Applied mathematics
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PolynomialBased Methods for TimeIntegration
This thesis is divided into two parts: The first introduces a new time integration framework that is based on interpolating polynomials, and the second extends exponential integration to the spectral deferred correction ... 
Veriﬁcation of cloud production in the Community Atmosphere Model: A comparison of two data assimilation techniques
Clouds play an important role in regulating our climate, and it is vital that we are able to accurately simulate them in global climate models. Data assimilation can be used to force the model towards simulating a specific ... 
Some Problems in Stochastic Dynamics and Statistical Analysis of SingleCell Biology of Cancer
With the development of experimental apparatus and data processing softwares, one now has easy access to cancer related data on a single cell, its genome and/or molecular compositions. At this level of description, ... 
Multiscale Techniques for Nonlinear Dynamical Systems: Applications and Theory
Most interesting real world systems can be understood at multiple scales of detail. A physical system such as a closed container of gas particles can be understood in terms of hydrodynamic flows, molecules and atoms exerting ... 
Green's Law and the Riemann Problem in Layered Media
The propagation of long waves onto a continental shelf is of great interest in tsunami modeling, where understanding the amplification of waves during shoaling is of significant importance. When the linearized shallow water ... 
Stochastic Dynamics: Markov Chains, Random Transformations and Applications
Stochastic dynamical systems, as a rapidly growing area in applied mathematics, has been a successful modeling framework for biology, chemistry and data science. Depending upon the origin of uncertainties in an application ... 
Skewt Information Matrix: Evaluation and Use
Azzalini’s skewt distributions, described in detail in Azzalini [2013], have become very popular because of their practical usefulness and the complete R package sn for skewnormal distributions that include extensive ... 
This Brain Is a Mess: Inference, Random Graphs, and Biophysics to Disentangle Neuronal Networks
At first glance, the neuronal network seems like a tangled web in many areas throughout the nervous system. Often, our best guess is that such “messy” connections are close to random, while obeying certain statistical ... 
Uncertainty Quantification Problems in Tsunami Modeling and Reduced Order Models for Hyperbolic Partial Differential Equations
In this thesis, we consider an uncertainty quantification (UQ) problem that arises from tsunami modeling, namely the probabilistic tsunami hazard assessment (PTHA) problem. The goal of PTHA is to compute the probability ... 
High order shock capturing methods with compact stencils for use with adaptive mesh refinement and mapped grids
This thesis focuses on several developments toward creating a high order shock capturing method that can be used on mapped grids with blockstructured adaptive mesh refinement (AMR). The discontinuous Galerkin (DG) method ... 
On neural encoding: its estimation, application, and development
The spiking activity of neurons encodes information about sensory stimuli and about planned or executed motor outputs. An important problem in computational neuroscience is the development of predictive models that describe ... 
Irreversibility in Stochastic Dynamic Models and Efficient Bayesian Inference
This thesis is the summary of an excursion around the topic of reversibility. We start the journal from a classical mechanical view of the “time reversal symmetry”: we look into the details to track the movements of all ... 
The stability and instabilities of stationary solutions to the nonlinear Schroedinger equation and the sineGordon equation
I present an analysis of the stability spectrum of all stationary elliptictype solutions to the focusing Nonlinear Schroedinger equation and the sineGordon equation. An analytical expression for the spectrum is given. ... 
Data assimilation problems in glaciology
Rising sea levels due to mass loss from Greenland and Antarctica threaten to inun date coastal areas the world over. For the purposes of urban planning and hazard mitigation, policy makers would like to know how much ... 
Energy and Charge Transfer in Open Plasmonic Systems
Coherent and collective charge oscillations in metal nanoparticles (MNPs), known as localized surface plasmons, offer unprecedented control and enhancement of optical processes on the nanoscale. Since their discovery in ... 
Multiscale modeling of paracrine PDGFdriven glioma growth and invasion
The most common primary brain tumor in adults, glioma claims thousands of lives each year. Despite efforts to improve survival rates, the standard of care has remain unchanged for more than a decade. Recent research has ... 
On computing shape: a study of the neural processes concerning naturalistic boundary conformation within the ventral visual pathway
The perception of shape is a remarkable computation, solved rapidly by the brain to extract boundary features within natural scenes while being robust against many visual obstacles. Interestingly, while observers typically ... 
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 ...