Seminars and Colloquia by Series

Meshfree finite difference methods for fully nonlinear elliptic equations

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 7, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Brittany FroeseNew Jersey Institute of Technology
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.

Massive data analysis helps modern medical datasets

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 15, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Hautieng WuUniversity of Toronto
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.

Noise is your friend, or: How well can we resolve state space?

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 25, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Predrag CvitanovićCenter for Nonlinear Science, School of Physics, GT
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 )

Mobile & Impeding Boundaries in Thermal Convection

Series
Applied and Computational Mathematics Seminar
Time
Monday, December 7, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Jun ZhangCourant Institute
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.

Modeling and Controllability issues for a general class of smart structures, a general outlook

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 30, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dr. Ahmet Özkan ÖzerUniversity of Nevada-Reno
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).

Shock dynamics in particle laden flow

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 23, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Li WangUCLA->SUNY Buffalo
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.

Stochastic models of collective motion

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 16, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Gil ArielBar-Ilan University
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.

Co-dimension One Motion and Assembly

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 2, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor James von BrechtCal State University, Long Beach
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.

Some algebraic techniques in the numerical analysis of ordinary differential equations

Series
Applied and Computational Mathematics Seminar
Time
Thursday, October 29, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Philippe ChartierINRIA Rennes, Université de Rennes I, ENS Rennes

Please Note: 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.

Relative Entropy Relaxations for Signomial Optimization

Series
Applied and Computational Mathematics Seminar
Time
Tuesday, October 27, 2015 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Venkat Chandrasekaran Cal Tech
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.)

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