Georgia Institute of Technology College of Sciences

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.

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).