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Some of the material in is restricted to members of the community. By logging in, you may be able to gain additional access to certain collections or items. If you have questions about access or logging in, please use the form on the Contact Page.
Artificial neural networks (ANNs) are very popular nowadays and offer reliable solutions to many classification problems. Recent research indicates that these neural networks might be overparameterized and different solutions have been...
New numerical algorithms are devised for high-order, efficient quadrature in domains arising from the intersection of a hyperrectangle and a manifold implicitly defined by level sets. By casting the manifold locally as the graph of a...
With today's trend toward ever growing amounts of data, deep learning approaches are increasingly deployed with impressive success, notably in image recognition and time series classification applications. State of the art performance of...
The default threshold framework for credit risk modeling developed by Garreau and Kercheval [SIAM Journal on Financial Mathematics, 7:642-673, 2016] enjoys the advantages of both the structural form models and the reduced form models, ...
This dissertation investigates the practical expressive power of machine learning models as general function approximators and their practical applications in optimal filtering problems. The practical expressive power of these models...
Symmetric positive definite (SPD) matrices have become fundamental computational objects in many areas. It is often of interest to average a collection of symmetric positive definite matrices. This dissertation investigates different...
Diseases such as tuberculosis, chronic pneumonia, and inner ear infections are caused by bacterial biofilms. Biofilms can form on any surface such as teeth, floors, or drains. Many studies show that it is much more difficult to kill the...
Many phenomena in ecology, evolution, and organismal biology relate to how a system changes through time. Unfortunately, most of the statistical methods that are common in these fields represent samples as static scalars or vectors....
Iterative methods are widely used to solve algebraic equations and matrix equations. In fact, they can also be used to solve differential equations. Unlike discretization methods which generate point solutions, iterative methods generate...
Chan-Vese is a level set method that simultaneously evolves a level set surface and fits locally constant intensity models for the interior and exterior regions to minimize a Mumford-Shah integral. However, the length-based contour...
Overcoming Geometric Limitations in the Finite Element Method by Means of Polynomial Extension: Application to Second Order Elliptic Boundary Value and Interface Problems
In this dissertation, we present a new approach for approximating the solution of second order partial differential equations and interface problems. This approach is based on the classical finite element method, where instead of using...
Algorithmic Lung Nodule Analysis in Chest Tomography Images: Lung Nodule Malignancy Likelihood Prediction and a Statistical Extension of the Level Set Image Segmentation Method
Lung cancer has the highest mortality rate of all cancers in both men and women in the United States. The algorithmic detection, characterization, and diagnosis of abnormalities found in chest CT scan images can aid radiologists by...
We consider a classical risk process with arrival of claims following a non-stationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim...
The process of Data Assimilation in the Geosciences, in which real world data is used to improve the mathematical simulation of some event of interest, is a large field in applied mathematics. When that system is driven by dynamics, and...
Recently nonlocal continuum models have gained interest as alternatives to traditional PDE models due to their capability of handling solutions with discontinuities and their ease of modeling anomalous diffusion. The typical approach...
In this work, several nonlocal problems are studied. Analysis and computation have been done for these problems. Firstly, we consider the time-dependent nonlocal diffusion and wave equations, formulated in the peridynamics setting....
Next generation sequencing can rapidly analyze entire genomes in just hours. However, due to the nature of the sequencing process, errors may arise which limit the accuracy of the reads obtained. Luckily, modern sequencing technologies...
Quasi-Newton methods have gained popularity across various domains, providing efficient iterative algorithms for finding optimal solutions to unconstrained optimization problems. Their limited- memory variants offer advantages in terms...
Diffusion Tensor Imaging (DTI) provides a unique contrast based on the restricted directionality of water movement in an anisotropic environment. As such, DTI-based tractography can be used to characterize and quantify the structural...
Unsteady fluid flows have complex dynamics due to the nonlinear interactions amongst vortical elements. In this thesis, a network-theoretic framework is developed to describe vortical and modal (coherent structure) interactions in...
Living systems consist of several complex interacting components. Depending on the complexity of the organism, these components can span from molecules to tissues and organs. Systems biology is the interdisciplinary field of study that...
Pancreatic islet beta-cells play a vital role in regulating blood glucose levels by releasing insulin into the bloodstream. Insulin is released in pulses that parallel interacting beta-cell rhythms, including oscillatory glucose...
The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., ...
In astrophysical hydrodynamical objects, multiple physical processes take place on a wide variety of spatial and temporal scales simultaneously, making direct numerical simulation of such objects dicult computationally. Our work focuses...
In previous work, Striegel and Hurdal have developed a mathematical model for cortical folding pattern formation during development (Striegel). A Turing reaction-diffusion system and a prolate spheroid domain were used to model the shape...
The diagnosis and treatment of gliomas continues to pose a significant challenge for oncologists who not only have to contend with managing acute neurological symptoms, but also the almost inevitable development of resistance to...
Mass Conserving Hamiltonian-Structure-Preserving Reduced Order Modeling for the Rotating Shallow Water Equations Discretized by a Mimetic Spatial Scheme
Ocean modeling, in a climate-modeling context, requires long time-horizons over global scales, which when combined with accurate resolution in time and space makes simulations very time-consuming. While high-resolution ocean-modeling...
The aim of this work is to carry out convergence analysis and algorithm implementation of a novel sample-wise backpropagation method for training a class of stochastic neural networks (SNNs). The structure of the SNN is formulated as a...
Neuron morphology plays a central role in characterizing cognitive health and functionality of brain structures. The problem of quantifying neuron shapes, and capturing statistical variability of shapes, is difficult because axons and...
In this dissertation, we present several applications of polynomial chaos in Monte Carlo simulation.First, we investigate the use of polynomial chaos as a control variate method for Monte Carlo simulation. Specically, we analyze the mean...
We consider the heat equation forced by a space-time white noise and with periodic boundary conditions in one dimension. The equation is discretized in space using four different methods; spectral collocation, spectral truncation, finite...
Tolerant bacteria enmeshed in a biofilm causes several difficult to treat illnesses like tuberculosis, chronic pneumonia, and chronic inner ear infections. These diseases typically respond poorly to antibiotics due to high tolerance....
This dissertation considers the optimization problems that are in the form of minX∈Fv f(x)+λ∥X∥1, where f is smooth, Fv = {X ∈ Rn×q : XTX = Iq, v ∈ span(X)}, and v is a given positive vector. Clustering analysis is a fundamental machine...
This thesis studies a novel self- and mutual-exciting stochastic model to capture two essential features underlying a general type of discrete-time event data motivated by the practice: the dependence on the past event arrivals and...
The stability of soliton solutions in Ablowitz-Musslimani type equations is investigated. In particular, the robustness of line solitons to transverse perturbations is studied. The linear stability problem for perturbed solitons is...
Birdsong is a model system for the production of learned, serially ordered motor movements, such as playing a musical instrument or riding a bicycle. To this end, the neural mechanisms underlying birdsong have been studied in great depth...
Dissertation focuses on exploring the capabilities of the SRSF statistical shape analysis framework through various applications. Each application gives rise to a specific mathematical shape analysis model. The theoretical investigation...
Groundwater is a vital natural resource, and our ability to protect and manage this resource efficiently and effectively relies heavily on our ability to perform reliable and accurate computer modeling and simulation of subsurface...
This dissertation uses Riemannian optimization theory to increase our understanding of the role extraction problem and algorithms. Recent ideas of using the low-rank projection of the neighborhood pattern similarity measure and our...
Clustering is a widely used technique with a long and rich history in a variety of areas. However, most existing algorithms do not scale well to large datasets, or are missing theoretical guarantees of convergence. In this dissertation, ...
While modern medicine has made enormous strides in improving patient care, treatment outcomes, and overall quality of life. It has become increasingly more clear that collaborative and highly interdisciplinary research is necessary to...
The central purpose of this thesis is to explore the behavior of the numerical solution of the Cold- Ion model with shock forming conditions in one and two dimensions. In the one dimensional case, a comparison between the numerical...
In our work we analyze and implement numerical schemes for the infinite Prandtl number model for convection. This model describes the convection that is a potential driving force behind the flow and temperature of the Earth's mantle....
In many composite materials, rigid fibers are distributed throughout the material to tune the mechanical, thermal, and electric properties of the composite. The orientation and distribution of the fibers play a critical role in the...
This research provides theoretical and computational developments in statistical shape analysis of shape graphs, and demonstrates them using analysis of complex network-type object data such as retinal blood-vessel (RBV) networks. The...
This dissertation research extends and simplfiies existing piecewise-linear homotopy (PL) methods to solve G(x) = 0, with G : ℝⁿ → ℝ[superscript m]. Existing PL methods are designed to solve F(x) = 0, with F : ℝⁿ → ℝⁿ and some related...
This study explores whether or not tropical cyclone (TC) structure information may be retrieved from satellite total ozone observations and how to link total ozone with analysis fields for potential application to TC vortex...
Computer vision principles enable the analysis of fire, wind, and plume behavior from visual and infrared (IR) video as opposed to sparse measurements obtained with expensive instrumentation. Data that quantifies the transport of heat...
Improving the Accuracy of 3D Chromosome Structure Inference and Analyzing the Organization of Genome in Early Embryogenesis Using Single Cell Hi-C Data
This dissertation summarizes my graduate work on the structure and organization of mouse genome during preimplantation development. My research is divided into three different areas, which I will discuss in turn. To begin, I will discuss...
Novel Numerical Analysis Methods, Using the WENO and WENO-Z Algorithms, for Combining Observational Data with Model Predictions for Improving Forecasts
“Data assimilation is a mathematical discipline that seeks to optimally combine theory (usually in the form of a numerical model) with observations.” — Wikipedia Strong constraint 4D-Variational data assimilation (4D-Var) seeks to find...
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