Summer 2018 Student Talks

In this research, we studied few numerical methods for solving conic programming problems that involve constraints on Fourier modes.Pair interaction problems can be found in variety of natural phenomena like biological aggregation, colloids, liquid crystals, and self-assembly. Finding the global minimum and stable solution to a dynamical system is more challenging when there are many particles in the system, as the total energy from the interaction of all particles becomes highly non-convex. In our approach instead of minimizing the original non-convex functional, we find minimizers to the lower bound functional of it, which is convex. Now, the so-called relaxed problem can be solved using linear programming techniques. The nature of constraints in our problem allows us to exploit structure in order to find a faster algorithm.
 

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.We study a sheared system in stick-slip regime via two- dimensional discrete element simulations.Particular focus is on the evolution of force networks before and during slip events. We will show that some features of force network evolution could be used to gain insight into the occurrence of a slip event.

Sufficient Dimension Reduction (SDR) has become an essential tool in dealing with high-dimensional data in the past few years. In this paper, we attempt to tackle the issue of high-dimensionality for inhomogeneous spatial point processes. The spatial point processes have been treated as binary response. We have used an SDR method, called weighted principal support vector machine (WPSVM). Most SDR methods don't perform well when the response is binary. It is, hence, the desire of the researcher to develop an efficient method to reduce dimension of the data without loss of information with binary response. PSVM combines techniques of SDR and SVM and can extract sufficient predictors for both linear and non-linear models. However, PSVM suffers from estimating at most one direction of the central subspace for binary response. Shin et. al. has shown that the WPSVM can estimate more than one direction with binary response. We have conducted simulation studies to evaluate the finite sample performance of the SDR methods and WPSVM. WPSVM seems to work reasonably well for inhomogeneous spatial point processes when compared with other methods and hence is our suggested approach. 

In this talk, I briefly introduce the Monge-Ampere Equation and an approach to solving it as an eigenvalue problem. I then discuss other examples of eigenvalue problems for nonlinear PDEs and the specific example I have solved. Then, the development of mesh-free point clouds for computing eigenvalues of the Hessian matrix in order to find the solution to these problems is discussed. Implicit boundary conditions specific to the problem are outlined along with the introduction of quadtrees in order to build higher order schemes.

In this work, we use mathematical modeling to study the influence of a membrane's internal structure on its flow properties and adsorptive fouling behavior. Layered planar membrane structures with intra-layer connections are modelled. Comparisons between non-connected and connected models are presented. Additionally, the influence of spatial, in-plane, inhomogeneities on overall performance is modeled by adding noise perturbation to homogeneous membrane structures. Membrane performance is gauged via 1) the relative comparison of total throughput and flux evolution during filtration; and 2) control of concentration of foulants at membrane pore outlets. At last, we present a possible optimal control problem inspired by the numerical simulations. 

Some of the most fundamental physiological cell processes such as synaptic neurotransmitter release, endocrine hormone release, muscle contraction and cytotoxic immune cell responses are activated by Ca2+ influx through trans-membrane Ca2+ channels. Computer simulations have shown that local Ca2+ elevations produced by the opening of a Ca2+ channel, termed nanodomains, form and collapse very rapidly in response to channel gating. Previously, several approximations have been developed to estimate nanodomain Ca2+ concentration profiles, obviating computationally expensive solutions of partial differential equations describing buffered Ca2+ diffusion. However, these approximations have two limitations: (1) their accuracy is restricted to specific regions in buffering parameter space, and (2) they have been developed for simple, one-to-one Ca2+-buffer binding, and apart from RBA, these approximations are hard to extend to more realistic biological buffers have multiple Ca2+ binding sites. Here we present our preliminary work on developing new approaches to better approximate single-channel Ca2+ nanodomains with more accuracy in a wider range of model parameters. One of the new approximation methods is based on matching the coefficients of short-range Taylor series and long-range asymptotic series of the nanodomain Ca2+ distance dependence using a simple Ansatz that we choose in advance. A second method is based on the variational approach, and involves a global minimization of some functional (an analogue of an action or energy function) with respect to parameters of a chosen Ansatz. The advantage of the new methods compared to the previously published approximations is that their accuracy is less dependent on the buffering conditions.

Pleated membrane filters, which offer larger surface area to volume ratios than unpleated membrane filters, are used in a wide variety of applications. The performance of the pleated filter, typically characterized by a flux-throughput plot indicates, however, that the unpleated filter provides better performance under the same pressure drop. Earlier work (Sanaei and Cummings 2016) used a highly-simplified membrane model to investigate how the pleating effect and membrane geometry impact performance. In this work, we extend their investigation to consider in detail how membrane pore morphology affects performance, and how to optimize performance through varying the pore profile. Our optimization is designed to maximize the total throughput while simultaneously assuring adequate particle removal, which is an essential part of filtration.

We present an approach using optimal control theory that transforms an initial electric field in a dispersive media into a desired field. In this context, the index of refraction as a function of space is treated as the control parameter and the performance measure is varied over the length of the waveguide. Once the proposed performance measure is motivated, the numerical solution to the optimization problem will be discussed.

Pleated membrane filters, which offer larger surface area to volume ratios than unpleated membrane filters, are used in a wide variety of applications. The performance of the pleated filter, typically characterized by a flux-throughput plot indicates, however, that the unpleated filter provides better performance under the same pressure drop. Earlier work (Sanaei and Cummings 2016) used a highly-simplified membrane model to investigate how the pleating effect and membrane geometry impact performance. In this work, we extend their investigation to consider in detail how membrane pore morphology affects performance, and how to optimize performance through varying the pore profile. Our optimization is designed to maximize the total throughput while simultaneously assuring adequate particle removal, which is an essential part of filtration.

New York City's 'Stop-and-Frisk' policy is police strategy that police officers can briefly stop and search pedestrians in an effort to get weapons, drugs, and other contraband off the street. The purpose of this policy is to protect the police officers and the whole society. Data released from the New York City Police Departmet show that blacks and Hispanics make up more than 80% of the stops while they only constitute about 50% of the New York City Population. This racial disparities in stops have aroused a lot of criticisms. Several studies, from the aspects of statistics, social science, and economy, were conducted to investigate whether the racial disparities reflect the crime rate or the racial bias of police officers. However, different and sometimes controversial conclusions were draw from these studies. Thus, this is still an open problem. In this talk, I will first briefly introduce and compare sme of these studies and then mostly focus on the study conducted by S. Goel et al.. In this study, authors studied the possible violation of the Forth Amendment of the 'Stop-and-Frisk' by analyzing the ex ante crime rate with logistic regression model. They concluded that even after considering locations, racial bias still exists in these stops. Our research project is to study the racial disparities in 'Stop-and-Frisk' by spatial point process. Best to our knowledge, no one has analyzed 'Stop-and-Frisk' data with spatial point process before. Thus, our research will provide a new way to study this problem.