In 1990, Thurston proved that for any two-dimensional simply connected region, the space of tilings is flip-connected. Later, Yamzon reproved this result using a commutative …
Given a stationary state-space model that relates a sequence of hidden states and corresponding measurements or observations, Bayesian filtering provides a principled statistical framework for …
With the phrases “artificial intelligence” and “Deep Learning” surfacing in the news alongside projections of rampant automation, this paper aims to explore Deep Learning in …
This thesis concerns the formulation and derivation of a generalization of a collection of basis functions originally devised by Norbert Wiener for function approximation over …
Computer models for underwater sound propagation often rely on ray tracing to obtain solutions to the high frequency wave equation. While adequate for large scale …
Electroencephalography (EEG) signals are created by electrical currents from pyramidal neurons in the outer layer of the brain. These signals are easy to obtain and …
The first part of the thesis presents the surface respresentation of blood vessel walls extracted from medical images, sensitivity to the inlet/outlet boundary conditions, and …
We combine the spectral element method with the smoothed profile method (SPM) to obtain an efficient method for flows with moving boundaries in complex geometries. …
nheritable differences in DNA sequence are a major focus of the genetic community, as such differences are known to play a role in phenotypic variation, …
As a malaria patient takes and metabolizes weekly doses of an anti-malarial pyrimethamine, there is a periodic change in drug concentration. The growth rate, or …
Electronic Health Records (EHRs) have become the primary form of medical data-keeping across the United States. Federal law restricts the sharing of any EHR data …
High-order methods are emerging in the scientific computing community as superior alternatives to the classical finite difference, finite volume, and continuous finite element methods. The …
In this dissertation we propose, analyze and computationally implement finite element models to two different two-dimensional saddle point systems of partial differential equations with boundary …
The mesoscopic description of molecular systems is investigated by using the memory function approach, and uncertainty quantification methods are developed for the evaluation of time …
This thesis is a large deviations study for the performance of an interacting particle method for rare event estimation. The analysis is restricted to a …
In this thesis, we investigate two applications of finite element methods that employ macro-elements: discrete elasticity sequences and convergence of Lagrange elements for a Maxwell …
Abstract Introduction: Cancer is a leading cause of death worldwide, accounting for 8.8 million deaths in 2015, and among which Lung Cancer consists 1.69 million …
Maximum likelihood estimators have traditionally dominated discrete inference for a long time. In this work we apply statistical decision theory to derive a new contender …
This thesis is a mathematical study of paleoclimatology and computational biology.<br/><br/> <br/><br/> Part I gives an introduction and overview of this dissertation.<br/><br/> <br/><br/> Part II presents the …
Studies in paleoclimatology should be accompanied by uncertainty analyses, as uncertainty is found within the very first measurements of the proxies imprinted in geological records. …
Non-allelic homologous recombination (NAHR) plays a major role in genome rearrangement and is implicated in numerous genetic disorders. But detection of NAHR poses a serious …
This dissertation presents two topics concerning weighted essentially non-oscillatory (WENO) finite difference schemes for solving hyperbolic problems. In the first part, we develop a high …
In this thesis, we study Prandtl's boundary layer theory for 2D, stationary, incompressible Navier-Stokes flows posed on domains with boundaries. The boundary layer hypothesis posed …
In many real-world applications, e.g., brain imaging and or weather patterns, data are captured over particular periods or intervals, which we call time series. Time …
We begin by asking how to write down Brownian motion on certain quotient Riemannian manifolds. The projection formula for the Laplace-Beltrami operator through Riemannian submersions …
Motile bacteria swim in fluid environments propelled by one or more flagella, spiral filaments that are attached to the cell body and a rotary motor. …
Burgers turbulence (1-D inviscid Burgers equation with random initial data) is a fundamental non-equilibrium model of stochastic coalescence. In this work we demonstrate that at …
We consider a discrete attachment model for the self-assembly of polyhedra called the Building Game. The combinatorial configuration space of the model is computed for …
Meshless methods provide an ideal framework for scalably simulating Lagrangian hydrodynamics in domains undergoing large deformation. For these schemes, interfaces can easily be treated without …
The problem of detecting changes in images taken by a stationary camera has been well studied and many algorithms now exist which perform robustly in …
This thesis is a mathematical and statistical study of conditional modeling and conditional inference. Two applications are covered: (1) Stock Prices and the Statistics of …
Confluence is an important property in many combinatorial processes. A globally confluent process is one in which a fixed initial state always leads to a …
Representation and comparison of shapes is a central problem in computer vision. In this dissertation, we continue investigation of a relatively recent approach, based on …
A space filling curve (or mapping) is a parametric map from a one-dimensional unit interval to a n-dimensional hypercube that traverses every point in its …
Abstract of Data-Driven Mathematical Analysis with Applications in Dynamical Systems, Biology, and Social Justice, by Rebecca Santorella Ph.D., Brown University, May 2022. As data becomes …
This thesis takes aim at two directions. The first direction involves setting the foundations for a new type of data-driven scientific computing, essentially creating a …
A number of problems of interest in applied mathematics and biology involve the quantification of uncertainty in computational and real-world models. A recent approach to …
This thesis discusses two topics, both related to the goal of determining tumor blood flow parameters from time-sequenced contrast-enhanced medical imaging data in an effort …
This thesis is concerned with the applicationof a certain class of eigenvalue algorithmsto random matrix initial data. In particularwe study numerically the time to first …
This dissertation focuses on studies of two different discontinuous Galerkin (DG) methods for general convection-diffusion equations. One preserves the strict maximum principle for general nonlinear …
This thesis contains two parts, including the development of a modified total variation bounded (TVB) limiter applied to the discontinuous Galerkin (DG) methods, and the …
This thesis consists of two topics on discontinuous Galerkin (DG) finite element methods for time-dependent problems. In the first part of the thesis, we analyze …
Dissipative Particle Dynamics (DPD) is a Lagrangian type mesoscopic method widely applied in mesoscale hydrodynamics and complex fluids simulations. DPD can be understood as coarse-grained …
Dissipative Particle Dynamics (DPD) is a mesoscopic simulation method, potentially very effective in simulating mesoscale hydrodynamics and soft matter. This thesis addresses open theoretical and …
Micron-size paramagnetic particles suspended in viscous fluid will aggregate to form linear chains when subject to a uniform magnetic field. This process provides a way …
This thesis studies the application of the Wiener chaos expansion in the analysis of stochastic partial differential equations (SPDEs). Specifically, linear parabolic SPDEs and the …
Hamilton-Jacobi partial differential equations (HJ PDEs) have deep connections to a wide range of scientific disciplines including optimal control, differential games, imaging sciences, and machine …
In this work, we develop and analyze solvers and preconditioners designed for the implicit time integration of discontinuous Galerkin (DG) discretizations. The discontinuous Galerkin method …
In this thesis, we study the elliptic boundary value problems on irregular domains. We first obtain the $W^{2,p}$ and $C^2$ regularity theories for second-order, non-divergence …
Queuing systems are one of the most prominent technological evolution in today's digital world. Data centers and cloud networks operated by large companies like Microsoft, …
This paper measures both the correlation and causal relationship between startups and urban economic growth in the U.S. over the past thirty years. Correlation results …
Stochastic networks arise in a variety of real world applications including telecommunications, service systems such as call centers, computer networks, health care services and biological …
A Monte-Carlo Simulation is presented for Radiation Transport in water. This process is of outmost importance, having applications in oncology and therapy of cancer, in …
The problem of forming exact sequences of finite element spaces that discretize Hilbert complexes is central to the finite element exterior calculus. This framework offers …
Spatially localized structures, in which a spatially oscillatory pattern on a finite spatial range connects to a trivial homogeneous solution outside this range, have been …
This dissertation aims to study finite element methods for two-dimensional stationary second order interface model problems on homogeneous and heterogeneous media, under discretizations non-fitted with …
In this dissertation, we mainly discuss two topics of partial differential equations in fluid dynamics and kinetic theory: viscous surface wave and diffusive limit. With …
Motivated by the ubiquitous demand of leveraging both data and partial knowledge of physical laws for stochastic modeling and uncertainty quantification, in the dissertation, a …
Colloidal membranes are an example of a novel bioinspired soft material with rich properties that stem from the interplay of geometry and molecular order. A …
In recent years, there has been a great deal of interest surrounding the study of the asymptotics and global attractor structure for scalar parabolic PDEs …
Finite element methods (FEMs) are used to discretize partial differential equations in a wide array of applications, including fluid flow, solid mechanics, electromagnetism, optimal control, …
This dissertation consists of three topics on bound-preserving discontinuous Galerkin (DG) methods for time-dependent and stationary hyperbolic equations, and efficient finite difference weighted essentially non-oscillatory …
Part I introduces the discontinuous Galerkin (DG) method for solving hyperbolic equations. The introduction and the DG scheme will be given in the first two …
This thesis consists of two diverse topics on high order numerical methods for time-dependent partial differential equations (PDEs). In the first part, we develop a …
This dissertation focuses on the analysis and computation of a certain high order accurate numerical scheme which is used to solve problems in computational Cosmology. …
The probability density approach based on the response-excitation theory is developed for stochastic simulations of non-Markovian systems. This approach provides the complete probabilistic configuration of …
This thesis contains two topics on high-order accurate methods for solving Maxwell's equations. The first topic is the application of high-order accurate methods to the …
This thesis presents results aiming to enhance and broaden the applicability of the discontinuous Galerkin (''DG'') method in a variety of ways. DG was chosen …
Diabetes is the leading cause of chronic kidney disease and kidney failure worldwide, causing 44 percent of new cases of kidney failure each year. Although …
Despite the enormous amount of biological variation and technical uncertainty in sequence data, most phylogenetic workflows propagate a single point estimate throughout the numerous analysis …
Motivated largely by the problems of estimation error in investment portfolio optimization, this dissertation develops techniques to improve covariance forecasting and portfolio construction. Covariance forecasting …
The spread of infectious disease is strongly influenced by human interaction. In order to understand this connection and develop effective strategies against future outbreaks, it …
Glaciers have been melting and reforming on the Earth for millions of years. Over the last several decades, Geologists have created δ18O proxy records that …
Large-scale stochastic networks arise in a variety of real world applications such as telecommunications, service systems, computer networks, health care, and biological systems. Such networks …
As data structures become larger and more complex, methods of data analysis must evolve to harness the most accurate insights. Inspired by the challenges presented …
Lagrangian data assimilation is the process of estimating a velocity flow field, given observations of the locations of passive drifters whose trajectories are determined by …
The first two thirds of this thesis are concerned with the properties of a single server using the weighted-serve-the-longest queue policy (WSLQ). It addresses three …
We study a general class of mean field interacting particle systems with a finite state space. Particles evolve as exchangeable jump Markov processes, where finite …
This thesis considers a feed-forward network with a single server station serving jobs with multiple levels of priority. The service discipline is preemptive in that …
Large deviations are classical topics of study that enjoy great practical application under probability theory. In essence, it studies the properties lie in the tail …
In this thesis, we use the large deviations principle to characterize the rate of convergence of the empirical measures of Markov processes. An explicit formula …
This goal of this thesis is to understand patterns in the Swift{Hohenberg equation. Thepatterns studied are localized, stationary and radially symmetric in dimensions one throughthree. …
We analyze the self-rated health (SRH) of women in Cebu, Philippines. In three surveys conducted between 2002 and 2007, each woman in the study cohort …
Our work is motivated by the classical belief propagation algorithms which are well-known for obtaining state of the art results in certain settings, but often …
Many organisms need to establish spatial orientation during early development. In egg cells (oocytes) of the frog Xenopus laevis, spatial differentiation is achieved by localization …
The zebrafish (Danio rerio), a widely-studied model organism, is a small fish with distinctive black and yellow stripes on its body and fins. Numerous experimental …
Prevalent in sub-Saharan Africa and Asia, malaria, caused by the parasite Plasmodium affects over 200 million people yearly. One widely used antimalarial is the antifolate …
We prove a moderate deviations principle for the continuous time linear interpolation of discrete time recursive stochastic processes, and then investigate importance sampling schemes based …
In this work we develop and apply numerical methods for quantifying parametric uncertainty in mathematical models. In Part I, we introduce the Multi-Element Probabilistic Collocation …
Many scientific simulations rely on the solution of very large linear systems of equations, and multigrid and domain decomposition methods are widely used solvers. It …
Computational simulation pervades science and engineering, enabling new insights about complex physical phenomena. The advent of exascale computer systems, which will enable simulation at an …
This dissertation is composed around the subject of multiscale modeling of soft matter and biophysical systems with applications using large-scale computations. Specifically, it is expanded …
This thesis contains three related topics on the multiscale discontinuous Galerkin (DG) methods and applications. In the first part, we present a multiscale model for …