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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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).

References

[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).

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.