2022 Faculty and Student Summer Talks

David Mazowiecki

Numerical Simulation of Particles Undergoing Quincke Rotation

We outline a plan to solve an electrohydrodynamic problem, in which many dielectric spherical particles are immersed in a dielectric fluid at low Reynolds number and subjected to a uniform applied electric field. For the electrodynamic component of the problem, if the ratio of electric permittivity to conductivity in the particles is higher than that of the fluid, a sufficiently strong applied field can cause the particles to rotate. This is known as Quincke rotation. We employ boundary integral methods to solve the coupled electrodynamic and hydrodynamic problems for a system of many interacting particles.

Valeria Barra

Nonlinear Paths: Career Perspectives with a PhD in Math

In this talk, I am going to speak about my own academic and non-academic journey. I will take the audience with me across the different areas of research I have worked on: geometry and computational geometry, computational fluid dynamics, high-performance computing and most recently, climate science.

If you are not sure you have it all figured it out, if you are curious about learning different topics, across various disciplines, if you are not worried about jumping into new projects, then this talk might speak to you. I will introduce the world of Research Software Engineering and share how I “accidentally” became a Research Software Engineer. We will try to answer some questions together that many graduate students may find themselves asking throughout their academic journeys: what are the transferable skills that I need to acquire in grad school? Are there good habits and best practices that I should pick up now to save myself time later? What types of career paths are ahead of me? How can I adjust to career changes? How can I overcome impostor syndrome?

By sharing my own multi-disciplinary experiences, I hope I can give students multiple points of views and insights about different possible career perspectives and inspire them to carve their own unique path.

Connor Robertson

Data-driven Continuum Modeling and Simulation of Active Nematics via Sparse Identification of Nonlinear Dynamics

Data-driven modeling methods have recently shown great potential in determining accurate continuum models for complex systems directly from experimental measurements. One such complex system is the active nematic liquid crystal system consisting of microtubule-motor protein assemblies immersed in a fluid. Although several models have been proposed for the system, the governing equations remain under debate.

In this talk, I will present the process for extracting a continuum model directly from experimental image data via the "sparse identification of nonlinear dynamics" (SINDy) data-driven modeling technique. This will include methods for data extraction, symbolic generation of plausible terms, numerical differentiation of extracted data, and sparse regression. I will also discuss the physical implications of the learned model as compared to previously proposed models and show simulation results of the discovered model using a pseudospectral method.

Lauren Barnes

Modeling Phase Separation of Colloid-Polymer Mixtures in Microgravity

Colloidal particles are of great interest in industrial applications involving materials engineering, pharmaceutics, and electronics. Much of their value lies in their phase transition behavior, which exhibits striking similarities to phase transitions of systems on a molecular and even atomic scale and thus provides insight into systems that are otherwise difficult to observe. The growth of such colloidal crystals is often studied in a microgravity environment in order to minimize the effect of gravity on the system. For this reason, experiments on phase transitions of colloid-polymer mixtures have been performed by NASA onboard the ISS. We present a theoretical model to describe the phase transition behavior of a colloid-polymer mixture in microgravity, along with some preliminary simulation results.

Souaad Lazergui

High Frequency Asymptotic Expansion of the Helmholtz Equation Solutions Using Neumann to Dirichlet and Robin to Dirichlet Operators

This talk is concerned with the asymptotic expansions of the amplitude of the high-frequency scattering problem solution in the exterior of two-dimensional smooth convex scatterers. The original expansion was obtained by using a pseudo-differential decomposition of the Dirichlet to Neumann operator and a microlocal model of the Kirchhoff operator. In this work, we use Neumann to Dirichlet operator to derive the asymptotic expansion of the Kirchhoff amplitude over the scatterer boundary. Similarly to the Robin to Dirichlet operator, we seek the same ansatz for Robin boundary condition. we strongly believe that the resulting expansions can be used to appropriately choose the  ansatz in the design of  high-frequency numerical solvers, such as those based on  integral equations, in order to produce more accurate high-frequency asymptotic numerical solution of the Helmholtz equation. 

Nicholas Dubicki

A Micromagnetic Study of Skyrmions in Thin Film Ferromagnetic Bilayers

The dissertation will present a modeling, analytical, and computational study of magnetic skyrmions and then treat, in detail, the problem of skyrmions in two thin parallel ferromagnetic layers. Magnetic skyrmions are topologically protected, localized, nanoscale spin textures in thin ferromagnetic materials. At present they are of great interest to applied researchers due to their potential applications to information technology. We treat the bilayer system analytically using a thin-film approximation and an ansatz-based minimization of the energy, and aim to identify the stable states as they depend on material properties. There are two interesting types of behavior whereby skyrmions may form bound pairs or else repel/annihilate each other. Of particular interest is an apparent bifurcation between two types of stable states where the skyrmion in each layer is perfectly concentric with its counterpart, and states where they form off center pairings.

Samantha Evans

Towards a Boundary Integral Numerical Method for Two Phase Flow with Soluble Surfactant

Surfactant is a surface active agent important in microfluidic applications that lowers surface tension and introduces Marangoni force that opposes outer flow. However, it is difficult to adapt boundary integral methods to problems with soluble surfactant as the equations do not have a Green’s function formulation and the evolution leads to a boundary layer. In this talk we discuss two phase flow with soluble surfactant and how we plan to continue developing a hybrid method boundary integral to further resolve the boundary layer. The approach is based on methods previously implemented in Xu, Siegel and Booty (2013). This includes a Green’s function formulation for the advection-diffusion equation for the concentration of bulk soluble surfactant in the boundary layer. We introduce a fast method for computing a time convolution integral that arises in the Green’s function formulation of the advection-diffusion equation, based on a method originally introduced by Johannes Tausch. Lastly, results illustrating the speed and accuracy of the fast method will be presented.

Yuexin Liu

Low-Reynolds-Number Locomotion via Reinforcement Learning

The development of artificial microswimmers has sparked a huge potential for future applications in biomedical applications such as microsurgery and targeted drug delivery. However, locomotion at the microscopic scale encounters stringent constraints due to the absence of inertia. In this talk, we first apply a recent reinforcement learning-based framework that allows us to generate net mechanical rotation at low Reynolds numbers. Without prior knowledge of locomotion, the system develops effective policies based on its interactions with the surrounding environment. We compare and analyze those effective policies with traditional strategies. We will then discuss how reinforcement learning can be leveraged to enable a model microswimmer to self-learn effective locomotory gaits for translation, rotation and combined motions. We show that the strategy advised by policy gradientbased deep reinforcement learning techniques is robust to flow perturbations and versatile in enabling the swimmer to perform complex tasks such as path tracing without being explicitly programmed. Lastly, we illustrate how the deep reinforcement learning method can be conveniently adapted to a broader class of problems such as a microswimmer in a non-stationary environment. Results from our studies highlight a powerful alternative to current traditional methods for applications in unpredictable, complex fluid environments and open a route towards future designs of ”smart” microswimmers with trainable artificial intelligence.

Soheil Saghafi

Stochastic Inverse Problem with Markov Chain Monte Carlo Version of the Genetic Algorithm Differential Evolution

Differential Evolution (DE) is a stochastic global optimization technique that uses a simple genetic algorithm for numerical optimization in real parameter spaces. In the statistical context, the uncertainty distribution of the optimal value can be obtained using Markov Chain Monte Carlo (MCMC) simulation, a method from Bayesian inference. In this talk, I will mainly focus on an MCMC method known as the Metropolis-Hastings algorithm and go through some of the specifics of its implementation and calibration using examples animated in MATLAB. By introducing the Differential Evolution-Markov Chain (DE-MC) method (C.J.F Ter Braak (2006) paper), which is the integration of the essential ideas of DE and MCMC, we can see how this method is superior to the other MCMC approaches in some cases involving nonlinearity, high-dimensionality, and multimodality. In the end, I will discuss applying this method for doing parameter estimation using the Rosenbrock function as a test case.

Philip Zaleski

Variational Models of Charged Drops: The Effect of Charge Discreteness

We will discuss the stability of electrically charged drops. The classical model for charged droplets was first developed in 1882 by Lord Rayleigh, who performed linear stability analysis on a charged spherical droplet and found a critical charge to mass ratio, above which a spherical droplet would become unstable due to "linear" perturbations. This critical charge to mass ratio is known as the Rayleigh limit and was later experimentally verified in 1971. However, it was recently shown that a spherical droplet is never locally stable with respect to smooth perturbations, as energy can always be decreased by smooth nonlinear perturbations, which were not accounted for in Rayleigh's original work. Because of this, the classical model for charged droplets is mathematically ill-posed, and a regularization of the current model is needed. We analyze a new model that considers the discreteness of charges, which is a naturally appearing regularizing phenomenon. We classify existence and non-existence regimes for this new model. 

Diego Rios

Geoacoustic Inversion and Sediment Classification using Sound Waves

We investigate the inverse problem of determining seabed properties using sound waves propagating in the ocean. Extracting the seabed parameters, called geoacoustic inversion, is performed in three steps: estimating arrival times of the sound waves using a Monte Carlo method, developing a ray tracing algorithm for sound propagation, and estimating the seabed parameters combining the two previous tasks. We discuss the directions that need to be taken regarding the ray tracing algorithm and arrival time estimation.  Another approach to the inverse problem is also sought: sediment classification using decision trees. Decision trees are a type of supervised machine learning that is useful and efficient in classification problems. We will discuss what the design of decision trees entails for our problem and the steps need to be taken next.

Jake Brusca

Numerical methods for the Monge-Ampère equation and Optimal Transport

This talk will discuss numerical methods for solving Optimal Transport (OT) problems via finite difference methods for the related Monge-Ampère equation. Given some cost of transportation, OT is a geometric optimization problem seeking to move mass from one distribution to another in the most efficient way. In the case of a quadratic cost, it is well known that this optimization problem can be recast as a fully nonlinear PDE of Monge-Ampère type. There has been much work in recent years in developing a convergence framework for these PDEs, however they often rely on low accuracy wide stencil discretizations that can be prohibitively expensive just to evaluate, especially in higher dimensions. We introduce an integral representation of the local Monge-Ampère operator and show how numerical quadrature can be used to create convergent discretizations. The resulting schemes offer higher orders of accuracy and allow for narrower stencil widths, while maintaining efficient scheme evaluations. We will conclude by examining the results of some numerical experiments which demonstrate the improved accuracy and efficiency. 

Professor Amit Bose

Beyond the Limits of Circadian Entrainment: Non-24-h Sleep-wake Disorder, Shift Work, and Social Jet Lag

While the vast majority of humans are able to entrain their circadian rhythm to the 24-hour light-dark cycle, there are numerous individuals who are not able to do so due to disease or societal reasons. We analyze a well-established model of human circadian rhythms to address cases where individuals do not entrain to the 24-h light-dark cycle, leading to misalignment of their circadian phase. We use computational and mathematical methods based on the entrainment map to derive strategies to minimize circadian misalignment. The entrainment map tracks the phase of light onset during each 24-hour cycle. We will discuss in detail the derivation of the map before showing how it is applied to cases of non-entrainment.

Joseph D'Addesa

Phase Separation of Two-Fluid Mixtures using Surface Acoustic Waves

In this talk we will present some theoretical and computational aspects of flow of thin fluid films. The theoretical part involves basic fluid mechanics and presents a brief derivation of the thin film equation including gravity and acoustics using the long wave approximation. A simplified version of this equation is then analyzed numerically using the software Comsol Multiphysics and compared to experimental results. The mechanism which makes such separation possible is based on the application of surface acoustic waves (SAWs), which propagate in the solid substrate in contact with oil-water-surfactant mixtures. Reaching the proposed goals will lead to establishment of proof-of-principle for use of SAW for the purpose of phase separation.

Connor Greene

Function Extension Methods for Solving PDEs

We describe a method for solving PDEs with a body force on irregular domains. By smoothly extending the body force term over a bounding box one can compute a particular solution using simple fast algorithms. This talk will focus on methods for smoothly extending functions beyond the original, irregular domain. Moreover, I will discuss difficulties that arise when the domain does not have a smooth boundary surface, and methods for overcoming these.

Mark Fasano

Modeling Phase Separation in Oil-Water Mixtures Atop a Substrate Vibration

In this talk, we will present a Cahn-Hilliard-like model to describe the phase separation in oil-water mixtures due to a surface acoustic wave (SAW) propagating beneath. We begin by examining some experimental work on the behavior of oil and deionized water due to such an excitation. After differentiating between the responses of oil and water due to the SAW, we will describe a simplified model to capture the behavior of a mixture of the two fluids. Using the standard Cahn-Hilliard equation with a horizontal advective term, we utilize thin-film theory to make considerable simplifications. Finally, numerical simulations will be presented along with error estimates and convergence results before ending with a brief description of future work. 

Binan Gu

Stochastic Modeling of Flows in Membrane Pore Networks

An overview of the mathematical modeling of membrane filtration will be introduced. Three complete problems using a graphical representation of membrane networks are studied. First, a graphical representation is set up with a novel network generation protocol, while governing equations on the network (modeling filtration) are formulated with a tractable discrete calculus framework. Then the influence of pore-size variations on the performance of pore networks is investigated. Lastly, inter-layer pore-size variations manifested via a pore-size gradient are studied. If time allows, two more open problems that utilize the graphical structure will be discussed. 

Rituparna Basak

Application of Computational Topology to Analysis of Granular Material Force Networks in the Stick-Slip Regime.

We discuss the properties of force networks in a two-dimensional annular Couette geometry for both experiments and simulations in which a small intruder is pulled by a spring. In particular, the connection between topological features of the force network and fluctuations of the intruder velocity is studied in the stick-slip regime. The force networks are analyzed using persistent homology methods, focusing on the statistics of clusters and loops composed of particles experiencing strong forces. We find that the networks evolve in a nontrivial manner as the system approaches a slip event. The presentation will discuss this evolution for systems of disks and pentagonal particles.

Prianka Bose

Rhythm Detection and Generation of New Tabla Composition using RNN

In this talk, we will discuss a method that we provide for rhythm detection and generation of new tabla composition using Recurrent Neural Network (RNN) . We will focus on North Indian Classical Music. The onsets of an audio signal provides important information on the rhythm and the beat of a sound file. We propose to develop an algorithm using onset detection to classify isochronous rhythms even in the presence of perturbations. In addition, using clean tabla audio signals, we have performed onset detection and beat tracking using an extension of librosa function library in Python. We seek to develop a novel approach to generate fresh Tabla compositions employing and using  RNN and use our algorithm for classification of the type of taal.

Nicholas Harty

Multiple Scattering in Chaff Clouds

Given a random distribution of chaff (finite-sized wires with lengths much bigger than their diameter), one wants to know how an incident electromagnetic field is scattered by the chaff cloud. In this talk, we will present an overview of techniques for multiple scattering by circular objects in two dimensions. We will start by presenting the scattering formalism for the Helmholtz equation for three types of boundary conditions imposed on the individual objects. We then describe the numerical techniques used to form an approximate solution to the scattering problem, comparing a naive technique with a more sophisticated one. Lastly, we present numerical experimental results comparing these two techniques for a model problem.

Zhiwen Wang

Something About Spatial Analysis

The process of examining the locations, attributes, and relationships of features in spatial data through overlay and other analytical techniques in order to address a question or gain useful knowledge. Spatial analysis extracts or creates new information from spatial data.

Spatial Analysis skills have many uses ranging from emergency management and other city services, business location and retail analysis, transportation modeling, crime and disease mapping, and natural resource management. 

This talk is a brief intro of spatial analysis and something about the homogeneous poisson process.

Michael Luo

Modeling Action Potentials in Diurnal Rodent Species

Suprachiasmatic nuclei (SCN) located in the hypothalamus are responsible for controlling circadian rhythms. The neuronal and hormonal activities generated by the SCN regulate many different bodily functions on a 24-hour cycle. Unfortunately, much of our knowledge of the SCN comes from a small subset of nocturnal rodents. Therefore, the degree to which this knowledge extends to diurnal (day-active) animals is unclear. Previous studies recorded the electrical activity of single SCN neurons in the diurnal rodent species, Rhabdomys pumilio. Their studies revealed that R. pumilio SCN neurons had a prominent suppressive response that is not present in the SCN of nocturnal rodents. However, at the end of their study, some of the model parameters remained unidentified. To remedy this, we want to approach this problem both computationally (using deep learning methods) and experimentally (using dynamic clamp). 

Zheng Zhang

Information-Theoretic Limits for Testing Community Structure of Bipartite Network

Abstract linked here.

Jose Pabon

On the Study of the Role of Hydrodynamic Interactions in the Collective Motion of Swarms of Active Matter

Our brief exposition will involve the mathematical modeling of the collective motion of swarms of active matter, i.e.‘swimmers’ or ‘flapping wings’. Our projects seek to elucidate the role of hydrodynamic interactions on such motion. Our models will study scenarios typically dealing with collections of swimmers, such as fish propelling themselves forward with flapping motion. We consider under what conditions and parameters would the hydrodynamic interactions by themselves lead to stable swimming configurations with an optimized swimming speed. We will discuss some of our models, simulations and exciting preliminary results, as well as our direction for our ongoing efforts.

Nastaran Rezaei

Interfacial Instability of Two-Layer Flows

In this talk, we will present an overview of two-layer flows and viscoelastic liquids; they have been classical topics for decades because of their industrial applications like material producing with desirable characteristics. The main issue that the industry tries to avoid is interfacial instability between layers. The Volume of Fluid method has been used to track the movement of the interface and investigate the wave evolution between layers. The important factor that stabilizes or destabilizes the interface in Newtonian fluids is viscosity. However, in viscoelastic liquids, stability is impacted by both fluid elasticity and viscosity.