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Applied Mathematics Seminar

The Applied Mathematics Seminar has talks on a wide range of topics, including but not limited to approximation theory and practice, numerical linear algebra, numerical optimization, numerical aspects of computer science, theoretical and applied partial differential equations and their numerical solutions, and mathematical biological models.

For more information contact Cameron Browne.

Fall 2017

For the Fall 2017 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.

  • 26 September 2017
    An Introduction to Bernstein polynomials
    Dun Liu
    Abstract: Bernstein polynomials have been playing a crucial role in approximation theory since the early 20th century. They were used to prove the Weierstrass theorem by S. Bernstein in 1910s, and later became the basis of Bézier Curves, which are now widely used in computer graphics to model smooth curves. With Bernstein polynomials, we can approximate a function over a finite domain and refine the approximation to any desired precision. In this presentation, we will look into the definition of Bernstein polynomials, some important properties of the polynomials, the Weierstrass approximation theorem, and Bézier Curves.
  • 10 October 2017
    The dynamics of an ion channel model
    Xiang-Sheng Wang
    Abstract: We study a time dependent Poisson-Nernst-Planck system which arises from the model of ion channels. The objective is to understand how the ion concentrations are distributed in the channel if there are ion fluxes on the channel boundaries. Assuming that the Debye length is small relative to the channel length, we derive an asymptotic formula for the dynamic solution by matching outer and Debye layer solutions. It is interesting to note that for the time-dependent problem, the outer solution has a boundary layer that does not occur in time-independent problems.
  • 31 October 2017
    Competitive Exclusion through Discrete Time Models
    Paul Salceanu
  • 7 November 2017
    Numerical Algorithms for PDE-Constrained Optimization Problems
    Jun Liu
    Southern Illinois University Edwardsville
    Abstract: PDE-Constrained optimization problems arise in many different scientiffic and engineering applications. In this talk, we will first present several efficient optimize-then-discretize algorithms for iteratively solving the first-order optimality KKT system from both parabolic and wave PDE-Constrained optimal control problems. Second, we will discuss some interesting numerical issues regarding the discretize-then-optimize algorithms, which are also widely used in practice. Numerical results will be shown to illustrate the effectiveness of our proposed numerical algorithms.
  • 28 November 2017
    Identifiability Issues of an Immuno-Epidemiological Model: The case of Rift Valley Fever Virus
    Maia Martcheva
    University of Florida
    Abstract: We discuss the structural and practical identifiability of a nested immuno-epidemiological model of arbovirus diseases, where host–vector transmission rate, host recovery, and disease-induced death rates are governed by the within-host immune system. We incorporate the newest ideas and the most up-todate features of numerical methods to fit multi-scale models to multi-scale data. For an immunological model, we use Rift Valley Fever Virus (RVFV) time-series data obtained from livestock under laboratory experiments, and for an epidemiological model we incorporate a human compartment to the nested model and use the number of human RVFV cases reported by the CDC during the 2006–2007 Kenya outbreak. We show that the immunological model is not structurally identifiable for the measurements of time-series viremia concentrations in the host. Thus, we study the scaled version of the immunological model and prove that it is structurally globally identifiable. After fixing estimated parameter values for the immunological model derived from the scaled model, we develop a numerical method to fit observable RVFV epidemiological data to the nested model for the remaining parameter values of the multi-scale system. For the given (CDC) data set, Monte Carlo simulations indicate that only three parameters of the epidemiological model are practically identifiable when the immune model parameters are fixed. Alternatively, we fit the multi-scale data to the multi-scale model simultaneously. Monte Carlo simulations for the simultaneous fitting suggest that the parameters of the immunological model and the parameters of the immuno-epidemiological model are practically identifiable. (Research performed in collaboration with Necibe Tuncer, Hayriye Gulbudak, and Vincent Cannataro.)

Spring 2017

For the Spring 2017 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.

  • 31 January 2017
    Poisson-Nernst-Planck system with multiple ions
    Xiang-Sheng Wang
    Abstract: We study the Poisson-Nernst-Planck (PNP) system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges. Assuming point charges and that the Debye length is small relative to the domain size, we derive an asymptotic formula for the steady-state solution by matching outer and boundary layer solutions. The case of two ionic species has been extensively studied, the uniqueness of the solution has been proved, and an explicit expression for the solution has been obtained. However, the case of three or more ions has received significantly less attention. Previous work has indicated that the solution may be nonunique and that even obtaining numerical solutions is a difficult task since one must solve complicated systems of nonlinear equations. By adopting a methodology that preserves the symmetries of the PNP system, we show that determining the outer solution effectively reduces to solving a single scalar transcendental equation. Due to the simple form of the transcendental equation, it can be solved numerically in a straightforward manner. Despite the fact that for three ions, previous studies have indicated that multiple solutions may exist, we show that all except for one of these solutions are unphysical and thereby prove the existence and uniqueness for the three-ion case.
  • 22 February 2017 (WEDNESDAY 2:30 Maxim Doucet Hall 208)
    Directional Statistics for High Volatility and Big Data Science
    Ashis SenGupta
    Indian Statistical Institute, Kolkata, West Bengal, INDIA and
    Augusta University, Augusta, Georgia
    Abstract: In this era of emerging complex problems, both small and big data – linear and non-linear, exhibit challenging characteristics which need to be carefully modelled. Thus, multidisciplinary research in mathematical sciences has become indispensable. Marked presence of asymmetry, multimodality, high volatility, long and fat tails, non-linear dependency, etc. are common features of contemporary data. Notwithstanding pitfalls, ideas from several disciplines do enrich the contribution of the research work. Directional statistics is one such scientific “key technology” as which on one hand is developed from the conglomeration of the inductive logic of statistics, objective rigor of mathematics and the skills of numerical analysis of computer science. On the other hand, it possesses the richness to handle the need for providing statistical inference to a wide and emerging arena of applied sciences. Directional data (DD) in general refer to multivariate observations on variables with possibly linear, axial, circular and spherical components. Circular random variables are usually those which pertain to observations on directions, angles, orientations, etc. Data on periodic occurrences can also be cast in the arena of circular data. In general, DD may be mapped to smooth manifolds, e.g. circle, hyper-sphere, hyper-toroid, hyper-cylinder, or to axial and hyper-disc also. Analysis of such data sets differs markedly from those for linear ones due to the disparate topologies between the line and the circle. First, some methods of construction of probability distributions and statistical models for DD on smooth manifolds are presented. The need for applications of these abound for data in a variety of applied sciences. To illustrate this fact, we take up two important problems, one for linear and the other for directional data. With linear data, we take up the problem of obtaining probability distributions for modelling high volatility. The work of Mandelbrot has shown the appropriateness of the stable families of distributions for high volatility. However, in general, these families do not possess any analytical closed form for their probability density functions. This leads to the complexity of inference involving the parameters of such distributions. We overcome this problem of modelling high volatility data by appealing to the area of probability distributions for directional data. A new family of possibly multimodal, asymmetric and heavy-tail distribution is presented. The usual fat-tail, Cauchy and t, distributions are encompassed by this family and it has even tails comparable to that of the stable family. The second problem deals with data, possibly Big Data, on smooth manifolds. It is first noted that such data invariably exhibit multimodality and hence the possibility of underlying multiple component distributions. Thus, it would be prudent to “Divide and Conquer”, prior to proceeding for drawing statistical inference on such data. Here we deal with this problem by developing Hierarchical Unsupervised Learning or statistical Clustering techniques for manifold data. We illustrate our approach by a real-life example based on agricultural insurance data.
  • 4 April 2017 (208 Maxim Doucet Hall)
    Estimation of Distributed Delays
    Temi Gaudet
    University of Louisiana at Lafayette
    Abstract: Delay differential equations have been studied for several decades as they arise in many applications. A common approach is to transform a distributed delay system to a related ordinary differential equation system via the ‘linear chain trick’. This is due to the fact that the term involving the distributed delay can be replaced by a state variable that is coupled to other state variables in a linear system of ODEs. This transformation relies on the assumption that the delay follows a gamma distribution. We try to determine when one correctly or incorrectly assumes a gamma distribution and the implications of such assumption by estimating the parameters associated with the distribution followed by a time delay. The results when performing the estimations in the ODE system (this is equivalent to the delay system if a gamma distribution is assumed) are compared to the results in the delay system.

Fall 2016

For the Fall 2016 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.

  • 13 September 2016
    Traveling wave solutions of a diffusive epidemic model
    Xiang-Sheng Wang
    Mathematics Department
    University of Louisiana at Lafayette
    We study the traveling wave solutions of a diffusive epidemic model with standard incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and a minimal wave speed. Our proof is based on Schauder fixed point theorem and Laplace transform.
  • 27 September 2016
    The Dynamics of Vector-Borne Relapsing Diseases
    Cody Palmer
    Mathematics Department
    University of Louisiana at Lafayette
    Motivated by Tick-Borne Relapsing Fever (TBRF) we will be investigating the dynamics of various models for the spread of a relapsing disease by a vector. In particular we quantify the effect that relapses have on the disease spread and the how the number of relapses influence control strategies for the disease.
  • 4 October 2016
    Modeling Multi-Epitope HIV/CTL Immune Response Dynamics and Evolution
    Cameron Browne
    Mathematics Department
    University of Louisiana at Lafayette
    The CTL (Cytotoxic T Lymphocyte) immune response plays a large role in controlling HIV infection. CTL immune effectors recognize epitopes (viral proteins) presented on the surface of infected cells to mediate their killing. The immune system has an extensive repertoire of CTLs, however HIV can evolve resistance to attack at different epitopes. The ensuing arms race creates an evolving network of viral strains and CTL populations with variable levels of epitope resistance. Motivated by this, we formulate a general ODE model of multi-epitope virus-immune response dynamics. Some special cases for the HIV/CTL interaction network are considered, the case of a nested network and the general two-epitope case. We characterize the persistent viral strains and immune response in terms of system parameters and prove global properties of solutions via Lyapunov functions. The results are interpreted in the context of within-host HIV/CTL evolution and numerical simulations are provided. To conclude, we discuss extensions of the model to a PDE system which incorporates cell-infection age structure.
  • 11 October 2016
    Synchrony and the dynamic dichotomy in a class of matrix population models
    Amy Veprauskas
    Mathematics Department
    University of Louisiana at Lafayette
    In this talk, I will discuss the dynamics of a class of discrete-time structured population models called synchrony models. Synchrony models are characterized by the simultaneous bifurcation of a branch of positive equilibria and a branch of synchronous 2-cycles from the extinction equilibrium. These models exhibit a dynamic dichotomy in which the two steady states have opposite stability properties that are determined by the relative levels of competition in the population. I will also present an application that is motivated by observations of a population of cannibalistic gulls.