Professor Gadi Fibich, School of Mathematics, Tel-Aviv
University

New Singular Solutions of
the Nonlinear Schrodinger Equation (NLS)

The study of singular solutions of the NLS
goes back to the 1960s,with applications in
nonlinear optics and, more recently, in BEC.
Until recently, the only known singular
solutions had a self-similar "Gaussian-type''
profile that approaches a delta function near
the singularity. In this talk I will present new
families of singular solutions of the NLS
that collapse with a self-similar ring profile,
and whose blowup rate is different from the one
of the "old'' singular solutions. I will also
show, both theoretically and experimentally,
that these new blowup profiles are attractors
for large super-Gaussian initial conditions.

Professor David Hoff, Department of
Mathematics, Indiana University

Lagrangean
Structure and Propagation of Singularities in
Multidimensional Compressible Flow

I'll motivate and describe some results
concerning the Lagrangean structure and
propagation of singularities in solutions of the
Navier-Stokes equations of multidimensional
compressible flow: for solutions in the
regularity class under consideration, there is a
unique particle trajectory emanating from each
point of any open set in physical space in which
the initial density is strictly positive and
such open sets are convected homeomorphically by
the flow; as corollaries, Holder continuous
surfaces are transported into Holder continuous
surfaces, sectional continuity of the density
and the divergence of the velocity are
preserved, the Rankine-Hugoniot conditions hold
in a strict, pointwise sense across such
surfaces, and the strengths of singularities
decay exponentially in time when the pressure is
a monotone function of density. These results
require that initial velocities are in certain
fractional Sobolev spaces depending on the
dimension; I'll give an example indicating that
this requirement is really necessary. The talk
should be accessible to grad students with some
background in pde's and analysis.

Professor Efthimios Kaxiras, Department
of Physics, Harvard University

Simulations of Complex
Materials Across Multiple Scales

A variety of physical phenomena involve
multiple length and time scales. Some
interesting examples of multiple-scale phenomena
are:

the mechanical behavior of crystals and
in particular the interplay of chemistry and
mechanical stress in determining the
macroscopic brittle or ductile response of
solids.

the molecular-scale forces at interfaces
and their effect in macroscopic phenomena
like wetting and friction.

the alteration of the structure and
electronic properties of macromolecular
systems due to external forces, as in
stretched DNA nanowires or carbon nanotubes.

In these complex physical systems, the
changes in bonding and atomic configurations at
the microscopic, atomic level have profound
effects on the macroscopic properties, be they
of mechanical or electrical nature. Linking the
processes at the two extremes of the length
scale spectrum is the only means of achieving a
deeper understanding of these phenomena and,
ultimately, of being able to control them. While
methodologies for describing the physics at a
single scale are well developed in many fields
of physics, chemistry or engineering,
methodologies that couple scales remain a
challenge, both from the conceptual point as
well as from the computational point. In this
presentation I will discuss the development of
methodologies for simulations across disparate
length scales with the aim of obtaining a
detailed description of complex phenomena of the
type described above. I will also present
illustrative examples, including hydrogen
embrittlement of metals, DNA conductivity and
translocation through nanopores, and affecting
the wettability of surfaces by surface chemical
modification.

Professor Selim Esedoglu, Department of
Mathematics, University of Michigan

Finding Global Minimizers
of Segmentation and Denoising Functionals

Segmentation is a fundamental procedure in
computer vision. It forms an important
preliminary step whenever useful information is
to be extracted from images automatically. Given
an image depicting a scene with several objects
in it, its goal is to determine which regions of
the image contain distinct objects.

Variational segmentation models, such as the
Mumford-Shah functional and its variants, pose
segmentation as finding the minimizer of an
energy. The resulting optimization problems are
often non-convex, and may have local minima that
are not global minima, complicating their
solution. We will show that certain simplified
versions of the Mumford-Shah model can be given
equivalent convex formulations, allowing us to
find global minimizers of these non-convex
problems via convex minimization techniques. In
particular, we will show that a recent convex
duality based algorithm due to A. Chambolle,
which was originally developed for Rudin, Osher,
and Fatemi's total variation denoising model,
can be adapted to the segmentation problem.

Professor Nigel Goldenfeld, Department of
Physics, University of Illinois

Beyond Phase Field
Models: Renormalization Group Approach to
Multiscale Modeling in Materials Science

Dendritic growth, and the formation of
material microstructure in general, necessarily
involves a wide range of length scales from the
atomic up to sample dimensions. Phase field
models, enhanced by optimal asymptotic methods
and adaptive mesh refinement, cope with this
range of scales, and provide a very efficient
way to move phase boundaries. However, they fail
to preserve memory of the underlying
crystallographic anisotropy. Elder and Grant
have convincingly shown how one can use the
phase field crystal (PFC) equation -- a
conserving analogue of the Swift-Hohenberg
equation -- to create field equations with
periodic solutions that model elasticity, the
formation of solid phases, and accurately
reproduce the nonequilibrium dynamics of phase
transitions in real materials. In this talk, I
show that a computationally-efficient multiscale
approach to the PFC can be developed
systematically by using the renormalization
group or equivalent techniques to derive the
appropriate coupled phase and amplitude
equations, which can then be solved by adaptive
mesh refinement algorithms.

Joint CSCAMM/Math/Norbert Wiener Center
*THURSDAY*, October 19th, at 3:30 PM, in the Math Colloquium Room #3206

Professor Anna Gilbert, Department of Mathematics, University of Michigan

Fast Algorithms for Sparse Analysis

I will present several extremely fast algorithms for recovering a compressible signal from a few linear measurements. These examples span a variety of orthonormal bases, including one large redundant dictionary. As part of the presentation of these algorithms, I will give an explanation of the crucial role of group testing in each algorithm.

Professor Michael Vogelius, Department of Mathematics, Rutgers
University

Electromagnetic Imaging and Cloaking

I shall start with a brief review of the main results that are known concerning the identification of internal electric (and magnetic) parameters of an object based on boundary field measurements. Then I shall proceed to examine some examples of anisotropic conductivities originally introduced by Greenleaf, Lassas and Uhlmann, and more recently discussed in a Science article by Pendry, Schurig and Smith.

These conductivities are indistinguishable by electrostatic boundary measurements, and most importantly they allow (hiding) any anisotropic conductivity distribution inside a particular fixed set.

Dr. Rodrigo Platte, Department of Mathematics and Statistics, Arizona State University

A Hybrid Fourier-Chebyshev Pseudospectral Method for Non-Periodic Hyperbolic Problems

Fourier pseudospectral methods produce
outstanding results for smooth problems with
periodic boundary conditions. For non-periodic
problems, Chebyshev pseudospectral methods are
commonly used to obtain exponential rates of
convergence. However, while Fourier methods
require 2 points per wavelength, Chebyshev
methods require.
Moreover, the clustering of nodes near the
boundaries required by polynomial methods also
imposes an
restriction
on the time-step size when these methods are
used for hyperbolic problems. In order to
circumvent these difficulties in the solution of
non-periodic problems, we propose a hybrid
Fourier Chebyshev pseudospectral method. The
method requires a window function,
where
and
its derivatives are approximately zero at the
boundary. The target function
is
then replaced by the product
which
is periodic and can be accurately approximated
by a trigonometric series. A polynomial
approximation is needed only in the region where
is
close to zero (near the boundaries), since
cannot
be accurately recovered from
if
is
small. Numerical results confirm the spectral
accuracy and stability of the method. Because
may
have large gradients, the method is particularly
suitable for stiff problems.

Professor Boaz Nadler, Department of Computer Science and Applied Mathematics, Weizmann Institute of Science

Statistical Data Analysis and Stochastic Dynamical Systems - A Two Way Street

In this talk we present an interesting connection between various algorithms from the field of machine learning and statistical data analysis and the theory of stochastic processes.
In particular we show that some common data mining algorithms, such as spectral clustering and kernel based semi-supervised learning can be analyzed by standard tools of applied mathematics,
including asymptotic analysis and the theory of stochastic processes. Conversely, we also show that problems in the study of stochastic dynamical systems, such as
dimensional reduction of high dimensional dynamical systems and estimation of effective macroscopic dynamics, can be analyzed by applying data analysis tools inspired by spectral clustering.

Professor Teng Li, Department of Mechanical Engineering, University of Maryland

Modeling the Deformation of Organic/Inorganic Hybrids Mechanics of Flexible Macroelectronics

The advent of flat-panel displays has opened the era of macroelectronics. Currently, macroelectronics is being developed as a platform for many technologies, such as paper-like displays, printable solar cells, and electronic skins, indicating further desirable attributes for macroelectronic systems, including flexibility, portability and low-cost. Such flexible macroelectronic devices will have diverse architectures, hybrid materials, and small features. For example, such devices often consist of thin films of inorganic electronic materials (e.g., metals, dielectrics and semiconductors) on substrates of organic materials (e.g., polymers). The mechanical behavior of these large scale structures of hybrid materials poses significant challenges to the creation of the new technologies. For example, while polymers can sustain large deformation, thin films of most electronic materials fracture at small strains (less than ~1%). How to use these materials to make electronic devices with reliable deformability under cyclic loading remains uncertain.

This talk describes the ongoing work in the emerging field of research micro/nano mechanics of flexible macroelectronics. Particular focus is placed on how to make nanoscale thin films of various electronic materials mechanically deformable and electronically functional when the whole device is subject to large, cyclic stretching, bending or twisting, a key challenge confronted by this nascent technology. We first focus on understanding the tensile behavior of thin metal films on polymer substrates, gaining insight into the rupture mechanisms of such a representative architecture in flexible macroelectronics. We then identify and quantify the physical parameters governing the film rupture strains, shedding light on the materials optimization to achieve better device deformability. Next we broaden the focus onto general electronic materials and explore possible ways to enhance their deformability on polymer substrates. We identify the mechanisms of large, reversible deformability of thin metal films on elastomeric substrates, and then propose a general principle of making thin films of stiff materials deformable by suitably patterning. Such patterned films can serve as general platforms for flexible macroelectronics.