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Mathematics Department

Applied Math Seminar

Fall 2015

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

  • Dec
    08
  • Sparse Approximation In Computed Tomography
    Alireza Entezari
    University of Florida
    Location: CH351
    Time: 12:00 PM

    View Abstract

    Sampling and reconstruction are ubiquitous in many problems in computational mathematics. In this talk I present a spline framework, based on box splines, for discretization of Radon transform which is widely used in mathematical modeling of X-ray computed tomography. Using this spline representation, we formulate a sparse approximation approach to the tomographic reconstruction problem. As reconstruction from partial data (e.g., low-dose X-ray, limited view) is a major challenge in sparse signal processing, the advantages of the proposed framework for solving ill-posed inverse imaging problems are examined.
  • Dec
    01
  • The hybridized discontinuous Galerkin method for unsteady flows.
    Jochen Schuetz
    Institute of Geometry Science and Applied Mathematics
    Location: CH351
    Time: 12:00 PM

    View Abstract

    In this talk, we give an introduction to the hybridized discontinuous Galerkin (HDG) method applied to compressible Navier-Stokes equations and show recent results. In particular, we focus on the extension of HDG to unsteady flows using multiderivative time integration.
  • Nov
    24
  • Mathematical Modeling: Immune System Dynamics in the Presence of Cancer and Immunodeficiency in vivo
    Thomas Wester
    USNA (Trident Scholar)
    Location: CH351

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    The Human Immunodeficiency Virus (HIV) targets CD4 T-cells which are crucial in regulating the immune systems response to foreign pathogens, including cancerous cell development. Furthermore, several studies link HIV infection with the proliferation of specific forms of cancer such as Kaposi Sarcoma and Non-Hodgkin's Lymphoma; HIV infected individuals can be several thousand times more likely to be diagnosed with cancer. However, much remains unknown about the dynamic interaction between cancer development and immunodeficiency. During HIV-1 primary infection, we know that the virus concentration increases, reaches a peak, and then decreases until it reaches a set point. In this project, we studied longitudinal data from 18 subjects identified as HIV positive during plasma donation screening to examine the dynamics of primary HIV infection. In doing so, we applied several nonlinear ordinary differential equation HIV infection models and analyzed the behavior of the system. In our future work, we seek to integrate cancer-immune models to examine the interaction of both cancer and immunodeficiency within the immune system.
  • Nov
    19
  • Collective motion patterns of delay-coupled swarms: theory and experiment
    Klementyna Szwaykowska
    Time: 12:00 PM

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    Emergence of coherent motion patterns in swarms of interacting agents is a subject of active research in the study of complex dynamical systems. Understanding the physics of emergent behaviors is key to the development of functional swarms for a range of applications and scales, including autonomous long-term surveillance, environmental monitoring, defense, and smart medical detectors, to name a few. In this talk, we describe the emergent motion patterns of a group of individuals governed by a simple but general set of interaction rules. Individual agents are modeled as self-propelled, interacting particles communicating over a network with delay. We describe the change in patterns and emergence of new ones with the addition of communication delay; we also examine the effects of adding differences in the dynamical capabilities of individuals, and limited network communication. Our results are verified in a lab-based experiment.
  • Nov
    10
  • An introduction to discontinuous Galerkin methods (working seminar part 1)
    David Seal
    USNA
    Location: CH351
    Time: 12:00 PM

    View Abstract

    Abstract : This is the first in at least a couple of working seminars on the discontinuous Galerkin (DG) method. Of the modern numerical methods for hyperbolic conservation laws, the discontinuous Galerkin method has become one of the most popular high-order methods given its ability to obtain better resolution of the solution with far fewer unknowns than a low-order solver. It can be formulated to operate on unstructured grids (which is important for problems with geometry), and when compared to other high-order schemes such as the weighted essentially non-oscillatory (WENO) method, its theoretical foundation is much more well understood given its finite element roots. In addition, the DG method contains clean convergence (e.g. superconvergence) properties that other high-order methods do not have. The focus of this seminar is to present the minimal details necessary to implement the DG method for simple problems such as 1D advection and Burgers equation. Depending on interest, future seminars can be redirected to a number of different areas including time stepping options, limiters, shallow water or plasma applications, superconvergence properties, as well as elliptic or parabolic solvers that could include an introduction to the hybridized DG (HDG) method.
  • Nov
    03
  • An introduction to discontinuous Galerkin methods (working seminar part 1)
    David Seal
    USNA
    Location: CH351
    Time: 12:00 PM

    View Abstract

    Abstract : This is the first in at least a couple of working seminars on the discontinuous Galerkin (DG) method. Of the modern numerical methods for hyperbolic conservation laws, the discontinuous Galerkin method has become one of the most popular high-order methods given its ability to obtain better resolution of the solution with far fewer unknowns than a low-order solver. It can be formulated to operate on unstructured grids (which is important for problems with geometry), and when compared to other high-order schemes such as the weighted essentially non-oscillatory (WENO) method, its theoretical foundation is much more well understood given its finite element roots. In addition, the DG method contains clean convergence (e.g. superconvergence) properties that other high-order methods do not have. The focus of this seminar is to present the minimal details necessary to implement the DG method for simple problems such as 1D advection and Burgers equation. Depending on interest, future seminars can be redirected to a number of different areas including time stepping options, limiters, shallow water or plasma applications, superconvergence properties, as well as elliptic or parabolic solvers that could include an introduction to the hybridized DG (HDG) method.
  • Oct
    27
  • High order penalty methods: a Fourier approach to solving PDE's on
    David Shirokoff
    Location: CH351
    Time: 12:00 PM

    View Abstract

    Penalty methods offer an attractive approach for solving partial differential equations (PDEs) on domains with curved or moving boundaries. In this approach, one does not enforce the PDE boundary conditions directly, but rather solves the PDE in a larger domain with a suitable source or penalty term. The new penalized PDE is then attractive to solve since one no longer needs to actively enforce the boundary conditions. Despite the simplicity, these methods have suffered from poor convergence rates which limit the accuracy of any numerical scheme (usually to first order at best). In this talk I will show how to systematically construct a new class of penalization terms which improve the convergence rates of the penalized PDE, thereby allowing for higher order numerical schemes. I will also show that the new penalized PDE has the added advantage of being solved in a straightforward manner using Fourier spectral methods. Finally, I demonstrate that the method is very general and works for elliptic (Poisson), parabolic (heat), and hyperbolic (wave) equations and can be applied to practical problems involving the incompressible Navier-Stokes equations and Maxwell’s equations.
  • Oct
    16
  • Wall to wall optimal transport
    Charles Doering
    University of Michigan
    Time: 12:00 PM

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    How much stuff can be transported by an incompressible flow containing a specified amount of kinetic energy or enstrophy? We study this problem for steady 2D flows focusing on passive tracer transport between two parallel impermeable walls, employing the calculus of variations to find divergence-free velocity field with a given intensity budget that maximize transport between the walls. The maximizing velocity fields, i.e. the optimal flows, consist of arrays of (convection-like) cells. Results are reported in terms of the Nusselt number Nu, the convective enhancement of transport normalized by the flow-free diffusive transport, and the Péclect number Pe, the dimensionless gauge of the strength of the flow. For both energy and enstrophy constraints we find that as Pe increases, the maximum transport is achieved by cells of decreasing aspect ratio. For each of the two flow intensity constraints, we also consider buoyancy-driven flows the same constraint to see how scalings for transport reported in the literature compare with the absolute upper bounds. This work provides new insight into both steady optimal transport and turbulent transport, an increasingly lively area of research in geophysical, astrophysical, and engineering fluid dynamics. This work is joint with Gregory P. Chini (New Hampshire), Pedram Hassanzadeh (Harvard) and Andre Sousa (Michigan).
  • Oct
    13
  • Simulating and predicting statistical features of crystal growth in polycrystalline materials
    Maria Emelianenko
    George Mason University

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    Microstructure of polycrystalline materials undergoes a process referred to as coarsening (or grain growth), i.e. elimination of energetically unfavorable crystals by means of a sequence of network transformations, including continuous expansion and instantaneous topological transitions, when the material is subjected to heating. This talk will be focused on recent advances related to the mathematical modeling of this process. Two different strategies will be discussed, one describing the evolution of individual crystals in a 2-dimensional system via a vertex model focused on triple junction dynamics, and one providing a kinetic description for the evolution of probability density functions. Numerical characteristics and predictions obtained by both strategies will be discussed and contrasted.
  • Sep
    22
  • The design of shell structures in the built environment
    Samar Malek
    Department of Mechanical Engineering USNA

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    In the built environment, shell structures are used to span large areas like airports or stadiums because they require little or no intermediate support. Shells can be constructed as a continuous surface or from discrete elements following that surface, in which case they are called gridshells. In architecture there is a trend to design shell structures as more organic forms by imitating nature. As a structural engineer, part of the challenge is how to incorporate structural principles into the designs. And how to computationally generate such forms - how do you create a manufactured object which is not just a copy or pastiche of the natural form? This talk will present computational techniques to develop complex curved structural surfaces in the form of mathematics, computer algorithms and design case studies while maintaining an emphasis on practical applications to current engineering problems. This talk would be of interest to engineers, mathematicians and fans of architecture.
  • Sep
    08
  • Bipartite Community Detection
    Kelly Yancey
    University of Maryland

    View Abstract

    Community detection in data is a large and ongoing area of research. A community in a graph is a vertex set S such that there are many edges between the vertices of S. Recently the combinatorial Laplacian and the normalized Laplacian of a graph have been used to describe the community structure of the graph. Specifically, analyzing the smallest eigenpairs can be used to find a set of good communities. In this talk we are specifically interested in bipartite community detection, that is we are interested in finding two subsets of the graph S and S' where the number of edges between S and S' is significantly more than expected. This type of community detection has already been implemented in studying protein interactions. We will present the algorithm for detecting bipartite communities. We will also discuss the limits of the algorithm for finding bipartite communities. Specifically, we will discuss why one of these bounds is sharp. These graphs are also of independent interest as there construction has applications to such fields as coding theory.
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