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Monday, March 28, 2016 - 14:05 ,
Location: Skiles 005 ,
Zhilin Li ,
North Carolina State University ,
Organizer:

In this talk, I will introduce the Immersed Finite Element Methods (IFEM)

for one and two dimensional elliptic interface problems based on Cartesian

triangulations. The key is to modify the basis functions so that the

homogeneous jump conditions are satisfied in the presence of discontinuity

in the coefficients. Both non-conforming and conforming

finite element spaces are considered. Corresponding interpolation

functions are proved to be second order accurate in the maximum norm.

for one and two dimensional elliptic interface problems based on Cartesian

triangulations. The key is to modify the basis functions so that the

homogeneous jump conditions are satisfied in the presence of discontinuity

in the coefficients. Both non-conforming and conforming

finite element spaces are considered. Corresponding interpolation

functions are proved to be second order accurate in the maximum norm.

For non-homogeneous jump conditions, we have developed a new strategy to

transform the original interface problem to a new one with homogeneous jump

conditions using the level set function.

If time permits, I will also explain some recent progress in this direction

including the augmented IFEM for piecewise constant coefficient, and a SVD

free version of the method.

Monday, March 7, 2016 - 14:00 ,
Location: Skiles 005 ,
Prof. Brittany Froese ,
New Jersey Institute of Technology ,
Organizer:

The relatively recent introduction of viscosity solutions and the Barles-Souganidis convergence framework have allowed for considerable progress in the numerical solution of fully nonlinear elliptic equations. Convergent, wide-stencil finite difference methods now exist for a variety of problems. However, these schemes are defined only on uniform Cartesian meshes over a rectangular domain. We describe a framework for constructing convergent meshfree finite difference approximations for a class of nonlinear elliptic operators. These approximations are defined on unstructured point clouds, which allows for computation on non-uniform meshes and complicated geometries. Because the schemes are monotone, they fit within the Barles-Souganidis convergence framework and can serve as a foundation for higher-order filtered methods. We present computational results for several examples including problems posed on random point clouds, computation of convex envelopes, obstacle problems, Monge-Ampere equations, and non-continuous solutions of the prescribed Gaussian curvature equation.

Monday, February 15, 2016 - 14:05 ,
Location: Skiles 005 ,
Professor Hautieng Wu ,
University of Toronto ,
Organizer: Haomin Zhou

Explosive technological advances lead to exponential growth of massive data-sets in health-related fields. Of particular important need is an innovative, robust and adaptive acquisition of intrinsic features and metric structure hidden in the massive data-sets. For example, the hidden low dimensional physiological dynamics often expresses itself as atime-varying periodicity and trend in the observed dataset. In this talk, I will discuss how to combine two modern adaptive signal processing techniques, alternating diffusion and concentration of frequency and time(ConceFT), to meet these needs. In addition to the theoreticaljustification, a direct application to the sleep-depth detection problem,ventilator weaning prediction problem and the anesthesia depth problemwill be demonstrated. If time permits, more applications likephotoplethysmography and electrocardiography signal analysis will be discussed.

Monday, January 25, 2016 - 14:00 ,
Location: Skiles 005 ,
Predrag Cvitanović ,
Center for Nonlinear Science, School of Physics, GT ,
Organizer: Sung Ha Kang

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. What is the best resolution possible for a given physical system?It turns out that for nonlinear dynamical systems the noise itself is highly nonlinear, with the effective noise different for different regions of system's state space. The best obtainable resolution thus depends on the observed state, the interplay of local stretching/contraction with the smearing due to noise, as well as the memory of its previous states. We show how that is computed, orbit by orbit. But noise also associates to each a finite state space volume, thus helping us by both smoothing out what is deterministically a fractal strange attractor, and restricting the computation to a set of unstable periodic orbits of finite period. By computing the local eigenfunctions of the Fokker-Planck evolution operator, forward operator along stable linearized directions and the adjoint operator along the unstable directions, we determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive noise. The space of all chaotic spatiotemporal states is infinite, but noise kindly coarse-grains it into a finite set of resolvable states.(This is work by Jeffrey M. Heninger, Domenico Lippolis,and Predrag Cvitanović,arXiv:0902.4269 <http://arxiv.org/abs/0902.4269>, arXiv:1206.5506 <http://arxiv.org/abs/1206.5506> and arXiv:1507.00462 <http://arxiv.org/abs/1507.00462>)

Monday, December 7, 2015 - 14:05 ,
Location: Skiles 005 ,
Professor Jun Zhang ,
Courant Institute ,
Organizer: Martin Short

Thermal convection is ubiquitous in nature. It spans from a small

cup of tea to the internal dynamics of the earth. In this talk, I

will discuss a few experiments where boundaries to the fluid play

surprising roles in changing the behaviors of a classical Rayleigh-

Bénard convection system. In one, mobile boundaries lead to

regular large-scale oscillations that involve the entire system.

This could be related to the continental kinetics on earth over

the past two billion years, as super-continents formed and

broke apart in cyclic fashion. In another experiment, we found that

seemingly impeding partitions in thermal convection can boost the

overall heat transport by several folds, once the partitions are

properly arranged, thanks to an unexpected symmetry-breaking

bifurcation.

cup of tea to the internal dynamics of the earth. In this talk, I

will discuss a few experiments where boundaries to the fluid play

surprising roles in changing the behaviors of a classical Rayleigh-

Bénard convection system. In one, mobile boundaries lead to

regular large-scale oscillations that involve the entire system.

This could be related to the continental kinetics on earth over

the past two billion years, as super-continents formed and

broke apart in cyclic fashion. In another experiment, we found that

seemingly impeding partitions in thermal convection can boost the

overall heat transport by several folds, once the partitions are

properly arranged, thanks to an unexpected symmetry-breaking

bifurcation.

Monday, November 30, 2015 - 11:05 ,
Location: Skiles 006 ,
Dr. Ahmet Özkan Özer ,
University of Nevada-Reno ,
aozer@unr.edu ,
Organizer:

In many applications, such as vibration of smart structures (piezoelectric, magnetorestrive, etc.), the physical quantity of interest depends both on the space an time. These systems are mostly modeled by partial differential equations (PDE), and the solutions of these systems evolve on infinite dimensional function spaces. For this reason, these systems are called infinite dimensional systems. Finding active controllers in order to influence the dynamics of these systems generate highly involved problems. The control theory for PDE governing the dynamics of smart structures is a mathematical description of such situations. Accurately modeling these structures play an important role to understanding not only the overall dynamics but the controllability and stabilizability issues. In the first part of the talk, the differences between the finite and infinite dimensional control theories are addressed. The major challenges tagged along in controlling coupled PDE are pointed out. The connection between the observability and controllability concepts for PDE are introduced by the duality argument (Hilbert's Uniqueness Method). Once this connection is established, the PDE models corresponding to the simple piezoelectric material structures are analyzed in the same context. Some modeling issues will be addressed. Major results are presented, and open problems are discussed. In the second part of the talk, a problem of actively constarined layer (ACL) structures is considered. Some of the major results are presesented. Open problems in this context are discussed. Some of this research presented in this talk are joint works with Prof. Scott Hansen (ISU) and Kirsten Morris (UW).

Monday, November 23, 2015 - 14:05 ,
Location: Skiles 005 ,
Li Wang ,
UCLA-&gt;SUNY Buffalo ,
Organizer: Martin Short

We study the shock dynamics for a gravity-driven thin film flow with a

suspension of particles down an incline, which is described by a system

of conservation laws equipped with an equilibrium theory for particle

settling and resuspension. Singular shock appears in the high particle

concentration case that relates to the particle-rich ridge observed in

the experiments. We analyze the formation of the singular shock as well

as its local structure, and extend to the finite volume case, which

leads to a linear relationship between the shock front with time to the

one-third power. We then add the surface tension effect into the model

and show how it regularizes the singular shock via numerical

simulations.

suspension of particles down an incline, which is described by a system

of conservation laws equipped with an equilibrium theory for particle

settling and resuspension. Singular shock appears in the high particle

concentration case that relates to the particle-rich ridge observed in

the experiments. We analyze the formation of the singular shock as well

as its local structure, and extend to the finite volume case, which

leads to a linear relationship between the shock front with time to the

one-third power. We then add the surface tension effect into the model

and show how it regularizes the singular shock via numerical

simulations.

Monday, November 16, 2015 - 14:05 ,
Location: Skiles 005 ,
Gil Ariel ,
Bar-Ilan University ,
Organizer:

Collective

movement is one of the most prevailing observations in nature. Yet, despite

considerable progress, many of the theoretical principles underlying the

emergence of large scale synchronization among moving individuals are still

poorly understood. For example, a key question in the study of animal motion is

how the details of locomotion, interaction between individuals and the environment

contribute to the macroscopic dynamics of the hoard, flock or swarm. The

talk will present some of the prevailing models for swarming and collective

motion with emphasis on stochastic descriptions. The goal is to identify some generic characteristics

regarding the build-up and maintenance of collective order in swarms. In

particular, whether order and disorder correspond to different phases,

requiring external environmental changes to induce a transition, or rather meta-stable

states of the dynamics, suggesting that the emergence of order is kinetic.

Different aspects of the phenomenon will be presented, from experiments with locusts

to our own attempts towards a statistical physics of collective motion.

movement is one of the most prevailing observations in nature. Yet, despite

considerable progress, many of the theoretical principles underlying the

emergence of large scale synchronization among moving individuals are still

poorly understood. For example, a key question in the study of animal motion is

how the details of locomotion, interaction between individuals and the environment

contribute to the macroscopic dynamics of the hoard, flock or swarm. The

talk will present some of the prevailing models for swarming and collective

motion with emphasis on stochastic descriptions. The goal is to identify some generic characteristics

regarding the build-up and maintenance of collective order in swarms. In

particular, whether order and disorder correspond to different phases,

requiring external environmental changes to induce a transition, or rather meta-stable

states of the dynamics, suggesting that the emergence of order is kinetic.

Different aspects of the phenomenon will be presented, from experiments with locusts

to our own attempts towards a statistical physics of collective motion.

Monday, November 2, 2015 - 14:05 ,
Location: Skiles 005 ,
Professor James von Brecht ,
Cal State University, Long Beach ,
Organizer: Martin Short

In this talk, I will discuss mathematical models and tools for analyzing

physical and biological processes that exhibit co-dimension one

characteristics. Examples include the assembly of inorganic

polyoxometalate (POM) macroions into hollow spherical structures and the

assembly of surfactant molecules into micelles and vesicles. I will

characterize when such structures can arise in the context of isotropic

and anisotropic models, as well as applications of these insights to

physical models of these behaviors.

physical and biological processes that exhibit co-dimension one

characteristics. Examples include the assembly of inorganic

polyoxometalate (POM) macroions into hollow spherical structures and the

assembly of surfactant molecules into micelles and vesicles. I will

characterize when such structures can arise in the context of isotropic

and anisotropic models, as well as applications of these insights to

physical models of these behaviors.

Thursday, October 29, 2015 - 11:00 ,
Location: Skiles 006 ,
Philippe Chartier ,
INRIA Rennes, Université de Rennes I, ENS Rennes ,
Philippe.Chartier@inria.fr ,
Organizer: Molei Tao

Joint with School of Math Colloquium. Special time (colloquium time).

In this talk, I will introduce B-series, which are formal series indexed by trees, and briefly expose the two laws operating on them. The presentation of algebraic aspects will here be focused on applications to numerical analysis. I will then show how B-series can be used on two examples: modified vector fields techniques, which allow for the construction of arbitrarly high-order schemes, and averaging methods, which lie at the core of many numerical schemes highly-oscillatory evolution equations. Ultimately and if time permits, I will illustrate how these concepts lead to the accelerated simulation of the rigid body and the (nonlinear) Schrödinger equations. A significant part of the talk will remain expository and aimed at a general mathematical audience.