Joint CSCAMM/Math Seminar

One application is the Heil-Ramanathan-Topiwala conjecture that states that finitely many time-frequency shifts of one L² function are linearly independent. This turns to be equivalent to the absence of eigenspectrum for finite linear combinations of time-frequency shifts. I will prove a special case of this conjecture.

Next I present two applications in Wireless communications. One application is the channel equalization problem. I will show how to use the Wiener lemma and the Zak transform to effectively compute this inverse. A second application concerns canonical communication channels. The original canonical channel model allows for designing the Rake receiver. Recently Sayeed and Aazhang obtained a time-frequency canonical model and designed its Rake receiver. I will comment on its operator algebra meaning and ways to generalize it to other canonical models using time-scale or frequency-scale shifts.
 

Professor Yosef Yomdin, Department of Mathematics, Weizmann Institute of Science

Rescheduled to March 7, see below:
 

Professor Liliana Borcea, Department of Computational and Applied Mathematics, Rice University

Electrical Impedance Tomography with resistor networks.

We present a novel inversion algorithm for electrical impedance tomography in two dimensions, based on a model reduction approach.

The reduced models are resistor networks that arise in five point stencil discretizations of the elliptic partial differential equation satisfied by the electric potential, on adaptive grids that are computed as part of the problem. We prove the unique solvability of the model reduction problem for a broad class of measurements of the Dirichlet to Neumann map. The size of the networks (reduced models) is limited by the precision of the measurements. The resulting grids are naturally refined near the boundary, where we make the measurements and where we expect better resolution of the images. To determine the unknown conductivity, we use the resistor networks to define a nonlinear mapping of the data, that behaves as an approximate inverse of the forward map.

Then, we propose an efficient Newton-type iteration for finding the conductivity, using this map. We also show how to incorporate apriori information about the conductivity in the inversion scheme.
 

Professor Joel Smoller, University of Michigan

Stability of Black Holes

We consider decay of solutions of the Cauchy Problem for various fields in the Kerr (rotating Black Hole) geometry. We discuss the formulation of the problem in terms of the Teukolsky equation, a single second-order PDE depending on a real parameter s, the "spin". For various values of s, the Teukolsky equation describes different fields in the Kerr geometry: s=0,1/2,1, 2, correspond,respectively, to scalar waves, Dirac's equation, Maxwell's equations, and gravitational waves. We discuss our results for s=0,1/2, as well as our rigorous proof of Penrose's proposal (1969) for energy extraction from a Kerr BH. In the case of a Schwarzschild (non-rotating) BH, we discuss our decay results, and hence stability of this BH, for all spin.
 

Professor Yosef Yomdin, Department of Mathematics, Weizmann Institute of Science

Whitney C^k-extension problem, its lattice version, Hermite approximation, and numerical solution of PDEs

The C^k- extension problem posed by H. Whitney in 1932 is to provide conditions for a function f defined on a closed set A in Rⁿ, to be a restriction of a C^k-smooth function F on the entire space Rⁿ. For n=1 the answer is (roughly) given by the existing of a uniform bound for all the finite differences up to order k of f on A. In higher dimensions this problem was solved only very recently by Ch. Fefferman. There is a natural generalization of the Whitney's problem to the high-order data (Whitney fields). We show that the C^k-extension from regular lattices is closely related to the Hermite polynomial approximation of high-order data. The last problem can be studied with the tools of real Semi-Algebraic Geometry.

Finally, we discuss some high order numerical schemes, based on the Whitney extension of the high-order lattice data.
 

Professor Gui-Qiang Chen, Department of Mathematics, Northwestern University

Transonic Flow, Shock Reflection, and Free Boundary Problems

Abstract: When a plane shock hits a wedge head on, it experiences a reflection-diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. The complexity of reflection picture was first reported by Ernst Mach in 1878, and experimental, computational, and asymptotic analysis has shown that various patterns of reflected shocks may occur, including regular and Mach reflection. However, most fundamental issues for shock reflection have not been understood, including the transition of the different patterns of shock reflection, and there had been no rigorous mathematical result on the global existence and structural stability of shock reflection, especially for potential flow which has widely been used in aerodynamics.

In this talk we will start with various shock reflection phenomena and their fundamental scientific issues. Then we will describe how the shock reflection problems can be formulated into free boundary problems for nonlinear partial differential equations of mixed-composite type. Finally we will discuss some recent developments in attacking the shock reflection problems, including our results with M. Feldman on the global existence and stability of solutions of shock reflection by large-angle wedges for potential flow. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, and corner singularities when free boundaries meet degenerate elliptic curves.
 

Professor Charles Meneveau, Department of Mechanical Engineering, Johns Hopkins University

Lagrangian dynamics of turbulence: models and synthesis

Recent theoretical and numerical results on intermittency in hydrodynamic turbulence are described, with special emphasis on the Lagrangian evolution.

First, we derive the advected delta-vee system. This simple dynamical system deals with the Lagrangian evolution of two-point velocity and scalar increments in turbulence and shows how many known trends in turbulence can be simply understood from the proposed projection of the self-stretching effect coming from the nonlinear advective term. More detailed statistical information can be obtained from a model for the full velocity gradient tensor that uses a closure for the pressure Hessian and viscous terms.

Finally, we will describe efforts to use these insights in the generation of synthetic, multi-scale 3D vector fields with non-Gaussian statistics that reproduce many of know behavior of turbulence.
 

Professor Dionisios Margetis, Department of Mathematics, University of Maryland

From microscopic physics to continuum laws for crystal surfaces: Progress and challenges

In materials modeling the starting point ("truth") is atomistic, or takes the form of discrete schemes, by which evolution laws must be determined for the macroscale. In this talk I will focus on the evolution of crystal surfaces as a prototypical case of modeling across the scales, with implications in the design of novel solid-state devices.
I will discuss recent progress in three directions:

(i) Microscopic details of crystals enter free boundary problems, and affect evolution at the macroscale.
(ii) Continuum laws give signatures of instabilities that are due to the discrete nature of crystal surfaces.
(iii) Material parameters in a continuum description are determined as appropriate limits of microscopic, kinetic processes.

I will also discuss challenges and open questions that arise in this context.
 

Professor Saul Teukolsky, Cornell University

Simulations of Black Holes and Gravitational Waves

Gravitational wave detectors like LIGO are poised to begin detecting signals. One of the prime scientific goals is to detect waves from the coalescence and merger of black holes in binary systems. Confronting such signals with the predictions of Einstein's General Theory of Relativity will be the first real strong-field test of the theory. I will explain why multidomain spectral methods are ideally suited to tackle this problem.
Until very recently, theorists were unable to calculate what the theory actually predicts. I will describe recent breakthroughs that have occurred and that have set things up for an epic confrontation of theory and experiment.
 

Microbes live in fluctuating environments that are often limiting for growth. They have evolved several sophisticated mechanisms to sense changes in important environmental parameters such as light and nutrients. Most bacteria also have complex appendages that allow them to move, so they can swim or crawl into optimal conditions. This combination of sensing changes in the immediate environment and transducing these changes to the motion organisms, allows for movement in a particular direction: a phenomenon known as ''chemotaxis'' or ''phototaxis''.

Using time-lapse video microscopy we have monitored the movement of Cyanobacteria (which are phototaxis, i.e., bacteria that move towards light). These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important. The various patterns of movement that we observe appear to be a complex function of cell density, surface properties and genotype. Very little is known about the interactions between these parameters.

In this talk, we will present a hierarchy of new models for describing the motion of phototaxis that were constructed based on the experimental observations. The first model is a stochastic model that describes the locations of bacteria, the group dynamics, and the interaction between the bacteria and the medium in which it resides. The second model is a new multi-particle system that is obtained from a discretization of the first model. Our third model is obtained as the continuum limit of
the second model, and as such it is a system of nonlinear PDEs. Our main theorems clarify the sense in which the system of PDEs can be considered as the limit dynamics of the multi-particle system. We conclude with several numerical simulations that demonstrate the properties of our models.

This is a joint work with Devaki Bhaya (Department of Plant Biology, Carnegie Institute) and Tiago Requeijo (Math, Stanford)

CSCAMM WORKSHOP:
Nonequilibrium Interface and Surface Dynamics

Joint CSCAMM/Math Seminar
Professor Norbert Mauser, Wolfgang Pauli Institute,
University of Vienna
Nonlinear Schroedinger Equations : Semi-classics
and Blow Up : Numerical Studies

We present recent numerical methods and simulations of time dependent NLS with nonlinearities of the local type ("cubic NLS") or/and of the nonlocal type ("Hartree/Poisson equation") and of Davey Stewartson equations. Particular interest is laid on simulations of "blow up" of the critical (= 2-d cubic) NLS (*), depending on parameters of the data / equation, where a conjecture on monotonicity of the blow up time (Fibich) is shown not to hold. Also we show the Schroedinger-Poisson-X-alpha model, where we investigate properties of semi-classical asymptotics. We briefly present the equations, analysis and the numerical methods (time split spectral scheme and relaxation scheme implemented on a fixed grid on a parallel machine, thus allowing for 3-d simulations with up to 1000 points in each direction) and instructive simulation results.
 

Dr. Aziz Madrane, AIRBUS Bremen, Germany

3D Adaptive Central Schemes on Unstructured Staggered Grids

In this talk I present the implementation of adaptive central schemes on unstructured tetrahedral grids for approximating nonlinear hyperbolic conservation laws in 3D. The mesh adaptation algorithm is coupled to a cell vertex, finite volume scheme of the MUSCL type, employing a second order central schemes. First and second order central central schemes on staggered grids see figs. 1,2 were introduced by Nessyahu and Tadmor [7] in one spatial dimension and generalized to unstructured grids in two space dimensions by Arminjon, Viallon and Madrane [1], and in three dimensions by Arminjon, Madrane and StCyr [2,3,4]. In addition, a posteriori error estimates were obtained, by interpreting the central scheme on staggered grids as a upwind finite volume scheme in conservation form [5]. In this contribution we extend the idea of [5,6] to 3D to derive appropriate refinement indicators.
The adaptive scheme is then used to compute the shock waves around an NACA0012 airfoil for subsonic post-shock flow.
 

Professor Eliot Fried, Washington University in St. Louis

A Conjectured Hierarchy of Length Scales in a Generalization of the Navier-Stokes-alpha Equation for Turbulent Fluid Flow

We present a continuum-mechanical formulation and generalization of the Navier–Stokes-α theory based on a general framework for fluid-dynamical theories with gradient dependencies. Our flow equation involves two additional problem-dependent length scales α and β. The first of these scales enters the theory through the internal kinetic energy, per unit mass, α2|D|2, where D is the symmetric part of the gradient of the filtered velocity. The remaining scale is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity. When α and β are equal, our flow equation reduces to the Navier–Stokes-α equation. In contrast to the original derivation of the Navier–Stokes-α equation, which relies on Lagrangian averaging, our formulation delivers boundary conditions. For a confined flow, our boundary conditions involve an additional length scale l characteristic of the eddies found near walls. Based on a comparison with direct numerical simulations for fully-developed turbulent flow in a rectangular channel of height 2h, we find that α/β ~ Re^0.470 and l/h ~ Re^−0.772 , where Re is the Reynolds number. The first result, which arises as a consequence of identifying the internal kinetic energy with the turbulent kinetic energy, indicates that the choice α = β required to reduce our flow equation to the Navier–Stokes-α equation is likely to be problematic. The second result evinces the classical scaling relation η/L ~ Re^−3/4 for the ratio of the Kolmogorov microscale η to the integral length scale L. The numerical data also suggests that l≤ β. We are therefore led to conjecture a tentative hierarchy, l ≤ β < α, involving the three length scales entering our theory.
 

Joint CSCAMM/NWC Seminar
Dr. Jared Tanner, Department of Mathematics, University of Utah

The Surprising Structure of Gaussian Point
Clouds and its Implications for Signal Processing

We will explore connections between the structure of high-dimensional convex polytopes and information acquisition for compressible signals. A classical result in the field of convex polytopes is that if N points are distributed Gaussian iid at random in dimension n<<N, then only order (log N)^n of the points are vertices of their convex hull. Recent results show that
provided n grows slowly with N, then with high probability all of the points are vertices of its convex hull. More surprisingly, a rich "neighborliness" structure emerges in the faces of the convex hull. One implication of this phenomenon is that an N-vector with k non-zeros can be recovered computationally efficiently from only n random projections with n=2e k log(N/n).
Alternatively, the best k-term approximation of a signal in any basis can be recovered from 2e k log(N/n) non-adaptive measurements, which is within a log factor of the optimal rate achievable for adaptive sampling. Additional implications for randomized error correcting codes will be presented.

This work was joint with David L. Donoho.