Approximate separable representation of the Greenís functions for differential operators (with appropriate boundary conditions) is a basic and important question in analysis of differential equations and development of efficient numerical algorithms. It can reveal intrinsic complexity or degrees of freedom of the corresponding differential equation. Computationally, being able to approximate a Greenís function as a sum with few separable terms is equivalent to the existence of low rank approximation of the corresponding discretized system which can be explored for matrix compression and fast solution techniques such as in fast multiple method and direct matrix inverse solver. In this talk, two types of differential operators will be discussed. One is coercive elliptic equation in divergence form, which is highly separable. The other one is Helmholtz equation in high frequency limit for which
we developed a new approach to study the approximate separability of Greenís function based on an geometric characterization of the relation between two Green's functions and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors. We derive both lower bounds and upper bounds and show their sharpness and implications for computation setups that are commonly used in practice. This is a joint work with Bjorn Engquist.
Phantom jams and jamitons in macroscopic traffic models
Initially homogeneous vehicular traffic flow can become inhomogeneous even in the absence of obstacles. Such ``phantom traffic jams'' can be explained as instabilities of a wide class of ``second-order'' macroscopic traffic models. In this unstable regime, small perturbations amplify and grow into nonlinear traveling waves. These traffic waves, called ``jamitons'', are observed in reality and have been reproduced experimentally. We show that jamitons are analogs of detonation waves in reacting gas dynamics, thus creating an interesting link between traffic flow, combustion, water roll waves, and black holes. This analogy enables us to employ the Zel'dovich-von Neumann-Doering theory to predict the shape and travel velocity of the jamitons. We furthermore demonstrate that the existence of jamiton solutions can serve as an explanation for multi-valued parts that fundamental diagrams of traffic flow are observed to exhibit.
Global well-posedness and blow up for active scalar equations
We aim to study the behavior of solutions to a class of active scalar equations,
for which the two-dimensional surface quasi-geostrophic and the Burgers equations
are the main instances. When (nonlocal) dissipation is present, different scenarios
may occur depending on the dissipative operator: in the Burgers equation, the picture
is fairly clear, while for the SQG equation the global existence of regular solutions in the supercritical
(i.e. weakly dissipative) regime is an outstanding open problem. We prove that the SQG equation
in the supercritical case continuously depends on the dissipative operator, achieving regularity
for large initial data in critical spaces. Time permitting, we will discuss possible different (inviscid)
regularizations of such equations that lead to global regularity or blow-up of solutions in a similar fashion.
Nicolas Garcia Trillos, Department of Mathematical Sceinces, Carnegie Mellon University
Continuum limit of total variation on point clouds
We consider point clouds obtained as random samples of a measure
on a Euclidean domain. A graph representing the point cloud is obtained by
assigning weights to edges based on the distance between the points they
connect. We study when is the cut capacity, and more generally total
variation, on these graphs a good approximation of the perimeter (total
variation) in the continuum setting. We address this question in the
setting of Γ-convergence. Applications to the study of consistency
of cut based clustering procedures will be discussed.
The mathematics of three-dimensional structure determination of molecules by cryo-electron microscopy
Cryo-electron microscopy (EM) is used to acquire noisy 2D projection images of thousands of individual, identical frozen-hydrated macromolecules at random unknown orientations and positions. The goal is to reconstruct the 3D structure of the macromolecule with sufficiently high resolution. We will discuss algorithms for solving the cryo-EM problem and their relation to other branches of mathematics such as tomography, random matrix theory, representation theory, convex optimization and semidefinite programming.
Prof. Tao Tang, Department of Mathematics, Hong Kong Baptist University
Stablized and Adaptive Time-Stepping Methods for Phase-Field Models
Recent results in the literature provide computational evidence that
stabilized time-stepping method can efficiently simulate
phase field problems involving fourth-order nonlinear diffusion, with
typical examples like the Cahn-Hilliard equation and the thin film type
equation. The up-to-date theoretical explanation of the numerical stability
relies on the assumption that the derivative of the nonlinear potential
function satisfies a Lipschitz type condition. This talk will discuss
how to remove the Lipschitz assumption on the nonlinearity. On the
numerical algorithm side, we will demonstrate how to design
high-order and adaptive methods for the phase-dield equations.
Prof. Chunlei Liang, Department of Mechanical & Aerospace Engineering, George Washington University
Spectral Difference Method for Computational Fluid Dynamics : From Flapping/Rotary Wing Aerodynamics to Thermal Convection of the Sun
Since the beginning of the 21st century, high-order methods for Computational Fluid Dynamics (CFD) have been quickly adopted by the engineering community for various flow problems because they can produce accurate results on relatively coarse grids. The speaker will focus on the advancement of the high-order spectral difference method (SDM) for solving compressible Naiver-Stokes type equations on
unstructured grids. The first part of this lecture is about studies of flapping wing aerodynamics which have demonstrated the suitability of SDM for predicting unsteady vortex dominated flow on moving and
deforming domains. The second part of this lecture will discuss a simple, novel, and high-order curved sliding interface method to enable the SDM for simulating unsteady flow on coupled stationary and rotary domains for rotary wing aerodynamics. Finally, the SDM is further advanced for predicting stratified thermal convection in the sun. The speaker will report a successful effort for building a
Compressible High-ORder Unstructured Spectral-difference (CHORUS) code for predicting the convection zone of the sun in collaboration with the National Center for Atmospheric Research (NCAR).
From molecular dynamics to kinetic theory and fluid mechanics
In his sixth problem, Hilbert asked for an axiomatization of gas dynamics,
and he suggested to use the Boltzmann equation as an intermediate description
between the (microscopic) atomic dynamics and (macroscopic) fluid
models. The main difficulty to achieve this program is to prove the asymptotic
decorrelation between the local microscopic interactions, referred to as
propagation of chaos, on a time scale much larger than the mean free time.
This is indeed the key property to observe some relaxation towards local
This control of the collision process can be obtained in fluctuation regimes
[1, 2]. In , we have established a long time convergence result to the
linearized Boltzmann equation, and eventually derived the acoustic and incompressible
Stokes equations in dimension 2. The proof relies crucially on
symmetry arguments, combined with a suitable pruning procedure to discard
super exponential collision trees.
Keywords: system of particles, low density, Boltzmann-Grad limit, kinetic equation, fluid models
 T. Bodineau, I. Gallagher, L. Saint-Raymond. The Brownian motion as
the limit of a deterministic system of hard-spheres, to appear in Invent.
 T. Bodineau, I. Gallagher, L. Saint-Raymond. From hard spheres to the
linearized Boltzmann equation: an L2 analysis of the Boltzmann-Grad
limit, in preparation.
In this talk I will present new solitary waves of the two-dimensional nonlinear Schrodinger equation on bounded domains, which have a "necklace" structure. I will consider their structure, stability, and how to compute them.
Computational Modeling of the Eukaryotic Cytoskeleton
Cells of higher organisms contain a dynamically remodeling filamentous network, called cytoskeleton, comprised of actin, myosin and many other molecules. The cytoskeletons endue cells with their instantaneous shapes, providing a machinery for cells to move around, generate forces and also integrate both chemical and mechanical signaling. In terms of the physico-chemical mechanisms, the underlying acto-myosin network growth and remodeling processes are based on a large number of chemical and mechanical interactions, which are mutually coupled, and spatially and temporally resolved. To investigate the fundamental principles behind the self-organization of these networks, we have developed a detailed physico-chemical, stochastic model of actin filament growth dynamics, where the mechanical rigidity of filaments and their corresponding deformations under internally and externally generated forces are taken into account. Our work shedded light on the complex, non-linear feedbacks between the chemical and mechanical processes governing actomyosin network dynamics, highlighting, in particular, the importance of diffusional and active transport phenomena.
The atomistic origin of crystal growth: Study in 1+1 dimensions
Crystals are ubiquitous in nature and in manmade devices. The growth
of crystals at sufficiently low temperatures (e.g., room temperature) is
characterized by the existence of certain defects. "Mesoscale" theories
for the motion of such defects have been proposed since the 1950s but
they are largely phenomenological. In this talk, I discuss the derivation
of a mesoscale theory of crystals defects (steps) from a kinetic atomistic perspective.