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Mathematics Department

Applied Math Seminar

Fall 2018

All talks are from 12:00-1:00 p.m. in the Seminar Room CH351, unless otherwise specified.

  • Nov
    30
  • Stability Analysis for a swarm of self-propelled agents in the plane with parabolic coupling
    Kostya Medynets
    USNA, Math Department

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    In this talk, we will discuss the dynamics of the system of n agents in the plane whose motion is determined by the second-order differential equations: acceleration of the k-th agent = (1-|speed of the k-th agent |^2)* (velocity of the k-th agent) minus the vector connecting the center of mass and the position of k-th agent. In other words, every agent tries to maintain the speed equal to one and accelerates or slows down depending on whether the speed is less than or greater than one and, simultaneously, each agent experiences a force that pulls this agent towards the moving center of mass of the swarm. Previous numerical experiments have shown that for a large set of initial conditions the system converges to a rotating circular limit cycle with a fixed center of mass (the coordinates of the center of mass depend on the initial conditions), dubbed a ring state. We prove that a ring state is stable whenever the positions of the particles are not collinear, that is, they do not lie on a single straight line. Additionally, we show that every solution that starts near a stable ring state asymptotically approaches a stable ring state. The proofs are based on center manifold theory. We also provide the full description of limit cycles and the stability analysis, incl., the construction of a Lyapunov function, for the decoupled system (the center of mass = constant). We will also discuss the basics of center manifold theory and Lyapunov stability theory for dynamical systems. This is joint work with Carl Kolon (Class of 2018) and Irina Popovici.
  • Nov
    16
  • Dimension Breaking and Numerical Continuation
    Ben Akers
    Airforce Institute of Technology

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    Numerical methods for computing overturned traveling waves in the vortex sheet equations will be discussed. The extreme cost of simulation of this system motivates the use of dimension-breaking numerical continuation. The numerical method's strength, weaknesses and structure will be evaluated, both in the vortex sheet equations and weakly nonlinear models. The need for, and utility of, local and global bifurcation theorems in this area will be discussed. A sampling of these theorems in the vortex sheet setting will be reviewed.
  • Nov
    02
  • Mixed signals in the immunological escape of tumors
    Andrew Belmonte
    Department of Mathematics, Pennsylvania State University

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    One of the unique aspects of cancer as a disease is that it does not elicit the response of the immune system because it appears as "self". However, the details are more complex - and the recent development, with some success, of cancer immunotherapy indicates that the immune system can be alerted to the presence of a tumor. An analysis at the cellular level reveals an immensely complicated system of interactions and interlocutors, posing challenges for the development of mathematical models. In this talk I will review traditional phenomenological models, and present our own mathematical approach, as motivated by in vitro studies of simplified situations of immune cells and tumor cells under the imposed external stimulation of signaling chemicals, or cytokines.
  • Oct
    26
  • Least square discretization and preconditioning for mixed variational formulations
    Constantin Bacuta
    University of Delaware

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    We consider a least squares method for discretizing boundary value problems written as mixed variational formulations with different types of trial and test spaces. At the continuous level the only assumptions we require are LBB and data compatibility conditions. For the proposed discretization method a discrete $\inf-\sup$ condition is automatically satisfied by the natural choices of test spaces (first) and the corresponding trial spaces (second). The discretization and the iterative approach does not require nodal bases for the trial space. We present a multilevel conjugate gradient preconditioning approach that could take into considerations discontinuous coefficients and coupled physics of the problem to be solved. Applications of the method include discretizations of second order PDEs with variable coefficients, interface problems, and first order systems of parametric PDEs, such as the time-harmonic Maxwell equations.
  • Oct
    12
  • The impacts of 3D radiative transfer effect on cloud radiative property simulations and retrievals
    Scott Hottovy
    USNA, Mathematics Department

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    In this talk I will give a brief overview on the CyberTraining Class I participated in under the NSF Grant for Big Data + High-Performance Computing + Atmospheric Sciences (http://cybertraining.umbc.edu/). This course was 16 weeks and included studying parallel computing, atmospheric physics, and techniques in information sciences and big data. The course culminated in a 4 week research project. I will describe the research project and my contributions to it. Research Project: Satellite observations provide good opportunities to evaluate global cloud properties, but the 3D effects induced by the horizontal inhomogeneity of the medium cause possible uncertainties in the cloud remote sensing products. In this work, we we developed a method to generated synthetic cloud fields based on the inverse 2D Fourier transform and used it investigate the impacts of 3D effects on MODIS cloud property retrieval. Both the 3D and 1D radiative transfer simulation studies are conducted in order to understand the impacts of the 3D effects. We retrieve the cloud optical thickness(COT) and cloud effective radius(CER) from the simulated reflectance at 0.86\mum and 2.1\mum bands and compare between the retrieval and true values. The liquid water path (LWP) was obtained via CER and COT. The impacts in the cloud liquid water path retrieval are further studied and we find that the bias in the COT and CER will cause the over estimation of LWP estimations for both the illuminating and shadowy pixels.
  • Oct
    05
  • On the emergence of a turbulence model: Navier-Stokes-alpha model
    Jing Tian
    Towson University

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    Turbulence is a pervasive, complex family of phenomena observed in nature, and has been a great challenge to mathematicians, physicists, engineers and computational scientists. It is widely accepted by the scientific community that turbulent flows are governed by the Navier-Stokes equations, for large values of the Reynolds numbers. In this talk, we will begin with a brief introduction to turbulence, Navier-Stokes equations and the existence and smoothness problem. We then discuss a turbulence modelling method, the Navier-Stokes-alpha model. With the hypothesis that the turbulence described by the Navier-Stokes-alpha model partly due to the roughness of the walls, we present the transition from the Navier-Stokes equations into the Navier-Stokes-alpha model by introducing a Reynolds type averaging.
  • Sep
    21
  • Radial Basis Functions (RBFs) for Numerical Simulation of High Energy Lasers and Interfacial Fluid Dynamics
    Jonah Reeger
    USNA, Math Department
  • Sep
    14
  • PDE Constrained Optimization
    Harbir Antil
    George Mason University
    Time: 12:00 PM

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    Optimization problems with partial differential equations (PDEs) as constraints is known as PDE Constrained Optimization. In this talk, I will start with several applications of PDE constrained optimization, including, free boundary problems, magnetic drug targeting, and fractional nonlocal PDEs. I will focus on optimization problems under uncertainty. I will describe risk-measures and their role in such optimization problems. Several illustrative numerical examples will be discussed.
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