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.
 

Lagrangean  Structure and Propagation of Singularities in Multidimensional Compressible Flow  

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

1. 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.

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

3. 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.

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.

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.

*NOTE SPECIAL DAY*

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

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.

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.

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.

Thanksgiving
 

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.

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.