Monday, April 4, 2016 - 14:00 , Location: Skiles 005 , Wuchen Li , Georgia Tech Mathematics , Organizer: Martin Short
Fokker-Planck equations, along with stochastic differential equations, play vital roles in physics, population modeling, game theory and optimization (finite or infinite dimensional). In this thesis, we study three topics, both theoretically and computationally, centered around them.In part one, we consider the optimal transport for finite discrete states, which are on a finite but arbitrary graph. By defining a discrete 2-Wasserstein metric, we derive Fokker-Planck equations on finite graphs as gradient flows of free energies. By using dynamical viewpoint, we obtain an exponential convergence result to equilibrium. This derivation provides tools for many applications, including numerics for nonlinear partial differential equations and evolutionary game theory.In part two, we introduce a new stochastic differential equation based framework for optimal control with constraints. The framework can efficiently solve several real world problems in differential games and Robotics, including the path-planning problem.In part three, we introduce a new noise model for stochastic oscillators. With this model, we prove global boundedness of trajectories. In addition, we derive a pair of associated Fokker-Planck equations.
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 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
, arXiv:1206.5506 and arXiv: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.
Monday, November 30, 2015 - 11:05 , Location: Skiles 006 , Dr. Ahmet Özkan Özer , University of Nevada-Reno , email@example.com , Organizer: Chi-Jen Wang
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->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.
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.
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.