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Research Activities > Seminars > Spring 2008

Spring 2008 Seminars

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  • All talks are in the CSIC Bldg (#406) Room 4122 at 2.00pm (unless otherwise stated)
  • Directions can be found at: home.cscamm.umd.edu/directions
  • Refreshments will be served after the talk
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  • January 30

    4122 CSIC Bldg
    Professor Ellad Tadmor, Department of Aerospace Engineering and Mechanics, University of Minnesota

    A Quasicontinuum for Multilattice Phase Transforming Materials

    The quasicontinuum (QC) method is applied to materials possessing a
    multilattice crystal structure. Cauchy-Born (CB) kinematics, which accounts for the shifts of the crystal basis, is used in continuum regions to relate atomic motions to continuum deformation gradients. To avoid failures of the CB kinematics, QC is augmented with a phonon stability analysis that detects lattice period extensions and identifies the minimum required periodic cell size. This augmented approach is referred to as "Cascading Cauchy-Born kinematics". Applications of this methodology to one-dimensional test problems that highlight the salient features will be presented. Extension to higher dimensions is straightforward and is currently being pursued. Some preliminary results will be shown.

    February 6

    4122 CSIC Bldg
    Professor Alina Chertock, Department of Mathematics, North Carolina State University

    A Positivity Preserving Central-Upwind Scheme for Chemotaxis and Haptotaxis Models

    In this talk, I will present a new finite-volume method for a class of chemotaxis models and for a closely related haptotaxis model. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. The first step in the derivation of the new method is made by adding an
    equation for the chemoattractant concentration gradient to the original system. We then show that the convective part of the resulting system is typically of a mixed hyperbolic-elliptic type and therefore straightforward numerical methods for the studied system may be unstable.

    The new method is based on the application of the second-order central-upwind scheme, originally developed for hyperbolic systems of conservation laws, to the extended system of PDEs. We show that the proposed central-upwind scheme is positivity preserving, which is a very important stability property of the method. The scheme is applied to a number of two-dimensional problems including the most commonly used Keller-Segel chemotaxis model and its modern extensions as well as to a haptotaxis system modeling tumor invasion into surrounding healthy tissue. Our numerical results demonstrate high accuracy, stability, and robustness of the proposed scheme.

    This is a joint work with A. Kurganov, Tulane University.

    February 13

    4122 CSIC Bldg
    Professor Fadil Santosa, School of Mathematics and Institute for Mathematics and its Applications, University of Minnesota

    Modeling and Design of Optical Resonators

    We consider resonance phenomena for the scalar wave in an inhomogeneous medium. Resonance can be described as a solution to to the wave equation that is spatially localized while its time dependence is (mostly) harmonic except for decay due to radiation. The reciprocal of the decay rate is referred to as the quality of the resonator. We will discuss the problem of modeling resonators and propose a method for designing resonators which has high quality. We will start with an introduction to resonance and its computation, and describe a continuation approach that allows one to create a medium whose resonance has a high quality. Numerical examples which illustrate our method will be presented.

    February 20

    4122 CSIC Bldg
    Professor Jinchao Xu, Department of Mathematics, Penn State University

    Numerical Methods for High Order and Coupled PDE Systems

    In this talk, I will present some new numerical solution techniques for high order and coupled PDE Systems. I will first discuss some relevant numerical difficulty and subtlety for this type of problems, including the potential risk of reducing a higher order PDEs into a system of lower order PDEs. I will then report a family of finite element methods for three types of problems:
    (1) any 2m-th order elliptic boundary value problems in $R^n$ for any $1\le m\le n$
    (2) a 4-th order PDEs in terms curl operator (arising from magneto-hydrodynamics modeling) in $R^3$
    (3) Stokes-Darcy-Brinkman systems for coupled fluids and porpus-media (arising from fuel-cell and subsurface flow modeling).

    If time allows, I will also discuss algebraic techniques for solving the discretized systems.

    February 27

    4122 CSIC Bldg
    Professor Markus Püschel, Department of Electrical and Computer Engineering, Carnegie Mellon University

    Linear Transforms: Theory and Automatic Implementation

    Linear transforms, such as the discrete Fourier transform, discrete cosine transforms, and many others, are among the most important numerical kernels used in signal processing and many other disciplines.

    Most transforms possess a surprising number of fast algorithms, a fact established by the more than hundred publications on this topic. However, with few exceptions, the algorithms are derived through ingenious manipulation of the transform coefficients—a method that gives no insight into existence or structure of the algorithm. In the first part of this talk we sketch an algebraic theory of transform algorithms, which solves this problem for many transforms and also enables the derivation of many new algorithms, which could not be found with previous methods. (More info: http://www.ece.cmu.edu/~smart)

    In the second part of the talk we focus on the efficient implementation of transforms, which is a difficult problem on fast-changing and increasingly complex and parallel computing platforms. We present Spiral, a system that overcomes this problem by automatically generating highly optimized code directly from a problem specification. Optimization includes vectorization and parallelization for multicore platform. The performance of the generated code competes with and sometimes outperforms the best handwritten libraries. We show that Spiral’s framework extends beyond transforms and may serve as a prototype on how to teach computers to write fast numerical libraries. (More info: http://www.spiral.net)

    March 5

    4122 CSIC Bldg

    March 12

    4122 CSIC Bldg
    Professor Saswata Hier-Majumder, Department of Geology, University of Maryland

    Cross-scale Modeling in Magma Dynamics

    Thermal and chemical evolution of planetary bodies involves migration of magmatic melts through a viscous, rocky matrix. Magma migration and storage involves a set of coupled processes operating at different length scales. On the one hand, efficiency of melt extraction, thickness of melt-rich layers, distribution of radioactive elements within the mantle, and formation of oceanic crust are processes that are deeply related to magma migration over length scales of hundreds to thousands of kilometers. On the other hand, effective physical properties such as permeability, total interfacial tension, viscosity and elastic moduli are deeply influenced by the microstructure of the partially molten aggregates. Coupled motion of the viscous rocky matrix and the magmatic melt over large length scales is modulated by the physical properties arising from the microstructure of the rock. A robust description of magma migration and planetary evolution thus requires coupled modeling of microstructure in the grain scale as well as melt migration modeling in the planetary scale. In this talk, I will present some recent results from both scales of modeling, and some directions for future work.

    March 19

    March 26

    4122 CSIC Bldg
    Dr. Jim Purser, National Centers for Environmental Prediction (NCEP)
    Presentation Slides

    Discrete and Differential Geometry applied to the Efficient Numerical Synthesis of Spatially Adaptive Covariances

    The assimilation of spatially distributed meteorological data into the gridded form needed to initialize the time integration of a numerical forecasting model involves several specialized statistical and numerical procedures. These are designed to exploit what is known about the distribution of errors in the pre-existing best estimate, or ‘background’ field of dynamical variables. Of particular importance is the synthesis of the errors’ covariances, modeled as spatially adaptive filtering operators that act upon appropriate adjoints of the field in question.

    Recent developments illustrate the considerable value to be gained by employing advanced methods of both discrete and continuous geometry in the efficient synthesis of the covariance operators. The discreteness involves the systematic search for special sets of generalized lines threading the computational lattice. These line sets are suitable for the application of low-pass line filters which, when applied sequentially, will reproduce the desired degree of local anisotropy and coherence scale. The normalization of the resulting synthetic covariances is achieved by asymptotic methods derived from the tools provided by non-Euclidean geometry, with the scale and anisotropy of spatial covariance taken as an effective Riemannian metric in which the collective application of the aforementioned filters is interpreted as the action of a constant isotropic diffusion. The talk will emphasize the fundamental role played by symmetry in successfully orchestrating these geometrical ideas.

    April 2

    Professor Michael J. Shelley, Applied Mathematics Lab, The Courant Institute

    Dynamics and Transport in Active Suspensions

    Fluids with suspended micro-structure -- complex fluids -- arise commonly in micro- and bio-fluidics, and can have fascinating and novel dynamical behaviors. I will discuss some interesting examples of this, but will concentrate on my recent work on "active suspensions", motivated by recent experiments of Goldstein, Kessler, and their collaborators, on bacterial baths. Using large-scale particle-based simulations of hydrodynamically interacting swimmers, as well as a recently developed kinetic theory, I will investigate how hydrodynamically mediated interactions lead to large-scale instability, coherent structures, and mixing.

    April 9

    4122 CSIC Bldg
    Professor Alexander Gorban, Applied Mathematics, University of Leicester

    Limiters in lattice Boltzmann methods

    The lattice Boltzmann method has been proposed as a discretization of Boltzmann's kinetic equation and is now in wide use in fluid dynamics and beyond. Instead of fields of moments (hydrodynamic fields), the lattice Boltzmann method operates with fields of discrete distributions. This allows us to construct very simple limiters that do not depend on slopes or gradients.

    We construct a system of nonequilibrium entropy limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious oscillations without blurring of shocks, and do not affect smooth solutions. In general, they do the same work for LBM as flux limiters do for finite differences, finite volumes and finite elements methods, but for LBM the main idea behind the construction of nonequilibrium entropy limiter schemes is to transform a field of a scalar quantity - nonequilibrium entropy.

    There are two families of limiters: (i) based on restriction of nonequilibrium entropy
    (entropy "trimming") and (ii) based on filtering of nonequilibrium entropy (entropy filtering).

    The physical properties of LBM provide some additional benefits: the control of entropy production and accurate estimate of introduced artificial dissipation are possible.

    The constructed limiters are tested on classical numerical examples: 1D athermal shock tubes with an initial density ratio 1:2 and the 2D lid-driven cavity for Reynolds numbers Re between 2000 and 7500 on a coarse 100*100 grid. All limiter constructions are applicable both for entropic and for non-entropic equilibria.

    Joint work with R. A. Brownlee and J. Levesley (Leicester)

    April 16

    4122 CSIC Bldg
    Professor James M. Greenberg, Department of Mathematics, Carnegie Mellon University

    Two-Dimensional Sloshing Flows for the Shallow-Water Equations

    In this talk I’ll discuss “sloshing” 2-Dimensional flows for the “shallow-water” equations. The model describes the motion of a finite volume of viscous fluid taking place in container whose bottom is described by a paraboloidal like surface of the form
    z = (αx2 +βy2)/2,  α> 0,  β> 0 , or more generally, z = a(x,y) where a-> as (x2 +y2) ->∞. The model includes gravity, Coriolis, and viscous forces.  

    April 23

    4122 CSIC Bldg
    Professor Irene Fonseca, Department of Mathematical Sciences, Carnegie Mellon University

    Variational Methods in Materials and Imaging

    Several questions in applied analysis motivated by issues in computer vision, physics, materials sciences and other areas of engineering may be treated variationally leading to higher order problems and to models involving lower dimension density measures. Their study often requires state-of-the-art techniques, new ideas, and the introduction of innovative tools in partial differential equations, geometric measure theory, and the calculus of variations. In this talk it will be shown how some of these questions may be reduced to well understood first order problems, while in others the higher order plays a fundamental role.

    Applications to phase transitions, to the equilibrium of foams under the action of surfactants, imaging, micromagnetics and thin films will be addressed.

    April 30

    4122 CSIC Bldg
    Professor David Gottlieb, Division of Applied Mathematics, Brown University

    A Modified Optimal Prediction Method and Application to the Particle Method

    In the numerical solution of nonlinear PDE's there are always small scales that can not be resolved. Often the small scales themselves are not important but their influence on the large scales is crucial.

    Chorin suggested the modified prediction method as a tool to statistically model the impact of the small scales. In this talk a different variant of the optimal prediction method will be presented.

    Also we introduce and compare several approximations of this method. We apply the original and modified optimal prediction methods to a system of ODEs obtained from a particle method discretization of a hyperbolic PDE and demonstrate their performance in a number of numerical experiments.

    This is a joint work with Alina Chertock and Alex Solomonoff.

    May 7

    4122 CSIC Bldg
    Professor Lisa Fauci, Department of Mathematics, Tulane University

    Interaction of Elastic Biological Structures with Complex Fluids

    The bio-fluid-dynamics of reproduction provide wonderful examples of fluid-structure interactions. Peristaltic pumping by wave-like muscular contractions is a fundamental mechanism for ovum transport in the oviduct and uterus.

    While peristaltic pumping of a Newtonian fluid is well understood, in many important applications the fluids have non-Newtonian responses. Similarly, mammalian spermatozoa encounter complex, non-Newtonian fluid environments as they make their way through the female reproductive tract. The beat form realized by the flagellum varies tremendously along this journey. We will present recent progress on the development of computational models of pumping and swimming in a complex fluid. An immersed boundary framework is used, with the complex fluid represented either by a continuum Oldroyd-B model, or a Newtonian fluid overlaid with discrete visco-elastic elements.

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