Applied mathematics
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Stochastic Modeling of Reversible Biochemical ReactionDiffusion Systems and HighResolution ShockCapturing Methods for Fluid Interfaces
My thesis contains two parts, both of which are motivated by biological problems. One is on stochastic reactiondiffusion for biochemical systems and the other on shockcapturing methods for fluid interfaces. In both parts, ... 
Dimensionality hyperreduction and machine learning for dynamical systems with varying parameters
This work demonstrates methods for hyper reduction and efficient computation of solutions of dynamical systems using optimization and machine learning techniques. We consider nonlinear partial differential equations that ... 
Machine learning and data decompositions for complex networked dynamical systems
Machine learning has become part of our daily lives. Its applications include personalized advertisements, stock price predictions, and selfdriving cars. The goal of this thesis is to study ways to apply machine learning ... 
Dynamic, convex, and robust optimization with Bayesian learning for responseguided dosing
Medical treatment commonly involves the administration of drug doses at multiple timepoints. Intuitively, the higher the doses, the higher the likelihood of disease control as well as the risk of adverse effects and of ... 
Variability in Modified Estimators of VaR and ES
Modified ValueatRisk (mVaR) and Modified Expected Shortfall (mES) are risk estimators that can be calculated without modelling the distribution of asset returns. These modifided estimators use skewness and kurtosis ... 
Multiscale Modeling of Esophageal Adenocarcinoma
Over the past three to four decades, esophageal adenocarcinoma (EAC) incidence has increased dramatically in the Western world due to causes that are not well understood. Current screening strategies for early detection ... 
Interface Problems using the Fokas Method
Interface problems for partial differential equations are initial boundary value problems for which the solution of an equation in one domain prescribes boundary conditions for the equations in adjacent domains. These ... 
Coordinated neural activity: Mechanistic origins and impact on stimulus coding
How does the activity of populations of neurons encode the signals they receive? Since neurons in vivo are inherently variable, each fixed input to a population will elicit not a deterministic response, but rather a ... 
Closing the Loop: Optimal Stimulation of Neuronal Networks via Adaptive Control Algorithms
The Caenorhabditis elegans (C. elegans) worm is a wellstudied biological organism model. The nervous system of C. elegans is particularly appealing to study, since it is a tractable fully functional neuronal network for ... 
Integrating DataDriven Methods in Nonlinear Dynamical Systems: Control, Sparsity and Machine Learning
The goal of my thesis is to provide a theoretical demonstration of how dimension reduc tion, control and machine learning techniques can be applied to optimize the performance of complex nonlinear systems. Specifically, ... 
SelfOptimizing Metamaterial Antennas
The reconfigurable holographic metamaterial antenna is an attractive new technology for satellite communications, particularly in mobile applications. This antenna is thin, lightweight, consumes little power to operate, ... 
Mathematical modeling of focal axonal swellings arising in traumatic brain injuries and neurodegenerative diseases
There is a broad need in the neuroscience, neurological and biomedical engineering communities to better classify, quantify and diagnose Focal Axonal Swellings (FAS) and their impact on cognitive deficits and/or neural ... 
On the instability of water waves with surface tension.
We analyze the stability of solutions to Euler's equations in the presence of surface tension. First we compute stationary solutions to periodic Euler's equations in a travelling frame of reference and then we analyze their ... 
Lagrangian coherent structures and the dynamics of inertial particles
Dynamics of inertial particles in twodimensional planar flow have been investigated by evaluating finitetime Lyapunov exponents (FTLE). The first part of our work deals with inertial particle dynamics. The MaxeyRiley ... 
Spectral Methods for Partial Dierential Equations that Model Shallow Water Wave Phenomena
Mathematical models for waves on shallow water surfaces has been of interest to researchers dating back to the 1800's. These models are governed by partial differential equations, and many of them have rich mathematical ... 
Collective Activity in Neural Networks: the Mathematical Structure of Connection Graphs and Population Codes
Correlated, or synchronized, spiking activity among pairs of neurons is widely observed across the nervous system. How do these correlations arise from the dynamics of neural networks? The interconnectivity of neurons is ... 
Analysis of an Aggregationbased Algebraic Multigrid Method and its Parallelization
The interests of this thesis are twofold. First, a twogrid convergence analysis based on the paper [ \textit{Algebraic analysis of aggregationbased multigrid } by A. Napov and Y. Notay, Numer. Lin. Alg. Appl. 18 (2011), ... 
Finite volume methods for Tsunamis genereated by submarine landslides
(20140430)Submarine landslides can generate tsunamis, and the generated waves can be catastrophic when a large volume of landslide material is involved. Moreover, large earthquakes are often accompanied by submarine landslides that ... 
On driven neural assemblies: synchrony, chaos and entropy.
(20140224)In this dissertation, I address mathematical problems arising from the field of theo retical neuroscience that share a common theme: temporally driven neural networks treated as perturbed, coupled nonlinear dynamical ... 
Value function approximation methods for Linearlysolvable Markov Decision Process
(20140224)Optimal control provides an appealing machinery to complete complicated control tasks with limited prior knowledge. Both global methods and online trajectory optimization methods are powerful techniques for solving optimal ...