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 , firstname.lastname@example.org , 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.
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.
Tuesday, October 27, 2015 - 12:30 , Location: Skiles 005 , Venkat Chandrasekaran , Cal Tech , Organizer: Greg Blekherman
Due to its favorable analytical properties, the relative entropy function plays a prominent role in a variety of contexts in information theory and in statistics. In this talk, I'll discuss some of the beneficial computational properties of this function by describing a class of relative-entropy-based convex relaxations for obtaining bounds on signomials programs (SPs), which arise commonly in many problems domains. SPs are non-convex in general, and families of NP-hard problems can be reduced to SPs. By appealing to representation theorems from real algebraic geometry, we show that sequences of bounds obtained by solving increasingly larger relative entropy programs converge to the global optima for broad classes of SPs. The central idea underlying our approach is a connection between the relative entropy function and efficient proofs of nonnegativity via the arithmetic-geometric-mean inequality. (Joint work with Parikshit Shah.)
Monday, October 26, 2015 - 14:00 , Location: Skiles 005 , Professor Maarten de Hoop , Rice University , email@example.com , Organizer:
We consider an inverse problem for an inhomogeneous wave equation with discrete-in-time sources, modeling a seismic rupture. We assume that the sources occur along an unknown path with subsonic velocity, and that data is collected over time on some detection surface. We explore the question of uniqueness for these problems, and show how to recover the times and locations of sources microlocally first, and then the smooth part of the source assuming that it is the same at each source location. In case the sources (now all different) are (roughly speaking) non-negative and of limited oscillation in space, and sufficiently separated in space-time, which is a model for microseismicity, we present an explicit reconstruction, requiring sufficient local energy decay. (Joint research with L. Oksanen and J. Tittelfitz)
Simultaneous Random and Optimized Sources and Detectors for Efficient Optimization in Inverse ProblemsMonday, October 19, 2015 - 14:00 , Location: Skiles 005 , Eric de Sturler , Department of Mathematics, Virginia Tech , firstname.lastname@example.org , Organizer: Sung Ha Kang
In nonlinear inverse problems, we often optimize an objective function involving many sources, where each source requires the solution of a PDE. This leads to the solution of a very large number of large linear systems for each nonlinear function evaluation, and potentially additional systems (for detectors) to evaluate or approximate a Jacobian. We propose a combination of simultaneous random sources and detectors and optimized (for the problem) sources and detectors to drastically reduce the number of systems to be solved. We apply our approach to problems in diffuse optical tomography.This is joint work with Misha Kilmer and Selin Sariaydin.