Jing Zou, Center for Scientific Computation and Mathematical Modeling, University of Maryland

Super Fast Fourier Transform

The superlinear O(N log N) time requirement is the bottleneck for the Fast Fourier
Transform (FFT) to process huge amount of data. We analyze a sublinear RAlSFA
(Randomized Algorithm for Sparse Fourier Analysis) that finds a near-optimal
B-term Sparse Representation R for a given discrete signal S of length N, in
time and space poly(B,log(N)) (instead of O(N logN)). A straightforward
implementation of the RAlSFA, as presented in the theoretical paper by
Gilbert, Guha, Indyk, Muthukrishnan and Strauss, turns out to be very slow in
practice. Our main result is a greatly improved and practical RAlSFA (more
than a factor of 4000 times faster than original algorithm!). It beats the
FFTW for reasonably large N. We also extend the algorithm to higher dimensional
cases. The crossover point lies at N~70000 in one dimension, and at N~ 900 for
data on a N*N grid in two dimensions for small B signals. We also find this
algorithm is very robust to the noise.

Dr. Tomoya Tatsuno, CSCAMM, University of Maryland

Degenerate Continuous Spectra and
Secular Behavior: Linear Dynamics in Fluids and
Plasmas with Sheared Flow

Shear flow in fluids and plasmas brings about non-Hermiticity in the linear dynamics of fluctuations. As a consequence, a degeneracy of continuous spectra may lead to the algebraic growth of perturbation (secularity) even if all eigenvalues of generator are real. We will show that Rayleigh equation (governing equation for Kelvin-Helmholtz instability) with piecewise linear shear-flow profile contains a ``resonance' (frequency overlapping or degenerate frequency) between surface wave and ballistic response when the system is stable for Kelvin-Helmholtz mode. Normally, perturbed field may be stable in spite of the resonance, while it must be retained in the rigorous expression of the generator. When plasma oscillation (or internal gravity wave) is coupled with Rayleigh equation, another ballistic mode couples with the resonance and the perturbed density (or vorticity) asymptotically shows locally secular behavior. The secularity is decelerated to linear growth in time while the degeneracy of the spectra looks third order between single point and two continua. It is also shown by renormalization technique that the deceleration is attributed to the continuum damping of the point spectra.

Allen Tannenbaum, Departments of
Electrical & Computer and Biomedical
Engineering,
Georgia Institute of Technology

Geometric Registration Methods in Medical Imaging

In this talk, we will describe some key
problems in medical imaging especially
registration and surface warping. This will be
based on the Monge-Kantorovich theory of optimal
mass transport. Further, many of the algorithms
in medical imaging are based on curvature driven
flows implemented via level set methods. We will
describe a new stochastic interpretation of such
flows based on the theory of hydrodynamic
limits. We will try to make the talk accessible
to a broad audience with an interest in
mathematical medical imaging. All of the methods
will be illustrated with real image data.

Prof. Chi-Wang Shu, Division of Applied
Mathematics, Brown University

Anti-diffusive High Order Weighted Essentially Non-Oscillatory
Schemes for Sharpening Contact Discontinuities

In this talk we will first describe the general framework of
high order weighted essentially non-oscillatory (WENO) finite
difference schemes for solving hyperbolic conservation laws and
in general convection dominated partial differential equations.
We will then discuss our recent effort in designing anti-diffusive
flux corrections for these high order WENO schemes. The objective
is to obtain sharp resolution for contact discontinuities, close
to the quality of discrete traveling waves which do not smear
progressively for longer time, while maintaining high order
accuracy in smooth regions and non-oscillatory property for
discontinuities. Numerical examples for one and two space
dimensional scalar problems and systems demonstrate the good
quality of this flux correction. High order accuracy is maintained
and contact discontinuities are sharpened significantly compared
with the original WENO schemes on the same meshes. We will also
report the extension of this technique to solve Hamilton-Jacobi
equations to obtain sharp resolution for kinks, which are derivative
discontinuities in the viscosity solutions of Hamilton-Jacobi
equations. This is joint work with Zhengfu Xu.

Beginning from a kinetic description for a fully ionized
Hydrogen plasma with a simple collisional model, fluid approximations are
derived in a scaling where the gyrofrequency are comparable to the
collision frequency. One obtains strongly anisotropic viscous and thermal
transport, and anisotropic dispersive corrections to classical MFD models.
Moreover, these approximations respect entropy relations.

Michael Brenner, Division of Engineering and Applied Sciences, Harvard University

Drying Mediated Self Assembly

Developing reliable strategies for controlling the assembly of small objects
into functional structures is of great interest. We will focus on our recent
efforts in this area and focus on a particularly beautiful example of V. Manahoran
and D. pine in which small spheres are assembled into precise configurations through
drying of interstital fluid. For a given number of spheres the structures that formed
form uniquely corresponded to the sphere packing that minimized the second moment of
the particle distribution. I will focus on our efforts to understand these results,
focusing on the important role of geometrical constraints.

The theory of fracture usually referred to as that of Griffith shows many drawbacks: it does not initiate cracks; it is powerless when trying to predict the crack path; it does not know how to handle sudden crack jumps, ..... Jean-Jacques Marigo and I have proposed a model based on energy minimization which does away with many of those obstacles, while departing as little as feasible from Grifffith's theory.

I will first describe the proposed model, show how it does away with the above mentioned drawbacks and evoke its specific shortcomings.

From a mathematical stanpoint, the model resembles a kind of evolutionary image segmentation problem in the sense of Mumford & Shah. Chris Larsen and I have shown the existence of a solution to the evolution for the weak -à la De Giorgi - formulation of the problem. I will briefly describe the result, the method that was used in the proof and also mention the non-trivial extensions to the case of a non-convex bulk energy, obtained in collaboration with Gianni Dal Maso and Rodica Toader.

The model is readily amenable to numerics through various regularization of the energy which Gamma-converge to the original energy. This is the work of Blaise Bourdin ( partly in collaboration with Antonin Chambolle). For lack of time, I will not discuss numerical issues, which are fascinating, but merely illustrate the talk with Bourdin's latest 2 and 3-d. computations.

If time permits, I will show how the adoption of a cohesive surface energy and of a more lenient minimization criterion cures the shortcomings of the variational Griffith model and even bridges the mysterious gap between fracture and fatigue. This is the path actively pursued by Jean-Jacques Marigo at the present time. The numerics and 1d-results are very promising but the mathematical results are yet very primitive.

Thierry Goudon, CNRS-Université des Sciences et Technologies de Lille

Modeling of fluid/particles interactions

We consider a cloud of particles, subject to a
friction force exerted by a
surrounding fluid. The evolution of the
particles is described by a distribution
function in phase space f(t, x, v) 0; the
evolution of which is governed by

δ_{t}f +
∇_{x}(vf) + δ_{v}(Ff) = Q(f)

The expression of the force term F(t,
x, v) is
given by

F(t, x, v) =
((6πμa)/M)
/(v − u(t, x))

where u(t, x) stands for the velocity field of
the fluid. This is the referred to as
the Stokes force. It is proportional to the
relative velocity (v − u), the proportionality
coefficient involving the viscosity μ > 0 of the
fluid, the radius a > 0
and the mass M of a particle: M = 4/3πa^{3} ρ_{P}, ρ_{P} being the mass density of the
particles.

We are concerned with two differents questions.
The former is concerned
with the effect of high variations of the
(given) fluid velocity, intended to mimic
turbulence effects. The latter is concerned with
coupled models involving evolution
equations for the fluid velocity; then, we
investigate various asymptotic
regimes, depending on the value of physical
constants.

This is a survey of joint works with Frédéric
Poupaud, Alexis Vasseur,
Pierre-Emmanuel Jabin, José-Antonio Carrillo
(ICREA-UAB, Barcelona).

Preconditioners for the Time-Harmonic Maxwell Equations in Mixed Form

We introduce a new preconditioning technique for iteratively solving
linear systems arising from finite element discretizations of the mixed
formulation of the time-harmonic Maxwell equations, with small wave
numbers. The preconditioners are based on discrete regularization, and are
motivated by spectral equivalence properties of the discrete operators. We
show that using the scalar Laplacian as a weight matrix for regularization
leads to an effective block diagonal preconditioner, for which fast
iterative solution methods can be applied. The main computational cost is
related to solving a linear system whose associated matrix is the discrete
curl-curl operator, shifted (approximately) by the vector mass matrix. The
analytical observations are accompanied by numerical results that
demonstrate the scalability of the technique.

I. Michael. Navon, C.S.I.T. and Department of Mathematics
Florida State University

On adjoint error correction and bounding using Largange form of truncation term

The refinement of quantities of interest
(goal or cost functionals) using adjoint (dual)
parameters and a residual is at present a well
established technology. The truncation error may
be estimated via the value of residual
engendered by the action of a differential
operator on some extrapolation of the numerical
solution. The adjoint approach allows accounting
for the impact of truncation error on a target
functional by summation over the entire
calculation domain.

Numerical tests demonstrate the efficiency of
this approach for smooth enough physical fields
(heat conduction equation and Parabolized
Navier-Stokes (PNS)). The impact of solution
smoothness on the error estimation is found to
be significant. However, the extension of this
approach to discontinuous field is also
feasible. We can handle the error of
discontinuous solution (Euler equations) using
the solution for viscous flow (PNS) as a
reference. The influence of viscous terms may be
accounted for using adjoint parameters. Results
of numerical tests demonstrate applicability of
this approach.

For error estimates we use numerical results
that are significantly less smooth then the
computed physical field. For non-monotonic
finite-difference schemes error bounds may be
too large. Thus, the applicability of method
considered above is restricted to numerical
schemes which do not exhibit nonphysical
oscillations. Finally applicability of the
approach to POD model reduction with dual
weighted residuals will be briefly addressed and
an implementation in adaptive mesh ocean
modelling.

Leila De Floriani, Department of Computer
Science, University of Maryland

Modeling Non-Manifold Multi-dimensional Shapes

We address the problem of representing and manipulating non-manifold multi-dimensional shapes described by simplicial complexes. Simplicial complexes are widely-used shape representations in a variety of applications, including computer graphics, solid modeling, terrain modeling, visualization of scalar and vector fields, finite element analysis. The major driving application for our work has been modeling finite element meshes generated from CAD models through an idealization process. In this talk, we present different representations for non-manifold shapes, described by simplicial complexes, that we have developed in our work.
We analyze and compare
such representations, based on their expressive power, on their storage requirements, and on their effectiveness in supporting navigation and update operations. We first discuss two compact and scalable representations specific for two- and three-dimensional simplicial complexes embedded
in the 3D Euclidean space, and a dimension-independent representation for simplicial complexes in arbitrary dimensions. We then present an approach to modeling non-manifold multi-dimensional shapes based on a unique decomposition of a shape into nearly manifold components. This decomposition is not only an effective model for developing compact and dimension-independent data structures for non-manifold multi-dimensional shapes, but it also provides a suitable tool for performing geometric reasoning on such shapes.

Doron Levy, Department of Mathematics at Stanford University

Post-Transplantation Dynamics of the Immune Response to Chronic Myelogenous Leukemia

We model the immune dynamics between T cells and cancer cells
in leukemia patients after bone marrow transplants.
Our approach incorporates time delays and accounts for the
progression of cells through different modes of behavior.
We explore possible mechanisms behind a successful cure, whether mediated
by a blood-restricted immune response or a cancer-specific
graft-versus-leukemia effect.
Characteristic features of this model include sustained proliferation of T cells
after initial stimulation, saturated T cell proliferation rate, and the possible
elimination of cancer cells, independent of fixed-point stability.
In addition, we use numerical simulations to examine the effects of varying
initial cell concentrations on the likelihood of a successful transplant.
Among the observed trends, we note that higher initial concentrations of
donor-derived, anti-host T cells slightly favor the chance of success,
while higher initial concentrations of general host blood cells more
significantly favor the chance of success. These observations lead to the
hypothesis that anti-host T cells benefit from stimulation by general
host blood cells, which induce them to proliferate to sufficient levels to
eliminate cancer. This is a joint work with R. DeConde, P. Kim, and P. Lee.

Prof. Sergiu Klainerman, Princeton University, Department of Mathematics

A Break-down criterion for the Einstein equations

I will present some recent work in collaboration with Igor Rodnianski
concerning a breakdown criterion in General Relativity. An
essential ingredient of the work is to get a lower bound on the radius of
injectivity of null hypersurfaces which are boundaries of future or past sets
of
points in Ricci flat Lorentzian metrics, This is used in conjunction with
a Kirchoff-Sobolev type formula in Lorentzian geometry.

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