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 …
The solutions to nonlinear wave equations and diffusion equations often exhibit complicated behavior but the solution is nevertheless quite smooth, which makes high order methods …
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 …
Interacting particle systems (IPS) are a class of pure jump stochastic processes used to model systems of individual entities, referred to as ``particles", that evolve …
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 …
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. …
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 …
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 …
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 …
The derivation of fluid equations from the Boltzmann equation is a fundamental problem in kinetic theory. In this paper, we investigate several results concerning the …
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 …
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 …