Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

 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.

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 )

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).

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.