts features of the flow. These models are the shallow-water model (SWM), the integral-boundary-layer (IBL) model, and the weighted residual model (WRM). The model predictions for the critical Reynolds number for flow over an even bottom are compared to the theoretical value, and the WRM is found to match the theoretical value exactly. The neutral stability curves predicted by the three models are compared to two sets of experimental data, and again the WRM most closely matches the experimental data. Numerical solutions of the IBL model and the WRM are compared to numerical solutions of the full Navier-Stokes equations; both models compare well, although the WRM matches slightly better. Finally, the critical Reynolds numbers for the IBL model and the WRM for flow over a wavy incline are compared to experimental data. Both models give results close to the data and perform equally well. These comparisons indicate that the WRM most accurately models the flow. In the second part of the thesis, the WRM is extended to include the effects of bottom heating and permeability. The model is used to predict the effect of heating and permeability on the stability of the flow, and the results are compared to theoretical predictions from the Benney equation and to a perturbation solution of the Orr-Sommerfeld equation from the literature. The results indicate that the model does faithfully predict the theoretical critical Reynolds number with heating and permeability, and both effects destabilize the flow. Finally, numerical simulations of the model equations are compared to full numerical solutions of the Navier-Stokes equations for the case with bottom permeability. The results are found to agree, which indicates that the WRM remains appropriate when permeability is included.

This work seeks, through numerical simulations as well as analysis, to derive from relatively simple models an intuitive understanding of the dynamics and behaviour of flow in lakes near the bottom boundary. The main body is divided into two equally important sections, the analysis and simulation of the effects of nonlinear (quadratic) bottom drag on the flow, and the simulation of the effects of topography on lake dynamics as it relates to the redistribution of sediment from the lakebed. The simulations all follow a structured scheme, beginning with relatively simple one-dimensional models to build intuition and proceeding to full two-dimensional simulations using the weakly nonhydrostatic shallow water equations. Thus this work seeks to build an understanding of the behaviour of the modified shallow water equations (a good representation of lake behaviour) and to analyze the effects of nonlinear drag and bottom topography on these systems. The nonlinear drag chapters demonstrate that the addition of a nonlinear friction term, while very efficient at removing energy from the system, also causes interesting new behaviour. In the pendulum (a good one dimensional analogy to the shallow water equations) the presence of nonlinear drag alters the parameter space enough to induce or destroy chaotic behaviour. A phenomenon worth considering in relation the shallow water equations. Additionally, the presence of drag causes as a cascade in spectral space, similar to the classical turbulent cascade. This work considers this effect and seeks to differentiate it from the turbulent cascade wherever and whenever possible. The final section of the thesis deals with the presence and effects of bottom topography (namely protrusions from the lake bed) on wave velocities due to a basin-scale seiche. This section examines both the dynamics of the system, through deflection about topography and the modification of the wave due to nonlinearity and bathymetry, as well as the relationship between lake dynamics and sediment redistribution. Finally, possible future directions are suggested as natural extensions to the work already done, as well as more sophisticated numerical models which could provide further insight into the problems discussed herein.

abstract: This dissertation is intended to tie together a body of work which utilizes a variety of methods to study applied mathematical models involving heterogeneity often omitted with classical modeling techniques. I posit three cogent classifications of heterogeneity: physiological, behavioral, and local (specifically connectivity in this work). I consider physiological heterogeneity using the method of transport equations to study heterogeneous susceptibility to diseases in open populations (those with births and deaths). I then present three separate models of behavioral heterogeneity. An SIS/SAS model of gonorrhea transmission in a population of highly active men-who-have-sex-with-men (MSM) is presented to study the impact of safe behavior (prevention and self-awareness) on the prevalence of this endemic disease. Behavior is modeled in this examples via static parameters describing consistent condom use and frequency of STD testing. In an example of behavioral heterogeneity, in the absence of underlying dynamics, I present a generalization to ``test theory without an answer key" (also known as cultural consensus modeling or CCM). CCM is commonly used to study the distribution of cultural knowledge within a population. The generalized framework presented allows for selecting the best model among various extensions of CCM: multiple subcultures, estimating the degree to which individuals guess yes, and making competence homogenous in the population. This permits model selection based on the principle of information criteria. The third behaviorally heterogeneous model studies adaptive behavioral response based on epidemiological-economic theory within an $SIR$ epidemic setting. Theorems used to analyze the stability of such models with a generalized, non-linear incidence structure are adapted and applied to the case of standard incidence and adaptive incidence. As an example of study in spatial heterogeneity I provide an explicit solution to a generalization of the continuous time approximation of the Albert-Barabasi scale-free network algorithm. The solution is found by recursively solving the differential equations via integrating factors, identifying a pattern for the coefficients and then proving this observed pattern is consistent using induction. An application to disease dynamics on such evolving structures is then studied. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2012

abstract: In vertebrate outer retina, changes in the membrane potential of horizontal cells affect the calcium influx and glutamate release of cone photoreceptors via a negative feedback. This feedback has a number of important physiological consequences. One is called background-induced flicker enhancement (BIFE) in which the onset of dim background enhances the center flicker response of horizontal cells. The underlying mechanism for the feedback is still unclear but competing hypotheses have been proposed. One is the GABA hypothesis, which states that the feedback is mediated by gamma-aminobutyric acid (GABA), an inhibitory neurotransmitter released from horizontal cells. Another is the ephaptic hypothesis, which contends that the feedback is non-GABAergic and is achieved through the modulation of electrical potential in the intersynaptic cleft between cones and horizontal cells. In this study, a continuum spine model of the cone-horizontal cell synaptic circuitry is formulated. This model, a partial differential equation system, incorporates both the GABA and ephaptic feedback mechanisms. Simulation results, in comparison with experiments, indicate that the ephaptic mechanism is necessary in order for the model to capture the major spatial and temporal dynamics of the BIFE effect. In addition, simulations indicate that the GABA mechanism may play some minor modulation role. / Dissertation/Thesis / Ph.D. Mathematics 2012

<p> This thesis focuses on the topics of sparse and non-local signal and image processing. In particular, I present novel algorithms that exploit a combination of sparse and non-local data models to perform tasks such as compressed-sensing reconstruction, image compression, and image denoising. The contributions in this thesis are: (1) a fast, approximate minimum mean-squared error (MMSE) estimation algorithm for sparse signal reconstruction, called Randomized Iterative Hard Thresholding (RIHT). This algorithm has applications in compressed sensing, image denoising, and other sparse inverse problems. (2) An extension to the Block-Matching 3D (BM3D) denoising algorithm that matches blocks at different rotation angles. This algorithm improves on the performance of BM3D in terms of both visual quality and quantitative denoising accuracy. (3) A novel non-local, causal image prediction algorithm, and a corresponding codec implementation that achieves state of the art lossless compression performance on 8-bit grayscale images. (4) A deep convolutional neural network (CNN) architecture that achieves state-of-the-art results in bilnd image denoising, and a novel non-local deep network architecture that further improves performance.</p><p>

<p> The purpose of this thesis is to investigate possibly interesting classes of polygons within quadrilaterals and hexagons. We utilize zero sets of polynomials using the vertices of these polygons to find characteristics of two new quadrilateral classes and four hexagon classes. We will refer to the polynomials as &ldquo;forms&rdquo;. These forms are invariant under translation and rotation and scaled by a factor under dilation. We define three functions: X- a reflection across the <i> x</i>-axis, <i>&fcy;</i>- a relabeling of vertices across the <i> AC</i> diagonal (or in a hexagon, across a long diagonal AD), and <i> &rho;</i>- a relabeling of the vertices by rotating them clockwise. </p><p> We find forms that characterize our classes of polynomials based on how they interact with these functions. For these particular classes, there is one form (up to constant multiples) of order 1 that interacts with the functions in the manner that characterizes the class of polygon. A form of order one is scaled by <i>r</i>² if the polygon is scaled by <i> r</i>. For each class then, we found several forms that are all equivalent, because they all interact with the functions in the same way.</p><p>

<p> Computational models are aiding in the advancement of science &ndash; from biological, to engineering, to social systems. To trust the predictions of computational models, however, we must understand how the errors in the models&rsquo; inputs (i.e., through measurement error) affect the output of the systems: we must quantify the uncertainty that results from these input errors. Uncertainty quantification (UQ) becomes computationally complex when there are many parameters in the model. In such cases it is useful to reduce the dimension of the problem by identifying unimportant parameters and disregarding them for UQ studies. This makes an otherwise intractable UQ problem tractable. <i> Active subspaces</i> extend this idea by identifying important linear combinations of parameters, enabling more powerful and effective dimension reduction. Although active subspaces give model insight and computational tractability for scalar-valued functions, it is not enough. This analysis does not extend to time-dependent systems. In this thesis we discuss time-dependent, dynamic active subspaces. We develop a methodology by which to compute and approximate dynamic active subspaces, and introduce the analytical form of dynamic active subspaces for two cases. To highlight these methods we find dynamic active subspaces for a linear harmonic oscillator and a nonlinear enzyme kinetics system.</p><p>

abstract: Advances in experimental techniques have allowed for investigation of molecular dynamics at ever smaller temporal and spatial scales. There is currently a varied and growing body of literature which demonstrates the phenomenon of \emph{anomalous diffusion} in physics, engineering, and biology. In particular many diffusive type processes in the cell have been observed to follow a power law $\left<x^2\right> \propto t^\alpha$ scaling of the mean square displacement of a particle. This contrasts with the expected linear behavior of particles undergoing normal diffusion. \emph{Anomalous sub-diffusion} ($\alpha<1$) has been attributed to factors such as cytoplasmic crowding of macromolecules, and trap-like structures in the subcellular environment non-linearly slowing the diffusion of molecules. Compared to normal diffusion, signaling molecules in these constrained spaces can be more concentrated at the source, and more diffuse at longer distances, potentially effecting the signalling dynamics. As diffusion at the cellular scale is a fundamental mechanism of cellular signaling and additionally is an implicit underlying mathematical assumption of many canonical models, a closer look at models of anomalous diffusion is warranted. Approaches in the literature include derivations of fractional differential diffusion equations (FDE) and continuous time random walks (CTRW). However these approaches are typically based on \emph{ad-hoc} assumptions on time- and space- jump distributions. We apply recent developments in asymptotic techniques on collisional kinetic equations to develop a FDE model of sub-diffusion due to trapping regions and investigate the nature of the space/time probability distributions assosiated with trapping regions. This approach both contrasts and compliments the stochastic CTRW approach by positing more physically realistic underlying assumptions on the motion of particles and their interactions with trapping regions, and additionally allowing varying assumptions to be applied individually to the traps and particle kinetics. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2014

abstract: Analysis of social networks has the potential to provide insights into wide range of applications. As datasets continue to grow, a key challenge is the lack of a widely applicable algorithmic framework for detection of statistically anomalous networks and network properties. Unlike traditional signal processing, where models of truth or empirical verification and background data exist and are often well defined, these features are commonly lacking in social and other networks. Here, a novel algorithmic framework for statistical signal processing for graphs is presented. The framework is based on the analysis of spectral properties of the residuals matrix. The framework is applied to the detection of innovation patterns in publication networks, leveraging well-studied empirical knowledge from the history of science. Both the framework itself and the application constitute novel contributions, while advancing algorithmic and mathematical techniques for graph-based data and understanding of the patterns of emergence of novel scientific research. Results indicate the efficacy of the approach and highlight a number of fruitful future directions. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics for the Life and Social Sciences 2015