2020 Faculty and Student Summer Talks | Department of Mathematical Sciences

Meeting number (access code): 925 459 487

Meeting password: 68VBq84UJhF

Meeting Link : https://njit.webex.com/njit/j.php?MTID=m52fd905ec633bb19220408ae482d22db

Date	Day	Speaker, Title, and Abstract
June 2	T	Ryan Allaire Thermal Effects in Nanoscale Thin Liquids Heated by a Laser with Applications to Liquid Metal Assembly Thin film dynamics, particularly on the nanoscale, is a topic of extensive interest. The process by which thin liquids evolve is far from trivial and can lead to dewetting and drop formation. Not only does it involve resolving fluid mechanical aspects of the problem, but also requires the coupling of other physical processes, including liquid-solid interactions and dependence of material parameters on temperature and material composition. The research we describe here focuses both on (a) single and (b) alloy metal assembly. In (a) we focus on continuum theory and formulate a leading order “asymptotic model” to simulate heat conduction in thin liquid metals melted with a pulsed laser. The experimental setup described here is performed at University of Tennessee Knoxville (UTK) and Oak Ridge National Laboratory (ORNL). We validate our model by comparing to both (i) full heat conduction, and (ii) 1D heat conduction, and show that our model agrees well with (i) compared to (ii), with a significant numerical speedup. The model is then used to simulate an experimental setup, by allowing variability in the material parameters and is shown to be a good model for describing film evolution in the case of a thin substrate. In (b) we focus on the assembly mechanism of NiAg alloys by using molecular dynamics (MD) simulations. We create phase-separated rivulets with wavelengths corresponding to Rayleigh-Plateau (RP) type of instabilities. The (dewetting) process by which these lines evolve into droplets is explored and compared to traditional RP behavior. Tadanaga Takahashi Decomposition for a Wave Scattering Problem Multi-medium EM wave scattering poses numerical challenges due to the duality of the unbounded domain and the non-uniform bounded domains. Domain decomposition addresses the problems with the ill-posed system, spurious solutions, and poor scaling with wave frequency. The formulation and discretization of the scattering problem is discussed.
June 4	R	Professor David Shirokoff Stability and Numerics in Differential Equations In this talk, I will survey several ongoing projects advancing the stability and accuracy in numerical differential equations. The first part of the talk will present a new unconditional stability theory for implicit-explicit (IMEX) time integration methods. IMEX methods have become a popular choice for integrating large scale PDEs as they allow for some terms to be treated implicitly (to ensure stability) while leaving other terms explicitly (for efficiency). The theory characterizes the loss of stability in high order methods and can be used to overcome stability limitations in existing methods. The second part will outline a new theory (weak stage order) to overcome order reduction in Runge-Kutta (RK) schemes. Order reduction currently limits the use of high order RK schemes in scientific computing problems. The third part of the talk will present results in applying polynomial optimization to compute (first order) phase transitions in molecular dynamics. First order phase transitions are difficult to characterize as they arise as instabilities (similar to a global bifurcation) in stochastic differential equations.
June 9	T	Professor Enkeleida Lushi Micro-swimmers Moving in Complex Confinement Interactions between micro-swimmers and solid boundaries play an important role in many biological and technological processes. I will discuss recent results in experiments and simulations that aim to understand the motion of micro-swimmers such as bacteria, micro-algae, spermatozoa or active colloids in various confinements or structured environments. Our results highlight the complex interplay of the fluidic and contact interactions of the individuals with each-other and the boundaries to give rise to complex individual and collective behavior. Lauren Barnes Image Analysis 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, providing plentiful images showing the formation and evolution of crystal structures in time. By analyzing these images in such a way as to characterize the size of crystalline domains, we seek to obtain a quantitative description of the time evolution of the colloidal crystal growth process. Emel Khan Dynamics of a Cyanobacterial Circadian Clock Model Circadian rhythms are daily oscillations that occur in a variety of living organisms including animals, plants, fungi, and cyanobacteria. Our focus is on circadian rhythms in cyanobacteria, whose core clock is comprised of just three proteins, KaiA, KaiB, and KaiC. When these proteins are mixed with ATP in a test tube, KaiC phosphorylates and dephosphorylates in an oscillatory manner with a period near 24 hours. Rust et al. have developed an ODE model of this in vitro oscillator, however some aspects of the model’s behavior do not match experimental data. Specifically, when the concentration of the KaiA protein is increased, the period of circadian oscillations in the Rust model increases, while experimentally it has been observed that increasing KaiA shortens the period of oscillations. We address this issue by developing a lower-dimensional version of the Rust model and then using dynamical system tools such as phase plane analysis and geometric singular perturbation theory to identify which components of the model are responsible for the discrepancy between the model’s behavior and experiments.
June 11	R	Professor Cyrill Muratov The Mathematics of Charged Liquid Drops In this talk, I will present an overview of recent analytical developments in the studies of equilibrium configurations of liquid drops in the presence of repulsive Coulombic forces. Due to the fundamental nature of Coulombic interaction, these problems arise in systems of very different physical nature and on vastly different scales: from femtometer scale of a single atomic nucleus to micrometer scale of droplets in electrosprays to kilometer scale of neutron stars. Mathematically, these problems all share a common feature that the equilibrium shape of a charged drop is determined by an interplay of the cohesive action of surface tension and the repulsive effect of long-range forces that favor drop fragmentation. More generally, these problems present a prime example of problems of energy driven pattern formation via a competition of long-range attraction and long-range repulsion. In the talk, I will focus on two classical models - Gamow's liquid drop model of an atomic nucleus and Rayleigh's model of perfectly conducting liquid drops. Surprisingly, despite a very similar physical background these two models exhibit drastically different mathematical properties. I will discuss the basic questions of existence vs. non-existence, as well as some qualitative properties of global energy minimizers in these models, and present the current state of the art for this class of geometric problems of calculus of variations. Axel Turnquist Elliptic PDE and Some Regularity Questions We desire to solve the Monge-Amp\'{e}re, a fully nonlinear second-order elliptic PDE, on the unit sphere using numerical schemes. In order to prove a convergence of the scheme, it is important to understand the function space we expect the solution to be in: this is the realm of regularity theory of elliptic PDE. In order to understand the regularity theory for our PDE, it requires a fundamental understanding of how nonlinearity, the degree of the PDE, the property of uniform ellipticity, and the geometry of the sphere all affect the regularity of the solution. This leads us to an examination as to what it means fundamentally to be an elliptic PDE, and why the conditions we commonly see in the literature are so important conceptually. This also leads us to examine some more technical cases that extend beyond the usual classical literature, and their ramifications.
June 16	T	Professor Linda Cummings Wetting and Dewetting of Nanoscale Nematic Liquid Crystal Films The evolution of ultra-thin films (tens of nm) of nematic liquid crystals (NLCs) is considered. Such free-surface films can undergo complex dewetting behavior, as observed in experiments. We present a simplified thin-film model for the free surface evolution that includes strong spatially-varying planar anchoring at the substrate, and weak antagonistic anchoring at the free surface. A number of large-scale simulations are presented, showing good qualitative agreement with experiments. Ongoing work including the effect of spatially-varying electric fields on film evolution is briefly highlighted. Rituparna Basak Application of Machine Learning Techniques for the Stick Slip Dynamics of a Particulate Media The stick-slip transition of granular systems is related to earthquakes and avalanches, and therefore understanding the conditions leading to slip events is of general importance. We studied different machine learning techniques to the analysis of topological data such as force network and persistence diagrams evolving from discrete element simulations of granular systems to understand the stick-slip behavior. We will discuss the potential of machine learning-based methods to predict slips in the considered system.
June 18	R	Chao Cheng The Force Network Precursors to Slip Events in Sheared Granular Systems The stick-slip transition of granular systems is related to earthquakes and avalanches, and therefore understanding the conditions leading to slip events is of general importance. Although stick-slip behavior has been studied extensively, what triggers a slip event still remains unclear. The purpose of our study is to explore the existence of precursors to slip events. For this purpose, we study a sheared system in a stick-slip regime via two-dimensional discrete element simulations. Particular focus is on the evolution of force networks before and during slip events. We apply the persistence diagrams and other classic measures to the force networks' evolution of slip events. Persistence homology to granular systems can reveal the changes in the topological structure of the force networks, which can be shown that relate the macro behavior of the system. We will show that some features of force network evolution could be used to gain insight into the occurrence of a slip event. Guangyuan Liao Model Reduction Techniques for Coupled Circadian Oscillators The circadian rhythm refers to an internal body process that regulates many body processes including the sleep-wake cycle, digestion and hormone release. The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of the most important properties. There are several scenarios in which circadian oscillators do not directly receive light-dark forcing. Instead they are part of hierarchical systems in which, as “peripheral” oscillators, they are periodically forced by other “central” circadian oscillators that do directly receive light input. Such dynamics are modeled as hierarchical coupled limit cycle systems. Those models usually have a large population, and are non-autonomous. Direct simulations usually are incapable of understanding the full dynamics of such models. One topic of this talk is to apply proper mathematical methods on simplification of the original systems. A phase reduction method is applied for reducing the original system to phase model. A parameterization method is introduced for simplifying such systems, and it is also applied for computing invariant manifolds of some biological oscillators. Another topic of this talk is to develop new tools. A novel tool, entrainment map, is developed. Compared with direct simulations, the map has great advantages of describing the conditions for existence and stability of the limit-cycle solutions, studying the bifurcations on forcing strength and coupling strength. It is also more practical to calculate the entrainment times by just iterating the map than direct simulations. Diego Rios Sound Propagation Modeling for Geoacoustic Inversion A sound source is transmitting acoustic signals in an oceanic environment that are received at vertically separated hydrophones in the water column. Using ray theory, we focus on four travel paths the emitted signal travels by and generate replica arrival times, which will allow for the inverse problem to be solved. This problem is simply the estimation of parameters that affect sound propagation by employing mathematical propagation modeling and received data. The generated arrival times will eventually be compared with arrival time estimates obtained from recorded time-series, allowing us to estimate properties of the seabed such as thickness and sound speed. Challenges in the estimation of the arriving times of the travel paths (especially of the one that propagates through the sediment) will be discussed.
June 23	T	Connor Robertson Discovering the Governing PDE of an Active Nematic System from Video Data Recent experimental advances have uncovered complex systems that have challenged traditional modeling techniques. In parallel, various methods have been created to extract the governing equations of a system directly from experimental data. In this talk, I will discuss one such data-driven method called "PDE-Find," and will use it to model assemblies of microtubules and motor proteins at a fluid interface. Various PDE models based on physical principles have been proposed for this system, but none fully capture its dynamics. First, I will describe the accuracy and robustness of PDE-Find for the simplified task of reconstructing one of these models from simulation data with artificially added noise. I will then discuss future steps for producing a model directly from experimental data. Soheil Saghafi Entrainment of Periodically Forced FitzHugh Nagumo Model Entrainment is typically defined as the synchronization of a self-sustained oscillator to external periodic forcing. Thisphenomenon is ubiquitous in biological systems. The concept of entrainment is less well-defined when the unforced system does not exhibit self-sustained oscillations. In the context of neuroscience, neurons exhibit a variety of intrinsic dynamics including limit cycleoscillations, damped oscillations, and non-oscillatory depolarized or hyperpolarized steady states. We would like to compare and contrast the entrainment properties of neurons in these distinct regimes. To do this, we study aperiodically forced piecewise-linear model of FitzHugh-Nagumo type, which is a two-dimensional simplification of the Hodgkin-Huxley model of neuronal action potential generation.
June 25	R	Gan Luan Parameter Estimation and Inference of Spatial Autoregressive Model by Stochastic Gradient Descent Many data contain spatial components, and it is important to consider spatial correlation in modeling and parameter estimation. Spatial autoregressive (SAR) model is often used to modeling these data. Parameters of SAR model are mainly estimated by maximum likelihood method (based on profile likelihood). However, no closed form of MLE exits, and it cannot scale up well due to heavy computation involved in numerical methods used for parameter estimation. What’s more, since profile likelihood is used, estimators are often biased. Stochastic gradient descent (SGD) is a desirable method for model parameter estimation in large-scale data and online learning settings, since it goes through the data in only one pass. Although many studies regarding SGD have been conducted, application of SGD for spatial models is still not common. In this talk, I consider spatial lattice data and use averaged SGD for model parameter estimation. Genuine likelihood rather than profile likelihood is used. Also, a bootstrap procedure is used to conduct inferences based on SGD estimator. This inference procedure updates SGD estimates, and at the same time generates many randomly perturbed SGD estimates for each observation. These perturbed estimates can be used to produce confidence intervals. I will present results of simulation studies and the asymptotic properties of these procedures.
June 30	T	Professor Brittany Hamfeldt Convergent Numerical Methods for Optimal Transport The problem of optimal transportation, which involves finding the most cost-efficient mapping between two measures, arises in many different applications. However, the numerical solution of this problem remains extremely challenging and standard techniques can fail to compute the correct solution. Recently, several methods have been proposed that obtain the solution by solving the Monge-Ampere equation, a fully nonlinear elliptic partial differential equation (PDE), coupled to a non-standard implicit boundary condition. Unfortunately, standard techniques for analyzing numerical methods for fully nonlinear elliptic equations fail in this setting. We introduce a modified PDE that couples the usual Monge-Ampere equation to a Hamilton-Jacobi equation that restricts the transportation of mass. This leads to a simple framework for guaranteeing that a numerical method will converge to the true solution, which applies to a large class of approximation schemes. We describe some simple examples. A range of challenging computational examples demonstrate the effectiveness of this method, including the recent application of these methods to problems in beam shaping and seismic inversion. Yinbo Chen Closed-Form Approximations of Single-Channel Calcium Nanodomains in the Presence of Cooperative Calcium Buffers Calcium ion (Ca2+) elevations produced in the vicinity of single open Ca2+ channels are termed Ca2+ nanodomains and play an important role in triggering secretory vesicle exocytosis, myocyte contraction, and other fundamental physiological processes. Ca2+ nanodomains are shaped by the interplay between Ca2+ influx, Ca2+ diffusion and its binding to Ca2+ buffers, which absorb most of the Ca2+ entering the cell during a depolarization event. In qualitative studies of local Ca2+ signaling, the dependence of Ca2+ concentration on the distance from the Ca2+ channel source can be approximated with a reasonable accuracy by analytic approximations of quasi-stationary solutions of the corresponding reaction-diffusion equations. Such closed-form approximations help to reveal the qualitative dependence of nanodomain characteristics on Ca2+ buffering and diffusion parameters, without resorting to computationally expensive numerical simulations. Although a range of nanodomain approximations had been developed for the case of Ca2+ buffers with a single Ca2+ binding site, for example the Rapid Buffer Approximation, the Excess Buffer Approximation, and the Linear approximation, most biological buffers have more complex Ca2+-binding stoichiometry. Further, several important Ca2+ buffers and sensors such as calretinin and calmodulin consist of distinct EF-hand domains, each possessing two Ca2+ binding sites exhibiting significant cooperativity in binding, whereby the affinity of the second Ca2+ binding reaction is much higher compared to the first binding reaction. To date, only the Rapid Buffer Approximation (RBA) has been generalized to Ca2+ buffers with two binding sties. However, the performance of RBA in the presence of cooperative Ca2+ buffers is limited by the complex interplay between the condition of slow diffusion implied by the RBA, and the slow rate of the first Ca2+ binding reaction characterizing cooperative Ca2+ binding. To resolve this problem, we present modified versions of several Ansatzes recently introduced for the case of simple buffers, extending them to the case of Ca2+ buffers with 2-to-1 stoichiometry. These new approximants interpolate between the short-range and long-range distance-dependence of Ca2+ nanodomain concentration using a combination of rational and exponential functions. We examine in detail the parameter-dependence of the approximation accuracy and show that this method is superior to RBA for a wide ranges of buffering parameter values. In particular, the new approximants accurately estimate the distance-dependence of Ca2+ concentration in the case of calretinin or calmodulin.
July 2	R	Zhoncheng Lin Some Properties of Bivariate Archimedean Copula Models Under Left Censoring Many models have been proposed to model multivariate survival data, and Archimedean copula models have been more popular choices. In this talk, we will assume that two failure time variables can be modelled by Archimedean copula model and they are subject to dependent or independent left censoring respectively. Some distributional results for their joint cdf under different censoring patterns will be presented. Those results are expected to be useful in both model fitting and checking procedures for Archimedean copula models with bivariate left-censored data. As an application of the theoretical results we obtained, a moment estimator of the dependence parameter in Archimedean copula models will be proposed as well, and some simulation studies have been performed to demonstrate our parameter estimation method. Linwan Feng Numerical Methods for the Dispersive Shallow Water Equations This talk focuses on developing efficient and stable [high order] time-stepping strategies for the dispersive shallow water equations (DSWE)with different bathematry. The DSWE extends the regular shallow water equations to include the dispersive effect which can maintain the shape of the wave when solving the system. With high accuracy pseudo-spectral method to discretize the space, the dispersive effect limits the choice of the time-stepping method. Different bathematry also make the dispersive shallow water equations appear differently. By discussing the stiffness of the bathematry, there are several types of equations which can be obtained with corresponding bathematry assumptions. And all of them can be solved with the chosen time-stepping methods. In this talk, the system are treated as differential-algebraic equations. Then two strategies can be applied to solve the algebraic constrains: (i) preconditioning iterative methods to invert the semi-linear operator which contains the dispersive effect; (ii) semi-implicit time stepping (ImEx) methods that bypass a full inversion of the operator (and do not require sub-iterations). Guaranteeing stability for the semi-implicit approach is a nontrivial issue due to the fact that certain stiff terms in the equations are treated explicitly. A stability theory is provided which outlines how to choose the semi-implicit terms in such a way to guarantee numerical stability.
July 7	T	Professor Shahriar Afkhami Numerical Algorithms for Multi-Scale Viscoelastic Flows In everyday life, we encounter materials that show viscoelastic behavior; materials that exhibit both liquid and solid like behavior. Macro-molecules, such as synthetic polymers, and almost all biological tissues, such as cells, structural proteins, and skin tissue, for example, show viscoelasticity. Many models (constitutive laws) exist that describe the viscoelastic behavior; they generally represent the collective dynamical response of polymer micro-structures to macroscopic deformation. In this talk, I will present a multi-scale algorithm, by means of the coupling of the continuum formulation on the macroscopic level, without a particular constitutive law associated with the viscoelastic fluid, with the mesoscopic computations based on the microstructure dynamics of polymer molecules. This multi-scale approach poses significantly more challenging computational issues, since tracking individual molecules is computationally costly and requires solving a stochastic differential equation. I will propose a multi-resolution adaptive mesh refinement to reduces the number of individual molecules on the spatial grid. I will also propose to adapt the stochastic grid when solving the stochastic differential equation to adaptively refine the number of stochastic samplings required for a specified accuracy. Finally, I will discuss a Machine Learning approach to speed up the computations: the approach is based on using high-fidelity data to compute a set of initial training data, and then using a regression process to predict the constitutive function. The computed viscoelastic stresses from the previous step are then compared with the stresses computed from the high-fidelity data, to obtain the next optimal sampling data. Ziyan Guo Motivation & Brief Report on Projects of Competing Risks In multivariate analysis, it is often a very difficult problem to model nonnormal multivariate data. “Copulas” are multivariate probability distributions which are used to describe the dependence between random variables. Many different ways to construct nonsymmetric Archimedean copulas were introduced. However, most of them are not very flexible. We aim to construct general models allowing arbitrary selection of pairwise correlation which is desired in our practical applications. We also derive the copula-graphic estimator (Zheng and Klein 1996) for marginal survival functions using Archimedean copula models based on competing risks data subject to univariate right censoring and prove its uniform consistency and asymptotic properties. We then propose a parameter estimation strategy to analyze the semi-competing risks data using Archimedean copula models. Our method is flexible in that it allows us to determine dependence levels between competing risks when two dependent competing risks are subject to independent censoring. Based on our estimation strategy, we propose a new model selection procedure. We also describe an easy way to accommodate possible covariates in data analysis using our strategies. Simulation studies have shown that our parameter estimate outperforms the estimator proposed by Lakhal, Rivest and Abdous (2008) for the Hougaard model and the model selection procedure works quite well. Yasser Almoteri Bacterial Motion and Spread in Porous Media I will discuss a continuum model that describes the collective dynamics of micro-swimmers such as bacteria through a porous wet material. The motion of the swimmer suspension is coupled to the fluid dynamics that is modeled through a Stokes-Brinkman equation with an added active stress. The linear stability of the uniform isotropic state reveals that the suspension transitions from a long-wave instability to a mid-range one where the so-called bacterial “turbulence” is weakened. Simulations of the full nonlinear system confirm the analytical results. I will discuss the dynamics of a bacterial suspension through a structured surface. Last, I will talk about the chemotactic motion of a micro-swimmer suspension in a porous material and discuss similarities and differences with experimental results of E Coli spread in gel packings.
July 9	R	Yixuan Sun Membrane Filtration with Multiple Species of Particles Membrane filtration is widely used in many applications, ranging from industrial processes to everyday living activities. Fouling is an unavoidable part of filtration and understanding the particle fouling mechanism is critical for improving the filtration performance and avoiding filtration failure, hence this is a topic of much ongoing research. Experimental studies can be very valuable, but are expensive and time-consuming, therefore theoretical studies offer potential as a cheap and predictive way to improve on current filter designs. The majority of theoretical and experimental research focuses on filtration of suspensions that consist of chemically homogeneous particles. In this work we propose a model for filtration of a suspension containing an arbitrary number of particle species, each with different affinities for the filter membrane. We present preliminary results showing how the presence of additional species can change filtration outcomes. In addition, a model for screening (shielding) effect is proposed. Ruqi Pei A New Panel-based scheme for the Discretization of Boundary Integral Equations Boundary integral equations and Nystr\"om discretization methods provide a powerful tool for computing the solution of Laplace and Helmholtz boundary value problems (BVP). Using the fundamental solution (free-space Green's function) for these equations we can convert such problems into boundary integral equations, thereby reducing the dimension of the problem. The resulting geometric simplicity and reduced dimensionality allow for high-order accurate numerical solutions with greater efficiency than standard finite-difference or finite-element discretizations. One such approach is standard panel-based Generalized Gaussian Quadrature rule (GGQ) which takes a few nodes in neighboring panels when we integrate over singularities in the kernel function. Recently, we developed a new panel-based scheme by getting rid of the near-neighbor panels that are more efficient than GGQ under complicated geometry and high-frequency values.
July 14	T	Atefeh Javidialsaadi Model Checks for Two-Sample Location-Scale Two-sample location-scale refers to a model that permits a pair of standardized random variables to have a common distribution. Function-based hypothesis testing in these models refers to formal tests that would help decide whether or not two samples may have come from some location-scale family of distributions. For uncensored and censored data, a comparison between two approaches of testing, one based on empirical characteristic functions (ECFs) and another on plug-in empirical likelihood (PEL), is carried out. Sample means and standard deviations are used as plug-ins for both approaches. Results of numerical studies are reported. Beibei Li On Weighted Holm Procedures In many statistical applications, such as clinical studies and a genome-wide association study (GWAS), it often happens that some hypotheses are more important than the others, which suggests us to assign different weights to hypotheses according to their different importance. Recently, many weighted procedures have been developed for controlling the familywise error rate (FWER) when conducting multiple hypotheses testing. Among these procedures, weighted Holm procedures are the most popular and easy to be applied without any assumption of dependence structure. There are two common weighted Holm procedures. One is based on ordered weighted p-values that we called WHP; the alternative weighted Holm procedure that is based on ordered original p-values is named WAP. The objective of this project is to study these two weighted Holm procedures and make recommendations for their use. In this talk, by constructing and comparing their corresponding closed testing procedures, graphical approaches, and adjusted p-values, we show that WHP is more powerful than WAP. Also, we provide a theoretical result that WHP is an optimal procedure in the sense that the procedure cannot be improved by increasing even one of its critical values without losing control over the FWER. Simulations were conducted to provide numerical evidence of superior performance of WHP in terms of the FWER controlling and average power. Yuexin Liu How to Train Stokes Swimmer Using Machine Learning Machine learning has been applied to an increasing range of physics and engineering, and recently it has been applied in zero-Reynolds number flow for flow control and designing of swimmers. Artificial micro-swimmers in engineering and medical applications such as drug delivery and cell manipulation show great success, however, hydrodynamic interactions in low Reynolds number environments and the uncontrolled environmental factors will also influence the swimmer’s behavior. Here, we present a reinforcement learning paradigm to design a new set of self-learning, adaptive linear N-swimmer and N-sphere rotator (N=3, 4) in a viscous Stokes fluid. Different from the typical designed autonomous rotators, we do not prescribe any propulsion strategy but allow the rotator to self-learn its own rotating gaits based on its interactions with the surrounding environment through reinforcement learning. We show the ability of the N-sphere rotator to obtain the optimal propulsion policies. Our study illustrates the potential of reinforcement learning in fluid mechanics and provides a new way for designing smart artificial swimmers.
July 16	R	Erli Wind-Anderson Introduction to Convolution Quadrature This talk will explore solving the exterior wave equation in an unbounded domain. The method that will be used to do this is the Convolution Quadrature method (CQ) , which is a fast numerical solution. The benefits of CQ lies in its ability to transform the exterior wave scattering problem into many decoupled Helmholtz Equations, which can then be solved via a boundary integral formulation. The talk will focus on a basic derivation of the CQ scheme along with reductions in the number of Helmholtz equation solves that need to be computed.
July 21	T	Professor Yuan-Nan Young Cellular Dynamics and Fluid-Structure Interactions The first example is the oscillation of a centrosome due to the collective active pulling forces from the cortical shell. This oscillation is related to the positioning of spindle formation during cell division. I will give a quick overview of such oscillation and show how it might be captured using continuum modeling and dynamical system analysis. The second example is the dynamics of a primary cilium, which is a mechanosensor in almost all mammalian cells when they are not dividing. We use slender-body theory to examine the dynamics of a cilium under a fluid flow, and compare against the cilium profiles observed from the experiments. Such comparison shows that we need to incorporate a rotational spring at the cilium base, and we obtain very good agreement of the cilium profiles between experimentally observed profiles and the theoretical prediction. We then used this model to further probe the cilium structures to understand the mechanical properties of microtubule doublets inside the cilium. Kosuke Sugita Boundary Integral Equation Methods on Stokes Flow Equations in Two Dimensions We consider the numerical computation of the interior velocity of incompressible flow governed by the Stokes flow equations with mixed boundary conditions. The problem includes cases that have boundaries with corners. Boundary integral equation (BIE) methods have some promising advantages such as the reduction in the dimensionality of the problem and the capability to deal with the complex geometry on the boundary. On the other hand, the BIE formulation of this problem causes singularities at the intersections of mixed boundary conditions and the geometric corners. Standard quadrature rules for the discretized BIE give ill-conditioned linear systems due to the singularities, which result in severe loss of accuracy. An adaptive mesh refinement to the quadrature is a way of mitigating the situation, however, it is still difficult to obtain satisfactory accuracy. Besides, the size of the matrix in the linear system grows as the refinement level increases. To resolve such issues, we focus on two methods: Recursively compressed inverse preconditioning (RCIP), and the scaling technique. In this talk, we present applications of these methods to the problem and illustrate the performance with numerical experiments.
July 23	R	Professor James MacLaurin Emergent Dynamics in Interacting Particle Models with Random Connections I study the mean-field dynamics of large ensembles of interacting stochastic particles with random connections. The first model I study is that of interacting neurons in the primary visual cortex (in the brain). The connections are sparse, and sampled from a probability distribution with statistics resembling experimental data. The dynamics on the synaptic connections are delayed and stochastic. Upon taking the large size limit, I obtain novel macroscopic neural field equations. The sparseness of the connections leads to a Laplacian in the emergent dynamics. This is then used to explain slow emergent dynamics observed in experimental data: initial `bumps' of highly correlated neural activity slowly damp down through the diffusive action of the Laplacian. The second type of mean-field model I study is that of mean-field `spin glasses'. The connections are all-to-all, static, random and mean zero. The variance of each connection is of the order of the inverse of the system size (which is relatively high). The high variability in the connection strength is know to produce slow `glassy' emergent dynamics. I obtain an autonomous pde for the dynamics of the population density, and use this to investigate the phase transitions that occur as the temperature is lowered. Ryan Atwater Studies of Two-Phase Flow with Soluble Surfactant Surfactants, also known as surface active-agents, change the interfacial properties between fluids by reducing surface tension. This characteristic of surfactants allows for many applications, which include detergents and soaps, the prevention of embolisms, and the stabilization of emulsions. This project will serve as an extension to previous work on the effects that surfactants have on drops stretched by an imposed flow. A “hybrid” numerical method has been developed to study the effects that surfactant solubility has on the dynamics of interfacial flow in the limit of a large bulk Peclet number, Pe. In fluid systems with large Pe, there is a transition layer adjacent to the interface with a rapidly changing surfactant concentration. To accurately resolve the interface dynamics, the hybrid method uses a leading order asymptotic reduction of the governing equations in the limit as Pe tends to infinity. This solution of the hybrid method is then compared to a separate numerical solution of the full governing equations (with asymptotic reduction) for large yet finite Pe, which is referred to as a “traditional” method. Until now, the hybrid method has been implemented to simulate the dynamics of drops and bubbles where the bulk concentration of soluble surfactant is solely located in the exterior fluid. Although this accurately models some fluid systems, situations also arise where there is surfactant in the interior fluid. Hence, the goal of this project is to develop the hybrid and traditional methods for interior flow, which will use complex variable techniques for 2D Stokes flow. Binan Gu Graphical Representation of Membrane Filters We formulate the filtering problem in a random network where vertices and edges of the underlying graph represent pore junctions and throats. We study the adsorptive and sieving fouling mechanisms simultaneously and define each mode more carefully. The main goal is to relate membrane filter properties reflected in spectral graph theory to the performance of the filter. We also provide a formula for network tortuosity in terms of the transition matrix of a sieving particle which we assume to perform a weighted random walk on the network.
July 28	T	Brandon Behring Dances and Escapes of the Vortex Quartet The leapfrogging motion of two pairs of equal strength point vortices has fascinated researchers since its discovery 150 years ago by Grobli and by Love. This talk will focus on the mechanisms by which orbits transition from a neighborhood of this bounded (relative) periodic motion into orbits that either remain bounded but 'dance' or orbits in which the vortices escape to infinity as pairs along two transverse rays. By observing that the dances of vortex quartet are, in many parts of the phase space, close to those of a related integrable three-vortex system, we reframe the problem as a near-integrable two degree-of-freedom Hamiltonian system. By visualizing the geometry of this dynamical system, we can understand the role that phase space structures such as periodic orbits (along with their corresponding invariant manifolds) and KAM tori (and their breakup) play in allowing or forbidding the transition from dance to escape. Nicholas Dubicki Electrostatics Modeling of electrostatic fields. Basic mathematics of potential fields and complex analysis applied to problems of electrostatics. Ref: Landau and Lifshitz, "Electrodynamics of Continuous Media".
July 30	R	Professor Zuofeng Shang Statistical Optimality of Deep Neural Networks in Regression and Classification In this talk, I will discuss statistical optimality of deep neural network methods in regression and classification. In the first part, I will consider linear regression model with instrumental variables that characterize endogenous errors. Deep neural network provides a flexible way to characterize the relationship between the design variable and instrumental variable. Asymptotic distribution and semiparametric efficiency are established for the proposed estimator. In the second part, I will consider nonparametric classification in which classifiers are characterized by deep neural networks. Tight upper and lower bounds for the classification risk are established. The proposed methods enjoy the so-called intrinsic dimension phenomenon. Hewei Zhang Predicting CME and Solar Flares Using Machine Learning Methods The surface of the Sun is a very busy place. When it is active, the sun releases pent-up energy in the form of solar flares or coronal mass ejections (CME). Although, the earth’s magnetic field protects us from these energies. However, if they are very intense, the radiation they release could hit earth communications and could even have devastating and lingering effects on civilization. A key scientific goal is understanding how the interplay between the sun and the earth. In recent years, many scientists have applied machine learning techniques to predict solar flares and CME, which has great significance for the study of solar activities. This talk provides a brief introduction to the recent research findings.
August 4	T	Professor Christina Frederick Some Open Problems in Frame Theory Frame theory is a branch of harmonic analysis that provides the foundation for many mathematical tools used in signal processing, for example, wavelet expansions and sampling theory. Unlike the Fourier transform, time-frequency analysis enables localized representations in terms of elementary functions. In this tutorial, I will provide a background on frames and their applications. I will also describe how numerical methods can be used to shed insights on a few open problems in this field. Jake Brusca Finite Difference Method for The Monge-Ampère Equation The Monge-Ampère equation is a fully nonlinear elliptic partial differential equation that arises in various applications where the notion of a classical solution is no longer suitable, thus we must consider some form of weak solution. A basic introduction to finite difference methods for viscosity (weak) solutions is presented, along with a convergent method for the Elliptic Monge-Amère equation using wide stencil schemes. The finite difference scheme creates a system of nonlinear equations which are approximated using Newton’s method. Finally, we propose a new discretization that utilizes quadrature on a periodic domain. This holds promise for developing more accurate and efficient methods.
August 6	R	Professor Victor Matveev Deterministic vs Stochastic Modeling of First Passage Time to Calcium-Triggered Vesicle Release Like most physiological cell mechanisms, synaptic neurotransmitter vesicle release is characterized by a high degree of variability in all steps of the process, from Ca2+ channel gating to the final triggering of membrane fusion by the SNARE machinery. The associated fluctuations can be quite large since only a small number of Ca2+ ions enter the cell through a single channel during an action potential, and these fluctuations are further increased by the stochasticity in Ca2+ interactions with Ca2+ buffers and sensors. This leads to a widely-held assumption that solving mass-action reaction-diffusion equations for buffered Ca2+ diffusion does not provide sufficient accuracy when modeling Ca2+-dependent cell processes. However, comparative studies show a surprisingly close agreement between deterministic and trial-averaged stochastic simulations of Ca2+ diffusion, buffering and binding. I this talk I will present further analysis and comparison of stochastic and mass-action simulations, focusing on Ca2+ dynamics downstream of Ca2+ channel gating. Namely, we will compare the distributions of first-passage-times (FPT) to full binding of the model Ca2+ sensor for vesicle fusion, when calculated using stochastic and deterministic approaches. I will start with the definition of first-passage time using a highly simplified single compartment model, and then progress to a two-compartment model of Ca2+ dynamics which will allow rigorous comparison of deterministic (mass-action) and stochastic calculation of FPT distribution. I will show that the discrepancy between deterministic and stochastic estimates of FPT density can be surprisingly small even when only a few Ca2+ ions enter the cell per single channel-vesicle complex, despite the fluctuations caused by the Ca2+ binding and unbinding. The reason for the close agreement between the two methods is that the non-linearities in the exocytosis process involve only bi-molecular reactions. In this case the discrepancy between the two approaches is primarily determined by the size of correlations between reactant molecule number fluctuations rather than the fluctuation amplitudes. The small size of reactant correlations is in turn determined by the relationship between the rate of diffusion relative the rate of Ca2+ buffering and binding. Finally, I will show that several open question still remain in fully understanding the relationship between mass-action and stochastic FPT computation. This work is supported by NSF grant DMS-1517085. Jose Pabon Research Work Update - on the Hydrodynamics of Interacting Swimmers, Potential Flow Around Slender Bodies and Related Uniform Asymptotic Solutions. Alternate Title: Three Papers in Three and a Third Minutes Each. Our presentation will be a succinct, brief traversal through the following papers: ‘Lattices of Hydrodynamically Interacting Flapping Swimmers’ by Oza, Ristroph, Shelley, ‘Axially symmetric potential flow around a slender body’ by Handelsman and Keller, and ‘Uniform Asymptotic Solutions for Potential Flow around a thin airfoil and the Electrostatic Potential about a thin conductor’ by Geer and Keller. Given the work is in early stages of literature review, our brisk exposition will highlight some of the aspects of the work we hope to build on. Subhrasish Chakraborty Classification of imbalanced data & SMOTE : An introduction Classification is one of the most researched problems in machine learning, since pose of a classification algorithm, also known as a ‘classifier’, is to identify what class, or category an observation belongs to. In many real-world scenarios, datasets tend to suffer from class imbalance, where the number of observations belonging to one class greatly outnumbers that of the observations belonging to other classes. Class imbalance has been shown to hinder the performance of clas- sifiers, and several techniques have been developed to improve the performance of imbalanced classifiers. In this presentation a very popular re-sampling technique SMOTE (Syn- thetic Over-sampling Technique) has been discussed. This introduces SMOTE as well as discusses the distributional changes caused to the minority class on its application. It also deals with few publications with relevant works in the literature.

Updated: August 5, 2020