Research Activities > Seminars > Fall 2008

Fall 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:
  • Refreshments will be served after the talk
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  • September 3

    2.00 PM,
    4122 CSIC Bldg
    Professor Tom Hou, Department of Computational and Applied Mathematics, CalTech

    On the Stabilizing Effect of Convection in 3D Incompressible Flows

    Convection and incompressibility are two important characteristics of incompressible Euler or Navier-Stokes equations. In 3D flows, the convection term is responsible for generating the vortex stretching term, which leads to large growth of vorticity and possibly a finite time blowup of the solution. Here we reveal a surprising nonlinear stabilizing effect that the convection term plays in regularizing the solution. We demonstrate this by constructing a new 3D model which is derived for axisymmetric flows with swirl using a set of new variables. The only difference between our 3D model and the reformulated Navier-Stokes equations in terms of these new variables is that we neglect the convection term in the model. If we add the convection term back to the model, we will recover the full Navier-Stokes equations. This model preserves almost all the properties of the full 3D Euler or Navier-Stokes equations. In particular, the strong solution of the model satisfies an energy identity similar to that of the full 3D Navier-Stokes equations. We prove a non-blowup criterion of Beale-Kato-Majda type as well as a non-blowup criterion of Prodi-Serrin type for the model.
    Moreover, we prove a new partial regularity result for the model which is an analogue of the Caffarelli-Kohn-Nirenberg theory for the full Navier-Stokes equations.

    Despite the striking similarity at the theoretical level between our model and the Navier-Stokes equations, the former has a completely different behavior from the full Navier-Stokes equations. We will present convincing numerical evidence which seems to support that the 3D model develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new 3D model and how the convection term in the full Euler and Navier-Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time.
    September 10

    2.00 PM,
    4122 CSIC Bldg
    Professor Michael Shelley, the Courant Institute, New York University

    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.
    September 11

    3:30 PM,
    3206 Math Bldg (note special time and place)
    Professor Kevin Zumbrun, Department of Mathematics, Indiana University

    Joint Seminar with Department of Mathematics

    Stability of Strong Viscous Shock Layers in an Ideal Gas

    By a combination of asymptotic ODE estimates and numberical Evans function computations, we examine the spectral stability of shock-wave solutions of the compressible Navier--Stokes equations with ideal gas equation of state, for arbitrary strength waves.

    Our main results are that, in appropriately rescaled coordinates, the Evans function associated with the linearized operator about the wave, an analytic function analogous to the characteristic polynomial whose zeros correspond to eigenvalues of L, (i) converges in the strong shock limit to the Evans function for a limiting shock profile of the same equations, for which internal energy vanishes at one endstate; and (ii) has no unstable (positive real part) zeros outside a uniform ball. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with the (nonphysical) limiting case, form a compact family of boundary-value problems that may be conveniently studied numerically. An intensive numerical study then yields unconditional stability, independent of amplitude, for a range of parameter values including all common gases.

    Besides its physical interest, we believe that this analysis has interest as an example where it is possible to carry out a rigorous globl stability analysis by numerical techniques, the obvious obstace being the need to treat an unbounded parameter range using finitely many operations.

    September 17

    2.00 PM,
    4122 CSIC Bldg
    Professor Gadi Fibich, Department of Applied Mathematics, Tel-Aviv University

    Applied Math Approach to Auction Theory

    The study of auctions began with Vickry in the 1961. It is nowadays a very active research area, driven by the huge popularity of auctions as "efficient", "unbiased" selling mechanisms. In this talk I will give a brief introduction to auction theory, and then show some applications of applied math techniques to problems in auction theory, such as an extension of the revenue equivalence theorem to the case of asymmetric auctions, and the effect of risk-aversion and asymmetry in large auctions.

    Joint work with Arieh Gavious and Aner Sela
    September 24

    2.00 PM,
    4122 CSIC Bldg
    Dr. Yonatan Sivan, School of Physics and Astronomy, Tel-Aviv University

    Qualitative and Quantitative Analysis of Stability and Instability Dynamics of Positive Lattice Solitons

    We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multi-dimensional focusing nonlinear media with a potential (lattice), which can be periodic, periodic with defects, quasi-periodic, single waveguide, etc. We show that when the solitons are unstable, the type of instability dynamic that develops depends on which of two stability conditions is violated. Specifically, violation of the slope condition leads to a focusing instability, whereas violation of the spectral condition leads to a drift instability.
    We also present a quantitative approach that allows to predict the stability and instability strength.
    October 1

    2.00 PM,
    4122 CSIC Bldg
    Professor Tony Chan, National Science Foundation & Department of Mathematics, UCLA

    Images, PDEs and Wavelets

    Wavelets and PDEs have had profound impacts on imaging sciences. Their successes rely on their remarkable mathematical properties, many of which are complementary to each other. In this talk, I will present an overview of our work along the direction of merging them to further improve the performance, as well as to model new applications in image processing. A main goal is to handle sharp discontinuities stably and robustly. The main approach combines regularity control using PDEs while manipulating coefficients in wavelet space. Applications include image compression, denoising, and wavelet inpainting. Connections with compressed sensing will be made.

    Based on Joint work with Hao-Min Zhou at Georgia Tech and Jackie Shen at Barclays
    October 8

    2.00 PM,
    4122 CSIC Bldg

    No Seminar

    October 15

    2.00 PM,
    4122 CSIC Bldg
    Professor Hailiang Liu, Department of Mathematics, Iowa State University

    Alternating Evolution, Flux Refinement, and the Level Set Method

    High resolution computation of convection, diffusion and dispersion is important in many applied PDEs, ranging from the Euler, Navier-Stokes to Schrödinger equations. In this talk I shall present some recent results on numerical methods for problems involving these terms, including:

    i) the alternating evolution (AE) method for convection -- based on sampling of a refined description of the underlying equation on alternative grids,

    ii) the direct discontinuous Galerkin(DDG) method for diffusion -- based on a novel numerical flux formula for the solution gradient, and

    iii) the level set method for capturing zero dispersive limits.

    For each method I shall highlight the essential step --- a step in which the `physics' is incorporated into the method via a pre-refinement of the model. The discretization of the refined one is then purely of numerical nature. Some numerical results will be presented to show the quality of these methods.
    October 22

    2.00 PM,
    4122 CSIC Bldg
    Professor Selim Esedoglu, Department of Mathematics, University of Michigan

    New Algorithms for Multi-phase Flow and High Order Geometric Motions

    Threshold dynamics, also called diffusion generated motion, of Merriman, Bence, and Osher generates the motion by mean curvature of an interface by alternating two very simple and computationally efficient operations: Convolution and thresholding. I will describe new variants that generate high order geometric motions (such as motion by surface diffusion) and how to improve the accuracy of the method on uniform grids. Applications include problems such as inpainting from image processing and the simulation of grain boundary motion in polycrystalline materials with many grains.

    The talk is based on joint works with Steve Ruuth and Richard Tsai and, separately, Matt Elsey and Peter Smereka.
    October 29

    2.00 PM,
    4122 CSIC Bldg
    Professor Jonathan Sherratt,Department of Mathematics, Heriot-Watt University, Edinburgh

    Nonlinear Dynamics and Pattern Bifurcations in a Model for Vegetation Stripes in Arid Environments

    In many semi-arid environments, vegetation is self-organised into spatial patterns. The most striking examples of this are on gentle slopes, where striped patterns are typical, running parallel to the contours. I will discuss pattern solutions of a mathematical model for this phenomenon, of reaction-diffusion-advection type. I will describe the use of numerical bifurcation methods of both the pattern odes and a discretised version of the model pdes. I will show that patterns exist for a wide range of rainfall levels. Moreover, for many rainfall levels, patterns with a variety of different wavelengths are stable, with mode selection dependent on initial conditions. This raises the possibility of hysteresis, and I will present numerical solutions of the model which show that pattern selection depends on rainfall history in a relatively simple way.
    November 5

    2.00 PM,
    4122 CSIC Bldg
    Dr. Burkhard Zink, Center for Computation & Technology, Louisiana State University

    Numerical Techniques for Accretion Flows around Black Holes

    I will discuss a research program to study the infall of stellar material into black holes, which is an important process used to explain a large class of energetic phenomena in astrophysics. To study these flows, three-dimensional general relativistic simulations of magnetohydrodynamics are needed, as well as a full treatment of Einstein's field equations. In this talk I will touch on three aspects of this program: First, the application of multi-block techniques to use adapted grid systems when modeling black holes and accretion disks. Secondly, I will describe recent efforts to develop schemes for radiation transport in these systems. And finally, I will share my perspective on the future role of many-core and GPU computing in astrophysics.

    November 7

    10:00am - 3:15pm
    4122 CSIC Bldg
    Special Event: CSCAMM Faculty & Post-Docs

    CSCAMM Research Overview

    Click here for list of talks
    November 12

    2.00 PM,
    4122 CSIC Bldg
    Professor Fabian Waleffe, Department of Mathematics, University of Wisconsin

    What is Turbulence?

    The Navier-Stokes equations describing fluid flows are notorioulsy difficult to solve, even on today's supercomputers. This is because common flows are typically turbulent: they develop a complex spatio-temporal structure with a broad range of interacting scales.

    In the Prandtl-von Karman view, turbulence is a random collision of `eddies,' akin to the random collision of molecules in a gas, leading to enhanced transport: increased friction, increased heat flux, increased mixing.

    In the Richardson-Kolmogorov view, turbulence is a cascade of energy from large to small scales, with a progressive loss of information about the large scales, the geometry and the nature of the forcing that drives the flow. That energy cascade picture is similar to the degradation of coherent large scale mechanical energy into incoherent small scale, thermal energy. For many decades, experiments and simulations in simple but realistic geometries have suggested a more coherent view of turbulence.

    These observations have inspired a search for coherent solutions of the Navier-Stokes equations: steady states, traveling waves and periodic solutions. Many such solutions have now been found, opening the way for a new attack on the turbulence problem.
    November 19

    2.00 PM,
    4122 CSIC Bldg
    Professor Tim Delsole, Department of Atmospheric, Oceanic, and Earth Sciences, George Mason University and Center for Ocean-Land-Atmosphere Studies (COLA)

    Accounting for Model Error in the Ensemble Kalman Filter

    The ensemble Kalman Filter has emerged as a powerful tool for incorporating realistic flow-dependent error statistics in assimilating observations into nonlinear forecast models. However, the Kalman Filter effectively assumes that the underlying forecast model is perfect, in the sense that it gives consistent estimates of forecast uncertainty.

    In practice, this assumption is rarely satisfied, and consequently modern applications include essentially ad hoc modifications, such as covariance localization and covariance inflation, to account for incorrect specification of forecast uncertainty.

    In this talk, I discuss three alternative approaches to dealing with imperfect models in the Kalman Filter. The first approach is a new type of filter called the Diffuse Ensemble Square Root Filter (DESRF). Traditional ensemble filters update the first guess and the forecast covariance only in the space spanned by the forecast ensemble. In other words, the space orthogonal to the ensemble is not modified by the filter, which is tantamount to assuming that the uncertainty in this space vanishes-- a highly unrealistic assumption. In Bayesian theory, the lack of prior information often is modeled by a distribution with arbitrarily large covariance matrix. Such a distribution is called a non-informative prior, or a diffuse prior. We discuss how this diffuse prior can be included in the ensemble filter. The second approach is to estimate model parameters simultaneously with the state variable. This combined state-parameter estimation approach has been shown to produce reasonable estimates, but can be problematic if the parameters are multiplicative with respect to the state variables. In the latter case, the filter may inadvertently modify the parameter to produce an unstable dynamical model, after which point the filter blows up. We suggest an approach for prevent such blow up.

    Finally, we discuss how the Ensemble Filter can be used to tune dynamical models that contain stochastic parameterizations. We argue that this latter problem cannot be solved by traditional parameter estimation methods, such as augmentation of the state vector, and discuss a promising new approach to this problem.
    November 26

    2.00 PM,
    4122 CSIC Bldg

    No Seminar, Thanksgiving Break
    December 3

    2.00 PM,
    4122 CSIC Bldg
    Dr. Silviu Niculescu, CNRS, France

    Delays in Interconnected Systems. Stability Analysis, Algorithms and Applications.

    It is well-known that the interconnection of two or more dynamical systems leads to
    an increasing complexity of the overall system’s behavior due to the effects induced by
    the emerging dynamics (in the presence or not of feedback loops) in strong interactions
    (sensing, communication) with the environment changes. One of the major problems
    appearing in such interconnection schemes is related to the propagation, transport, and
    communication delays acting ”through” and ”inside” the interconnections.

    The aim of this talk is to present briefly some ”user-friendly” methods and techniques
    (frequency-domain approaches) for the analysis and control of the dynamical
    systems in presence of delays. The presentation is as simple as possible, focusing
    on the main intuitive (algebraic, geometric) ideas to develop theoretical results, and
    their potential use in practical applications. Single and multiple delays will be both
    considered. Classical control schemes (as, for example, the well-known PI and PID
    controllers, or Smith predictors) will be also revisited.

    The talk ends with the analysis of some control schemes used in the motion synchronization in shared virtual environments.
    The presentation is mainly based on [1, 2] (and the references therein).


    [1] W. Michiels and S.-I. Niculescu: Stability and stabilization of time-delay systems.
    An eigenvalue based approach (SIAM: Philadelphia, 2007, to appear).
    [2] S.-I. Niculescu: Delay Effects on Stability: A Robust Control Approach (Springer-
    Verlag: Heidelberg, LNCIS, vol. 269, 2001).
    December 10

    2.00 PM,
    4122 CSIC Bldg
    Professor Dionisios Margetis, Department of Mathematics, University of Maryland

    On Kinetic Descriptions of Crystal Surface Evolution

    I will present recent progress on applying concepts of kinetic theory to the relaxation of crystal surfaces. The major goal is to bridge analytically two length scales:

    (i) the nanoscale, where line defects (``steps'') are evident; and

    (ii) the macroscale, where nonlinear PDEs for the height profile are applied.

    The starting point are coupled differential equations for the positions of steps. I will focus on two main results:

    (a) For one-dimensional morphologies, the motion of steps is described in terms of BBGKY hierarchies for correlation functions. I will show how these hierarchies connect to the continuum limit, i.e., a single PDE for the surface height.

    (b) For geometries with rotational symmetry, the parabolic PDE for the surface height approximately reduces to a hyperbolic PDE in Lagrangian coordinates. This PDE predicts shock-wave type solutions associated with moving boundaries of flat surface regions (facets). This formulation yields a nontrivial effect that the finite size has on the evolution of nanostructures.
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