# Summer 2016

## Schedule

Date Location Speaker and Title
May 31 CULM 611 Pejman Sanaei, Models for Membrane Filtration

The purpose of this talk is to formulate and investigate new mathematical models for membrane filtration. The work presented is divided into three parts. In the first part, a new mathematical model for flow and fouling in a pleated membrane filter is presented. Pleated membrane filters are widely used in many applications, and offer significantly better surface area to volume ratios than equal area unpleated membrane filters. However, their filtration characteristics are markedly inferior to those of equivalent unpleated membrane filters in dead- end filtration. While several hypotheses have been advanced for this poor performance, one possibility is that the flow field induced by the pleating leads to spatially nonuniform fouling of the filter, which in turn affects performance. In this work we investigate this hypothesis by developing a simplified model for the flow and fouling within a pleated membrane filter. Our model accounts for the pleated membrane geometry (which affects the flow), for porous support layers surrounding the membrane, and for two membrane fouling mechanisms: (i) adsorption of very small particles within membrane pores; and (ii) blocking of entire pores by large particles. We use asymptotic techniques based on the small pleat aspect ratio to solve the model, and we compare solutions to those for the closest-equivalent unpleated filter.

In the second part we propose mathematical models to describe the effects of filter mem- brane morphology on filtration efficiency. A reasonable question that membrane filter man- ufacturers may ask is: what is the optimal configuration of filter membranes, in terms of internal morphology (pore size and shape), to achieve the most efficient filtration? In order to answer this question, we must first propose a robust measure of filtration performance. Filter membrane performance can be measured in a number of different ways. As filtration occurs the membrane becomes blocked, or fouled, by the impurities in the feed solution, and any performance measure must take account of this. For example, one performance measure might be the total throughput – the amount of filtered feed solution – at the end of filtration process, when the membrane is so badly blocked that it is deemed no longer functional. Here we present a simplified mathematical model, which (i) characterizes membrane internal pore structure via pore or permeability profiles in the depth of the membrane; (ii) accounts for various membrane fouling mechanisms (adsorption, blocking and cake formation); and (iii) defines a measure of filter performance; and (iv) predicts the optimum pore or permeability profile for our chosen performance measure.

Finally, in the third part of the talk we discuss the future work that will be required to complete the research.
June 2 CULM 611 Michael Lam, Instabilities in Very Thin Nematic Liquid Crystal Films

The breakup of nematic liquid crystals (NLCs) films with thicknesses less than a micrometer is studied. NLCs are a state of matter intermediate between a fluid and a solid, where molecules have no positional ordering (fluid), but short range elastic interactions induce orientational ordering (crystal). Paying particular attention to the interplay between the bulk elasticity and the anchoring (boundary) conditions at the substrate and free surface, a fourth order nonlinear partial differential equation (PDE) is derived for the free surface height within the framework of the long wave approximation. Numerical simulations of a perturbed flat film show that, depending on the initial average thickness of the film, satellite droplets form and persist on time scales much longer than dewetting. Formulating the model in terms of an structural disjoining pressure (elastic response and van der Waals interaction), simulations further suggest that satellite droplets only form when the initial average film thickness corresponds to a positive effective disjoining pressure.
June 6 CULM 611 Richard Moore, Optimal Control in Nonlinear Waves

I will summarize my two primary research projects involving the application of large deviation theory to noisy nonlinear optical systems and the application of optimal control theory to guide autonomous vehicles so as to optimize the return of information in Lagrangian data assimilation.
June 7 CULM 611 Ensela Mema, Mathematical Models for Polymer-Nematic Interactions

In the first part of this presentation, we consider a mathematical model that consists of a nematic liquid crystal (NLC) layer sandwiched between two parallel bounding plates, across which an external field is applied. We investigate how the number and type of solutions for the director orientation within the layer change as the field strength, anchoring conditions and material properties of the nematic liquid crystal layer vary. In particular, we focus on how the inclusion of flexoelectric effects affects the Freedericksz and saturation thresholds. Next, we consider the interaction between nematic liquid crystal (NLC) and polymer substrates. Such substrates can interact with NLC if an prolonged electric/magnetic field is applied throughout the layer, exhibiting a phenomenon known as director gliding: the preferred orientation of the NLC molecules at the interface changes on timescales slow relative to the elastic relaxation timescale of the NLC. We present a mathematical model for gliding, inspired by experiments that investigate the interaction between the NLC and a polymer substrate.
June 9 CULM 611 Lenka Kovalcinova, Numerical Simulations of Dense Granular Systems With and Without Cohesive Effects

At the beginning of this talk, we discuss granular systems in 2D. For dense granular systems, we characterize the force networks and find that the scaling law of mean cluster size is not universal, contrary to the previous published results. Next, we focus on the source of the dissipated energy in granular systems that are sheared. It is found that additional attractive interaction (that may be due to cohesive effects in wet granular systems) play an important role in determining the source of energy loss. In the last part of the talk, we consider 3D systems and compare different force models for the interaction of spherical particles.
June 13 CULM 611 Ji Meng Loh, A Single-Index Model for Inhomogeneous Spatial Point Processes

I will talk about statistical modeling of spatial point processes and introduce a single-index model for the intensity function of the process in terms of spatial covariates. This new model complements the more commonly used log-linear model and allows the link function between the intensity and the linear combination of covariates to be unspecified and estimated from the data.
June 14 CULM 611 Szu-Pei Fu, Brownian Dynamics Simulations of Lipid Bilayer Membrane with Hydrodynamic Interactions in LAMMPS

Lipid bilayer membranes have been extensively studied by coarse-grained molecular dynamics (CGMD) simulations, where several lipids are coarse-grained into a particle of size 4~6 nm. I will introduce a pair-potential method which is able to capture the mechanical properties of a lipid bilayer membrane (such as gel-fluid-gas phase transitions of lipids and bending rigidity). In this work both explicit and implicit solvent CGMD simulations are performed. The explicit-solvent model uses Lennard-Jones (L-J) potential to account for hydrodynamic interactions (HIs) and His are incorporated via Rotne-Prager-Yamakawa (RPY) tensor in implicit solvent approach. We also consider the effects of cytoskeleton on the lipid membrane dynamics as a model for red blood cell (RBC) dynamics.

To demonstrate that the proposed methods can capture observed dynamics of vesicles and RBC, we focus on two sets of studies: 1. Vesicle shape transitions with varying enclosed volume; 2. RBC shape transitions with different enclosed volume.
June 16 CULM 611 Matthew Moye, Parameter Estimation Techniques of Neuron Models

Voltage-clamp protocols are a traditional means to identify the parameters in conductance-based neuron model, but are less suited for studies of neurons in the suprachiasmatic nucleus. We present a detailed model of SCN neurons for which we want to quantify parameters of those expressing or not expressing molecular clock gene Per1 (P and NP neurons) through parameter estimation using only an observed voltage trace. The primary focus of the talk will be on an introduction to and performance analysis of various methods when conducting twin experiments of the FitzHugh-Nagumo model. Estimation will be performed using a Kalman Filter, genetic algorithm, and linear regression. Lastly, the talk will conclude with discussion of alternative or hybrid approaches.
June 21
CULM 611 Aminur Rahman, A Brief History of Chaos and it's Appearance in Walking Droplets and Electronic Circuits

If Poincar ́e is the father of Dynamical Systems, Newton, Leibniz, and the Bernoullis, are its great grandfathers. The foundations were laid, in the late 1600s and early 1700s, through intimate connections to real world problems with works like “Nova Methodus”, “Principia”, “Methodus Fluxionum”, “Explicationes”, and many others. Two centuries later, Poincar ́e applied new mathematical techniques to his study of celestial mechanics, which came to be known as Dynamical Systems. He showed that information about a system can be extracted through its qualitative properties (phase space), i.e. without having to solve the system of equations. Meanwhile, Lyapunov developed ideas on the stability of dynamical systems. As qualitative analysis became more prevalent, mathematicians started studying more complex systems. Two well- known investigations, the three body problem studied by Poincar ́e and billiards by Hadamard, led to what we now know as chaos. However, many mathematicians were hesitant to study these types of systems until the 1950s.

In the 1950s and 1960s, Lorenz studied non-linear weather forecasting models through numerical simulations. He discovered that inputting two close initial con- ditions created very different simulation results. Over the years the study of chaos became more rigorous, and in 1967 Smale formalized a technique to prove a system was chaotic. More recently, chaos has been observed and studied in electronics, fluids, waves, population biology, epidemiology, and many other real world phenomena. In this talk, two such phenomena, chaotic logical circuits and walking droplets, are briefly presented.
June 23 Campus Center Conference Room 240 Casayndra Basarab, Hamiltonian Bifurcations in Schrodinger Trimers

We investigate the phase space of the three-mode discrete NLS in the nonlinear regime. We enumerate the families of standing waves and use normal forms to describe several families of relative periodic orbits whose topologies change due to Hamiltonian Hopf bifurcations, transcritical bifurcations and others.
June 27 CULM 611 Shidong Jiang, An Introduction to the Fast Multipole Method

In this talk, we will give an introduction to the fast multipole method (FMM). We first discuss in detail the original FMM developed by Greengard and Rokhlin in 1987. We then discuss its extension to the adaptive case, the acceleration of the FMM using plane wave expansions, the generalization of the FMM to other nonoscillatory kernels, and some recent related algorithms such as the butterfly algorithm and fast direct solvers. Finally, we will show some applications of the FMM and related algorithms.
June 28 CULM 611 Ivana Seric, Direct Computations of Marangoni-induced Flows Using a Volume of Fluid Method

The volume of fluid (VOF) interface tracking methods have been used for simulating a wide range of interfacial flows. An improved accuracy of the surface tension force computation has enabled the VOF method to become widely used for simulating surface tension driven flows. We present a new method for including variable surface tension in a VOF based Navier-Stokes solver. The tangential gradient of the surface tension is implemented using an extension of the classical continuum surface force model that has been previously used for constant surface tension simulations. Our method can be used for computing the surface gradients of surface tension that is temperature or concentration dependent.
June 30 CULM 611 Nanyi Dong, Thin Film Evolution Under Pulsed Laser Introduced Marangoni Effect

We consider thin fluid films placed on thermally conductive substrates and exposed to time-dependent spatially uniform heat source. The evolution of the films is considered within the long-wave framework in the regime such that both fluid/substrate interaction, modeled via disjoining pressure, and Marangoni forces, are relevant. We analyze the problem by the means of linear stability analysis as well as by time-dependent nonlinear simulations. The main finding is that when self-consistent computation of the temperature field is performed, a complex interplay of different instability mechanisms results. This includes either monotonous or oscillatory dynamics of the free surface. This oscillatory behavior is absent if the film temperature is assumed to be slaved to the current value of the film thickness. The results are discussed within the context of liquid metal films, but are of relevance to dynamics of any thin film involving variable temperature of the free surface, such that the temperature and the film interface itself evolve on comparable time scales.
July 4 N/A No Seminar - NJIT closed for 4th of July holiday
July 5 CULM 611 Horacio Rotstein, Inhibition-Based Theta Resonance in a Hippocampal Network: A Modeling Study

Rhythmic oscillations are ubiquitous in the nervous system, span a multitude of frequencies, and play important functional roles in cognition and motor behavior in both health and disease. A prominent rhythmic pattern is the hippocampal theta oscillations (4 - 10 Hz), which is evident in the rodent brain during motor activity and REM sleep, and provides a basis for temporal coding of spatial information and episodic memory.

The generation of hippocampal theta oscillations involves the interplay of various mechanisms including a septal pacemaker, circuit interactions, and the intrinsic properties of single neurons. Several neuron types including hippocampal pyramidal cells (PYR) exhibit subthreshold theta-band resonance: a peak in the voltage amplitude response to oscillatory current inputs at a preferred (resonant) frequency. These findings suggest that subthreshold resonance may underlie network theta oscillations. However, whether and how the subthreshold intrinsic oscillatory properties of single neurons affect the generation and properties of network oscillations is not well understood.

In recent work we have addressed these issues in the context of the hippocampal area CA1. Using optogenetic manipulations, we have found that, in the intact brain of the freely-moving animal, PYR do not exhibit theta-band spiking resonance by direct activation, apparently at odds with the 'in vitro' results. Instead, each PYR spikes at a different input frequency. In contrast, when PYR were indirectly activated through direct activation of PV+ interneurons (INT), PYR displayed theta-band resonance. Only input frequencies in the theta frequency band produced a spiking response. Blockade of the h-current (known to underlie subthreshold resonance in PYR) abolished inhibition-induced spiking resonance.

In this study we use mathematical modeling, numerical simulations and dynamical systems tools to investigate the underlying mechanisms. Specifically, we present a minimal biophysical (conductance-based) model of a CA1 hippocampal network that captures the above-mentioned experimental results. The basic model includes PYR, INT, AMPA synaptic excitation and GABA_A synaptic inhibition. The extended models include also OLM (orients-lacunosum moleculare) interneurons and synaptic depression (from INT to PYR). Both PYR and OLM include h-currents. The mechanisms of generation of subthreshold resonance in these cells involve the complex interaction between the voltage-dependent nonlinearities and the time scale separation between the voltage and the h-current gating variable.

The PYR subthreshold resonance fails to be communicated to the spiking regime by direct PYR activation because of the relatively strong effect of the oscillatory input amplitude that causes the spiking activity to spread over a broad range of input frequencies (for which the voltage response is above threshold). PYR theta-band resonance through direct INT activation results instead from a combination of rebound spiking and a timing mechanism. Rebound spiking is responsible for the spiking low-pass filter" (generation of spikes for input frequencies that are low enough for the voltage responses of both PYR and INT to be above threshold), but it is not enough to generate spiking resonance. The timing mechanisms are responsible for either ”erasing” spikes generated by input frequencies lower that theta or failing to produce spikes for these input frequencies. We identified three such mechanisms: (i) network-mediated inhibition from OLM, (ii) synaptic depression of INT synapses, and (iii) subthreshold gamma resonance in INT.

Overall, these results provide a mechanistic understanding of network resonance at theta frequencies and make several predictions. The principles identified in this study are applicable not only to CA1 networks, but also to other systems that exhibit theta resonance such as neocortical networks. Finally, the results and ideas that emerge from our study are seminal for the construction of a theoretical framework for the investigation of the preferred frequency responses of neuronal networks to oscillatory inputs at a variety of biophysically realistic frequency bands.

This project is in collaboration with E. Stark (Tel Aviv University) and G. Buzsaki (NYU Medical School).
July 7 CULM 611 Randolph Leiser, Network Response to Periodic Inputs: Heterogeneous vs. Homogeneous Cell Components

Human pancreatic beta cells are located within the pancreas and are responsible for producing the hormone insulin. They are found in clusters of endocrine cells known as the Islets of Langerhans. Despite being a heterogeneous population, all beta cells within an islet as well as all islets synchronize in phase. The goal of this project was to investigate what role this heterogeneity, or differences, between the cells’ properties plays in their function when the cells are electrically coupled with a gap junction. Different cells will respond differently to unique inputs, and this response can be better or worse depending on the scenario (e.g. insulin production activity). We have looked at a minimal network of electrically coupled cells and examined how differences in their parameter values affect the attributes of the network response to periodic input. The simplicity of the model allowed us to directly see the effects of parameter variation by examining the analytic solution. However, when multiple copies of a model are involved, the analysis becomes more complex. To understand the underlying dynamic mechanisms, we used dynamical systems tools (phase-plane analysis), noting how parameter changes move the nullclines and how these movements interact to produce the observed behavior.

July 11 CULM 611 Wooyoung Choi, On Strongly Nonlinear Long Wave Motions in Density-stratified Flows

In this talk, I will review our recent efforts to develop asymptotic models for strongly nonlinear long waves in density-stratified coastal oceans and describe their solitary wave solutions in comparison with laboratory experiments. In addition, surface expressions associated the internal wave motions observed on satellite images will be discussed.
July 12 CULM 611 Ruihua Cheng, Learning-Based Method with Valence Shifters for Sentiment Analysis

Automatic sentiment classification is becoming a popular and effective way to help online users or companies to process and make sense of customer reviews. In this talk, we extend learning-based methods for classification online reviews that achieves better classification accuracy. The method combines two recent developments. First, valence shifters and individual opinion words are combined as bigrams to use in an ordinal margin classifier. Second, relational information between unigrams expressed in the form of a graph is used to constrain the parameters of the classifier. By combining these two components, our method is able to extract more of the unstructured information present in the data than previous methods, like support vector machine, random forest, hence gaining the potential of better performance. Indeed our results show a higher classification accuracy on empirical real data with ground truth as well as on simulated data.
July 14 CULM 611 Yiming Yu, Rare Event Simulation for Exit Problem

We study rave event simulation with optimal control to estimate escape probability and mean first exit time. In this talk, we focus on estimating escape probability in finite time. Numerical simulations show that Importance Sampling (IS) with stochastic optimal control or a combination of stochastic and deterministic controls has a smaller variance than IS using deterministic control. We propose to show this analytically.

Meanwhile, the geometric minimum action method (GMAM) is a numerical technique to identify the minimizer of the Freidlin-Wentzell large deviations action functional for high dimensional system. We implemented an open loop feedback control based on the minimizer computed by GMAM for a reduction version of a mode-locked laser model with active feedback. For small $\epsilon$, the mean exit paths in finite time stay close to optimal path and the action functional approximated from importance sampling is close to the action functional computed by GMAM. As a result, the estimate probability $\sim \exp(-S/\epsilon^2)$ is very small. With relatively large $\epsilon$, the sample path deviates from the minimizer. In this case, we propose to create a better exit criterion, which is big enough to cover all possible exit points or compute a closed loop feedback control, which requires solving the Hamilton-Jacobi equations or update the optimal path by GMAM at each state.
July 18 Campus Center Conference Room 240 Valeria Barra, Numerical Study of Thin Viscoelastic Films

We present a computational investigation of thin viscoelastic films and drops on a solid substrate subject to the van der Waals interaction force, in two spatial dimensions. The governing equations are obtained within a long-wave approximation of the Navier-Stokes equations with Jeffreys model for viscoelastic stresses. We investigate the effects of viscoelasticity, Newtonian viscosity, and the substrate slippage on the dynamics of thin viscoelastic films. We also study the effects of viscoelasticity on drops that spread or recede on a prewetted substrate. For dewetting films, the numerical results show the presence of multiple secondary droplets for higher values of elasticity, consistently with experimental findings. For drops, we find that elastic effects lead to deviations from the Cox-Voinov law for partially wetting fluids. In general, elastic effects enhance spreading, and suppress retraction, compared to Newtonian ones. Finally, we introduce a preliminary computational investigation concerning three dimensional viscoelastic sheets in arbitrary geometries, starting from a discrete Lagrangian formulation for elastic thin shells.
July 19 Campus Center Conference Room 240 Haiyang Qi, Boundary Integral Equation Formulations and Nystrom Discretizations for the Solution of Helmholtz Problems

We present a regularization strategy that leads to well-conditioned boundary integral equation formulations of Helmholtz equations with boundary conditions in two-dimensional Lipschitz domains. The Nystrom method that we use for the discretization of the various integral equations under consideration are based on global trigonometric approximations, splitting of the kernels of integral operators into singular and smooth components, and explicit quadratures of products of singular parts (logarithms) and trigonometric polynomials.
July 21 Campus Center Conference Room 240 Michael Pedneault, Decomposition Methods for the Solution of Multiple Scattering Problems

(Abstract TBA)
July 25 Campus Center Conference Room 240 Roy Goodman, Managing Applied Math Research

In the course of your research in applied mathematics, you generate a lot of data: hand calculations, computer programs and their output, LaTeX manuscripts, etc. All this stuff can be hard to keep track of, but failure to do so can waste a lot of your time and even make it hard to find/remember/keep what you’ve done. Dr. Goodman will speak about his struggles with this throughout his 20 years as a student and researcher. He will introduce some practices and software tools that can help.
July 26 CULM 611 Michael Booty, Surfactant Solubility in the Large Peclet Number Limit

The presentation begins with an explanation of what a surfactant is and examples of what it can do. This is followed by overview of an analytical approach to the study of surfactant solubility effects that is based on matched asymptotics in the large Peclet number limit. The approach, or the method that results from it, can be implemented in a number of ways, and recent sample results will be explained and discussed.
July 28 CULM 611 Mahdi Bandegi, A Study of Minimizers for Pairwise Interaction Problems Using Convex Relaxation

In this talk, I will discuss both continuum and discrete energy models for a system of large particles. These models can be found in materials or biological aggregations. Finding minimizers of these models is important since the energy of systems of many particles approach a minimum value at equilibrium.

The new approach is to introduce a convex relaxation to the non-convex energy. This approach gives a linear programming problem which is computationally efficient to solve and can provide verification on a state's optimality.
August 1 CULM 611 Yassine Boubendir
August 2 CULM 611 Andrew DeStefan, Numerical Methods for Optimal Path Planning of Autonomous Vehicles

Autonomous surface vehicles and autonomous underwater vehicles (collectively referred to as AVs) are self-propelled waterborne drones which are commonly used to study various oceanographic properties. In this research, we are particularly interested in using AVs to better understand uncertain surface currents in oceanic domains. We hope to accomplish this by means of the Level Set Method, which is an algorithm capable of solving optimal control problems such as the minimum time-to-travel between two points in a given domain.

This presentation will be primarily focused on the Level Set Method; however, I will also discuss how this method may be adapted to address the aforementioned problem of interest.
August 4 CULM 611 Li Yu, A Generalized Graphical Approach to Sequentially Rejective Multiple Testing Procedures

Sequentially rejective, weighted Bonferroni-based multiple testing procedures (MTPs) are widely used in clinical trials for testing hypotheses with multiple endpoints, several doses of a new treatment, different subgroups, non-inferiority and superiority, etc. A Bonferroni-based iterative graphical approach was proposed to illustrate such Bonferroni-type tests. It facilitates the communication and visualization of theses tests to the clinical teams. However, this approach can only test one hypothesis at each step. Moreover, the validation of familywise error rate (FWER) control is complicated based on closure principle. These motivate us to propose a generalized graphical approach which can test more than one hypothesis at each step. The algorithm of the proposed graphical approach is based on sequentially rejective procedures, we can verify the FWER control by applying the proposed extended sequential rejection principle. A clinical trial example is used to compare the proposed approach and the original graphical approach.
August 4 CULM 611 Linwan Feng, Brief Introduce to Penalty Equation, Spectral Method and Shallow Water Wave

In the talk, I will briefly introduce the penalty equation as well as the convergence rate for both Dirichlet boundary condition and Neumann boundary condition. Then I am going to have several examples to discuss the spectral method based on the heat equation. The ongoing work is about the shallow water wave. I will simply show the models I will work on in the future.
August 8 CULM 611 Shahriar Afkhami, Moving Boundary Fluid Dynamics Problems

In this talk, I will first briefly survey the fluids systems involving moving boundaries. I will then introduce several popular numerical methods for solving moving boundary problems, classified as Eulerian, Lagrangian, and mixed methods, followed by a discussion of a widely used Eulerian Volume-of-Fluid method for effectively dealing with small to larger interface deformations. The Volume-of-Fluid method introduces a phase-characteristic function; tracking and locating the interface between the two fluids then amount to solving the advection equation for the phase-characteristic function. I will introduce some numerical schemes for solving the advection equation and show that while using first-order schemes results in excessive diffusion, second-order schemes are less diffusive but yield oscillations in the solution. I will then show a second-order Volume-of-Fluid method that circumvents both difficulties. Finally, I will show the derivation of geometrical quantities of the interface using the Volume-of-Fluid method and an extension of the method that formulates the problem as an optimization problem, yielding more accurate computation of geometrical quantities.
August 9 Campus Center Conference Room 240 Ryan Allaire, Ryan Atwater, Brandon Behring
August 11 Campus Center Conference Room 240 Subha Datta, Ziyan Guo