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Applied Mathematics Seminars

The main organizer of these seminars is Kai Yang.

2019 Applied Mathematics Seminars

The Applied Math Seminar takes place Mondays 2-3pm in DM 409A. It incorporates not only topics in applied math but also connections with various other disciplines and areas. For info, contact: Kai Yang (email: yangk@fiu.edu

Upcoming Seminars

Dynamics of Population Models with Two Physiological Structures

Speaker: Hao Kang, Bio-Math, University of Miami
Date: Oct 14

Abstract: It is well-known for a long time that the age-structure of a population affects the nonlinear dynamics of the species in ecology and the transmission dynamics of infectious diseases in epidemiology. In
modeling specific diseases, the age could be chronological age (the age of the population), infection age (the time elapsed since infection), recovery age (the time elapsed sine the last infection), class age (the
length of time in the present group), etc. Other physiological conditions or physical characteristics such as size, location, status, and movement have also been taken in consideration in population dynamical models.
Recently there are some studies taking into account the combined effects of two physiological characteristics (such as age-age, age-size, age-maturation, age-stage),  however there are very few theoretical studies on such models.

In this paper, we consider a scalar population model with two physiological structures and study its fundamental properties and dynamical behaviors. First, the semigroup will be defined based on the solutions and its infinitesimal generator will be determined; then the compactness of the solution trajectories will be analyzed; next spectrum theory will be employed to investigate stability of the zero steady state; when the zero steady state is unstable asynchronous growth of the solutions will be studied; finally we will apply theory of integral semigroup and non-densely defined operators to nonlinear models to establish the principle of linearized stability.

Conservative, positivity preserving and free energy dissipative numerical methods for the Poisson-Nernst-Planck equations

Speaker: Professor Zhongming Wang
Date: Oct 7

We design and analyze some numerical methods for solving the Poisson--Nernst--Planck (PNP) equations. The numerical schemes, including finite difference method and discontinuous Galerkin method, respect three desired properties that are possessed by analytical solutions: I) conservation, II) positivity of solution, and III) free-energy dissipation. Advantages of different types of methods are discussed.  Numerical experiments are performed to validate the numerical analysis. An application to an electrochemical charging system is also studied to demonstrate the effectiveness of our schemes in solving realistic problems. This is a joint work with H. Liu, D. Jie and S. Zhou.

Threshold solutions for the nonlinear Schrödinger equation

Speaker: Lucas Campos
Date: Sept 23

We consider the energy-subcritical and energy-critical nonlinear focusing Schrödinger equation. In previous works by Holmer-Roudenko in the subcritical setting, and by Kenig-Merle, the mass-energy of the ground state has been shown to be a threshold for the global behavior of solutions to the equation. In the present article, we study the dynamics at the threshold level and classify the corresponding solutions.

On the stochastic 2D nonlocal Cahn-Hilliard- Navier-Stokes model

Speaker: Professor Deugoue
Date: Sept 9

In this talk, we study a stochastic version of a well-known diffuse interface model. The model consists of the Navier-Stokes equations for the average velocity, nonlinearly coupled with a nonlocal Cahn-Hilliard equation for the order parameter. The system describes the evolution of an incompressible isothermal mixture of binary fluids excited by random forces in a two-dimensional bounded domain. For a fairly general class of random forces, we prove the existence and uniqueness of a variational solution.

Past Years

  • 2019

    Application in biomedical analytics to enhance cardiac tissue image analysis

    Speaker: Jennifer Rodgers
    Date: April 12

    Hypoxia (low oxygen) due to acute respiratory failure is the most common cause of ICU admission in patients. Oxygen therapy is a first-line treatment for patients with life-threatening hypoxia. High levels of oxygen is also known to result in harmful effects in patients. Despite this, there is still a tendency in critical care settings to provide hypoxic patients with excessive oxygen. Our lab has previously reported arrhythmias and cardiac remodeling in mice after 72 hours of high oxygen therapy. Based on this data, we hypothesized that there is a time-dependent nature to high oxygen-induced electrical disturbances and cardiac remodeling in mice. Therefore, we have studied oxygen therapy in mice, in a time course study, in order to generate incremental data regarding the onset of the cardiac impact of high oxygen exposure, using analytic techniques with machine learning applications. As a result, we have developed an application to enhance cardiac tissue image analysis. 

    Global regularity for a rapidly rotating convection model of tall columnar structure with weak dissipation

    Speaker: Yanqiu Guo
    Date: April 5

    This presentation is based on our analysis of a three-dimensional fluid model describing rapidly rotating convection that takes place in tall columnar structures. Global well-posedness for strong solutions is shown provided the model is regularized by a weak dissipation term. This is a joint project with Cao and Titi.

    Nonlinear waves and singularities in nonlinear optics, plasmas and biology

    Speaker: Pavel Lushnikov
    Date: March 8

    Many nonlinear partial differential equations have a striking phenomenon of spontaneous formation of singularities in a finite time (blow up). Blow up is often accompanied by a dramatic contraction of the spatial extent of solution,  which is called by collapse.  Near singularity point there is usually a qualitative change in underlying nonlinear phenomena, reduced models loose their applicability with diverse singularity regularization mechanisms become important such as optical breakdown and formation of plasma in nonlinear optical media, excluded volume constraints in bacterial aggregation or dissipation of breaking water waves. Collapses occur in numerous physical and biological systems including a nonlinear Schrodinger equation, Keller-Segel equation, Davey–Stewartson equation and many others. Wavebreaking is another example of spontaneous formation of singularities corresponding to the breaking of initially smooth smooth fluid's free surface. It can be reduced to the motion of complex singularities outside of fluid with wavebreaking resulting from the approach of these singularities to the free surface. The recent progress in collapse theory will be reviewed with multiple applications ranging from laser fusion to bacterial dynamics addressed.

    Numerical simulation for wave propagation and imaging

    Speaker: Kai Huang
    Date: March 1

    A direct imaging algorithm for point and extended targets is presented. 
    The algorithm is based on a physical factorization of the response matrix of a transducer array. 
    An efficient algorithm proposed for simulating wave propagation over long distance with both weak and strong scatters.

    A high order finite difference method with subcell resolution for stiff multispecies detonation in under-resolved mesh

    Speaker: Wei Wang
    Date: Feb. 22

    In this talk, we propose a high order finite difference WENO method with Harten's ENO subcell resolution idea for the chemical reactive flows. In the reaction problems, when the reaction time scale is very small, the problems will become very stiff. Wrong propagation of discontinuity occurs due to the underresolved numerical solutions in both the space and time. The proposed method is a modified fractional step method which solves the convection step and reaction step separately. A fifth-order WENO is used in convection step. In the reaction step, a modified ODE solver is applied but with the flow variables in the discontinuity region modified by the subcell resolution idea.

    Blind source separations in spectroscopy

    Speaker: Yuanchang Sun
    Date: Feb. 15

    We shall start with a framework for extracting spectral structures from mixed data when partial knowledge of the mixtures are available. Then computational modeling and methods will be introduced for data fitting with distortions.

    On the stochastic Navier Stokes equations

    Speaker: Annie Millet
    Date: Feb. 8

    This talk will give an overview on some results on the Navier Stokes equation subject to a stochastic perturbation: global well posedness for general 2D hydrodynamical models, anisotropic 3D Navier Stokes equation with some Brinkman Forchheimer regularization, exponential concentration of the distribution when the strength of the noise/ the viscosity  converges to 0, strong speed of convergence of time discretization schemes. These are joint results with H. Bessaih and I. Chueshov.

    An Introduction to Nonlinear Waves, Solitons and Collapses

    Speaker: Kai Yang
    Date: Feb. 1

    We demonstrate several solution profiles for different types of Nonlinear wave equations. 

    Introduction to Statistical Learning with an Emphasis on Bayesian Logic Multiple Classifications

    Speaker: Wensong Wu
    Date: Jan. 25

    This presentation has three parts. It will start with an introduction to Statistical Learning, which is a framework for machine learning from statistical perspective. Then I will talk about my current research on Two-Step Bayesian Multiple Classification with Logic Expressions, which utilizes Bayesian decision theoretic framework and Logistic regression together with some data mining methods. If time permits we will briefly talk about the route from Logistic regression to Neural Network and Deep Learning.  

    Some recent results on two phase flow models

    Speaker: Theodore Tachim Medjo
    Date: Jan. 18

    The purpose of this talk is to present some recent results on two phase flow models. We will consider both the Cahn-Hilliard-Navier-Stokes systems and the Allen-Cahn-Navier-Stokes system (deterministic and stochastic version).