PDE-Applied Math Archives for Academic Year 2013

A Riemann-Hilbert Problem in a Riemann Surface and Asymptotic Stability of the Periodic Toda Lattice

When: Thu, September 6, 2012 - 3:30am
Where: Math 3206
Speaker: Spyridon Kamvikiss (University of Crete) http://www.tem.uoc.gr/~spyros/
Abstract: We consider the stability of the periodic Toda lattice (and slightly more generally of the algebro-geometric finite-gap lattice) under a "short range" perturbation. We prove that the perturbed lattice
asymptotically approaches a modulated lattice that we describe explicitly.

Our method relies on the equivalence of the inverse spectral problem to a matrix Riemann-Hilbert problem defined in a hyperelliptic curve and generalizes the so-called nonlinear stationary phase/steepest descent method for Riemann-Hilbert problem deformations to Riemann surfaces.

Joint KI-Net/PDE Seminar: Analysis of PDE models for neuronal networks

When: Thu, October 25, 2012 - 3:30pm
Where: CSCAMM Seminar Room 4122, CSIC Bldg. #406
Speaker: Benoit Perthame ( Laboratoire J.-L. Lions, Universit\'e P. et M. Curie, CNRS, INRIA and Institut Universitaire de France) - http://www.ann.jussieu.fr/~perthame/
Abstract: Many integro-differential equations are used to describe neuronal networks or neural assemblies. Among them, the Wilson-Cowan equations are the most wellknown and describe spiking rates in different locations. Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. It has also been proposed to describe directly the spike time distribution which seems to encode more directly the neuronal information. This leads to a structured population equation that describes at time $t$ the probability to find a neuron with time s elapsed since its last discharge.

We will compare these models and perform some mathematical analysis. A striking observation is that solutions to the I&F can blow-up in finite time, a form of synchronization. We can also show that for small or large connectivity the 'elapsed time model' leads to desynchronization. For intermediate regimes, sustained periodic activity occurs which profile is compatible with observations. A common tool is the use of the relative entropy method.

Steady water waves with compactly supported vorticity

When: Thu, November 8, 2012 - 3:30pm
Where: Math 3206
Speaker: Samuel Walsh (Courant Institute, NYU) -
Abstract: In this talk, we discuss some recent results on the existence of two-dimensional traveling waves in water with the special property that the vorticity is a Dirac measure (a point vortex), or supported in a compact set (a vortical patch). Such waves arise naturally, for instance, if we think of a classical irrotational traveling wave with some interesting but localized vortex dynamics occurring below the surface.

Monge-Ampere meets Hamilton-Jacobi

When: Thu, November 15, 2012 - 3:30pm
Where: Math 3206
Speaker: Yanir Rubinstein (UMD) -

Joint KI-Net/PDE Seminar: Entropy, Entropy Production and trend to equilibrium in Kac's Model of any Dimension

When: Wed, November 28, 2012 - 2:00pm
Where: CSCAMM Seminar Room 4122, CSIC Bldg. #406
Speaker: Amit Einav (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge)
Abstract: In 1956 Marc Kac introduced a binary stochastic N-particle
model from which, under suitable condition on the initial datum (what we
now call 'Chaoticity') a caricature of the famous Boltzmann equation, in
its spatially homogeneous form, arose as a mean field limit. The
ergodicity of the evolution equation resulted in convergence to
equilibrium as time goes to infinity, for any N. Kac expressed hopes
that investigation of the rate of convergence can be expressed
independently in N and result in an exponential trend to equilibrium for
his caricature of Boltzmann equation. Later on, in 1967, McKean extended
Kac's model to a more realistic d-dimensional one from which the actual
Boltzmann equation arose, extending Kac's results and hopes to the real
Kac's program reached its conclusion in the 2000s in a series of papers
by Janvresse, Maslen, Carlen, Carvalho, Loss and Geronimo, however it
was known long before that the linear L^2 based approach of Kac will not
yield the desired result. A new method was devised, one that draws its
ideas from a conjecture by Cercignani's for the real Boltzmann equation:
investigate the entropy, and entropy production in Kac's model, in hope
to get a better rate of convergence.
In our talk we will discuss Kac models of any dimension, recall the
spectral gap problem and its conclusions as well as describe
Cercignani's many body conjecture. We will show that, while the entropy
and entropy production are more suited to deal with Kac's models, in
full generality the rate they produce is not much better than that of
the linear approach. We will conclude that more restrictions are need,
and share a few insights we may have in the subject.

C^1,α regularity of solutions of degenerate fully non-linear elliptic equations

When: Thu, November 29, 2012 - 3:30pm
Where: Cscamm seminar room
Speaker: Cyril Imbert (Univ. Paris Est Creteil) -

Mathematical approaches to stochastic signaling and pattern formation

When: Thu, December 6, 2012 - 3:30pm
Where: Math 3206
Speaker: Likun Zheng (University of California, Irvine) -
Abstract: Biological systems are often subject to external noise from signal stimuli and environmental perturbations, as well as noises in the intracellular signal transduction pathway. With many genes and proteins presenting in small numbers, the inherent fluctuations can be large. In this talk, we will present our analysis and computations of stochastic models, to understand what strategies are used to control noise propagation and refine gene expression regions. Our results imply a lot of interesting consistency between considerably fluctuating systems and their corresponding deterministic systems or systems with small fluctuations.

Parallel Replica Dynamics

When: Mon, December 10, 2012 - 3:00pm
Where: Math 3206
Speaker: Gideon Simpson (University of Minnesota) -
Abstract: Parallel replica dynamics was proposed by A.F. Voter as a numerical tool for accelerating molecular dynamics simulations characterized by a sequence of infrequent, but rapid, transitions from one state to another. An example would be the migration of a defect through a crystal. Parallel replica dynamics accelerates this by simulating many replicas simultaneously, concatenating the simulation time of the realizations, as though it were a single long trajectory. This motivates important questions: Is parallel replica dynamics algorithm doing what we hope? For what systems will it be useful? How do we implement it efficiently? In this talk, I will thoroughly describe the algorithm and report on progress towards rigorous justification. Open questions and related problems will also be discussed.

Swarm dynamics and equilibria for a nonlocal aggregation model

When: Thu, January 31, 2013 - 3:30pm
Where: Math 3206
Speaker: Razvan Fetecau (joint KiNet seminar) (Simon Fraser University) - http://people.math.sfu.ca/~van/
Abstract: We consider the aggregation equation ρt − ∇ · (ρ∇K ∗ ρ) = 0 in Rn , where the interaction
potential K models short-range repulsion and long-range attraction. We study a family of
interaction potentials with repulsion given by a Newtonian potential and attraction in the form
of a power law. We show global well-posedness of solutions and investigate analytically and
numerically the equilibria and their global stability. The equilibria have biologically relevant
features, such as finite densities and compact support with sharp boundaries. This is joint work
with Yanghong Huang and Theodore Kolokolnikov.

The dynamic and steady states for multi-dimensional Keller-Segel model with the diffusion exponent m>0

When: Thu, February 21, 2013 - 3:30pm
Where: Math 3206
Speaker: Jian-Guo Liu (Duke University) -
Abstract: In this talk, I will discuss infinite-time spreading and finite-time blow-up for the Keller-Segel system. For $0< m \leq 2-2/d$, the $L^p$ space for both dynamic and steady solutions are detected with $p:=\frac{d(2-m)}{2}$. Firstly, the global existence of the weak solution is proved for small initial data in $L^p$. Moreover, when $m>1-2/d$, the weak solution preserves mass and satisfies the hyper-contractive estimates in $L^q$ for any $p<q<\infty$.
Furthermore, for slow diffusion $1<m \leq 2-2/d$, this weak solution is also a weak entropy solution which blows up at finite time provided by the initial negative free energy. For $m>2-2/d$, the hyper-contractive estimates are also obtained. Finally, we focus on the $L^p$ norm of the steady solutions, it is shown that the energy critical exponent $m=2d/(d+2)$ is the critical exponent separating finite $L^p$ norm and infinite $L^p$ norm for the steady state solutions.

This is a joint work with Shen Bian of Tsinghua University.

Ground states and dynamics of the nonlinear Schrodinger/Gross-Pitaevskii equations

When: Thu, March 7, 2013 - 3:30pm
Where: CSCAMM seminar room
Speaker: Weizhu Bao, Department of Mathematics, Center for Computational Science & Engineering, National University of Singapore -
Abstract: In this talk, I begin with a brief derivation of the nonlinear Schrodinger/Gross-Pitaevskii equations (NLSE/GPE) from Bose-Einstein condensates (BEC) and/or nonlinear optics. Then I will present some mathematical results on the existence and uniqueness as well as non-existence of the ground states of NLSE/GPE under different external potentials and parameter regimes. Dynamical properties of NLSE/GPE are then discussed, which include conservation laws, soliton solutions, well-posedness and/or finite time blowup. Efficient and accurate numerical methods will be presented for computing numerically the ground states and dynamics. Extension to NLSE/GPE with an angular momentum rotation term and/or non-local dipole-dipole interaction will be presented. Finally, applications to collapse and explosion of BEC, quantum transport and quantized vortex interaction will be investigated.

Robust Subspace Modeling

When: Thu, March 28, 2013 - 3:30pm
Where: Math 3206
Speaker: Gilad Lerman (University of Minnesota) -
Abstract: Consider a dataset of vector-valued observations that consists of a modest number of noisy inliers, which are explained well by a low-dimensional subspace, along with a large number of outliers, which have no linear structure. We describe a convex optimization problem that can reliably fit a low-dimensional model to this type of data. When the inliers are contained in a low-dimensional subspace we provide a rigorous theory that describes when this optimization can recover the subspace exactly. We present an efficient algorithm for solving this optimization problem, whose computational cost is comparable to that of the non-truncated SVD. We also show that the sample complexity of the proposed subspace recovery is of the same order as PCA subspace recovery and we consequently obtain some nontrivial robustness to noise. This presentation is based on three joint works: 1) with Teng Zhang, 2) with Michael McCoy, Joel Tropp and Teng Zhang, and 3) with Matthew Coudron.

On rationality & optimality in navigating crowds

When: Thu, May 2, 2013 - 3:30pm
Where: Math 3206
Speaker: A. Vladimirsky (Cornell University) -
Abstract: In this talk I will discuss the qualitative features and consistency of several pedestrian flow models.
One recent popular approach is to derive a coupled system of nonlinear PDEs via the
"mean field games" theory: a conservation law models the evolution of the pedestrian density,
while a Hamilton-Jacobi-Bellman PDE is used to determine the directions of pedestrian flux.
My focus will be on anisotropic interactions between the pedestrians and
their implications for the resulting system of PDEs.

Self-Assembly: Mathematical Paradigms, Analysis and Computation

When: Thu, May 9, 2013 - 3:30pm
Where: Math 3206
Speaker: Rustum Choksi (McGill University) -
Abstract: "Self-assembly is a common theme in many physical systems.
In this talk, I will address a nonlocal perturbation of Coulombic-type to the well-known Ginzburg-Landau/Cahn-Hilliard free energy.
This mathematical paradigm has a very rich and complex energy landscape, and
I will present both rigorous asymptotic results concerning global minimizers and
numerical methods which attempt to access ground states (or at least states of lower energy).

I will also address a purely geometric paradigm for self-assembly:
centroidal Voronoi tessellation for generators which are curves and surfaces.
This work rests upon a recent iterative algorithm (joint with L. Larsson and J.C. Nave at McGill)
for the fast computation of measures of generalized Vororonoi regions."