Some Problems in Stochastic Control Theory Related to Inventory Management and Coarsening.

In this dissertation, we study two stochastic control problems arising from inventory management and coarsening. First, we study a stochastic production/inventory system with a finite production capacity and random demand. The cumulative production and demand are modeled by a two-dimensional Brownian motion process. There is a setup cost for switching on the production and a convex holding/shortage cost, and our objective is to find the optimal production/inventory control that minimizes the average cost. Both lost-sales and backlogging cases are studied. For the lost-sales model we show that, within a large class of policies, the optimal production strategy is either to produce according to an $(s,S)$ policy, or to never turn on the machine at all (thus it is optimal for the firm to not do the business); while for the backlog model, we prove that the optimal production policy is always of the $(s, S)$ types. Our approach first develops a lower bound for the average cost among a large class of non-anticipating policies, and then shows that the value function of the desired policy reaches the lower bound. The results offer insights on the structure of the optimal control policies as well as the interplay between system parameters. Then, we study a diffusive Carr-Penrose model which describes the phenomenon of coarsening. We show that the solution and the coarsening rate of the diffusive model converge to the classical Carr-Penrose model. Also, we demonstrate the relationship between the log concavity of the initial condition and the coarsening rate of the system. Under the assumption that the initial condition is log concave, there exists a constant upper bound on the coarsening rate of the diffusive problem. Our approach involves a representation of the solution using Dirichlet Green's function. To estimate this function, we exploit the property of a non-Markovian Gaussian process and derive bounds (both upper and lower) on the ratio between the Dirichlet and the full space Green's functions. The results shed light on the connection between the classical and diffusive Carr-Penrose models, and characterize the coarsening phenomenon under small noise perturbation.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/107311/1/wjch_1.pd

Phase Transitions

As in the past, the workshop brought together researchers with a background in physics, partial differential equations and continuum mechanics and statistical mechanics. Equilibrium and dynamic phase transitions were discussed. A wide range of systems from solid-solid phase transitions to the quantum Curie Weiss model were considered

Classical and Quantum Mechanical Models of Many-Particle Systems

The topic of this meeting were non-linear partial differential and integro-differential equations (in particular kinetic equations and their macroscopic/fluid-dynamical limits) modeling the dynamics of many-particle systems with applications in physics, engineering, and mathematical biology. Typical questions of interest were the derivation of macro-models from micro-models, the mathematical analysis (well-posedness, stability, asymptotic behavior of solutions), and “to a lesser extent” numerical aspects of such equations. A highlight of this meeting was a mini-course on the recent mathematical theory of Landau damping

The evolutionary game of pressure (or interference), resistance and collaboration

In the past few years, volunteers have produced geographic information of different kinds, using a variety of different crowdsourcing platforms, within a broad range of contexts. However, there is still a lack of clarity about the specific types of tasks that volunteers can perform for deriving geographic information from remotely sensed imagery, and how the quality of the produced information can be assessed for particular task types. To fill this gap, we analyse the existing literature and propose a typology of tasks in geographic information crowdsourcing, which distinguishes between classification, digitisation and conflation tasks. We then present a case study related to the “Missing Maps” project aimed at crowdsourced classification to support humanitarian aid. We use our typology to distinguish between the different types of crowdsourced tasks in the project and choose classification tasks related to identifying roads and settlements for an evaluation of the crowdsourced classification. This evaluation shows that the volunteers achieved a satisfactory overall performance (accuracy: 89%; sensitivity: 73%; and precision: 89%). We also analyse different factors that could influence the performance, concluding that volunteers were more likely to incorrectly classify tasks with small objects. Furthermore, agreement among volunteers was shown to be a very good predictor of the reliability of crowdsourced classification: tasks with the highest agreement level were 41 times more probable to be correctly classified by volunteers. The results thus show that the crowdsourced classification of remotely sensed imagery is able to generate geographic information about human settlements with a high level of quality. This study also makes clear the different sophistication levels of tasks that can be performed by volunteers and reveals some factors that may have an impact on their performance. In this paper we extend the framework of evolutionary inspection game put forward recently by the author and coworkers to a large class of conflict interactions dealing with the pressure executed by the major player (or principal) on the large group of small players that can resist this pressure or collaborate with the major player. We prove rigorous results on the convergence of various Markov decision models of interacting small agents (including evolutionary growth), namely pairwise, in groups and by coalition formation, to a deterministic evolution on the distributions of the state spaces of small players paying main attention to situations with an infinite state-space of small players. We supply rather precise rates of convergence. The theoretical results of the paper are applied to the analysis of the processes of inspection, corruption, cyber-security, counter-terrorism, banks and firms merging, strategically enhanced preferential attachment and many other

Fast, slow and super slow quantum thermalization

Thermalization is ubiquitous to all physical systems and is an essential assumption for the postulates of statistical mechanics. Generally, every system evolves under its own dynamics and reaches thermal equilibrium. In the quantum realm, thermal equilibrium is described by the Eigenstate Thermalization Hypothesis (ETH); hence every system that thermalizes is expected to follow ETH. Moreover, the thermalization process is always manifested as transport of matter and quantum information across the system. Thermalizing quantum systems with local interactions are expected to show diffusive transport of global conserved quantities and ballistic information spreading. The vast majority of many-body systems show the typical behavior described above. In this thesis, we study two mechanisms that break the standard picture of quantum thermalization. On the one hand, information spreading may be faster in the presence of long-range interactions. By simulating the Lieb-Robinson bounds in a spin chain with power-law decaying interactions, we distinguish the regime where the long-range character of the interactions becomes irrelevant for information spreading. On the other hand, the interplay of disorder and interactions can slow down transport, entering a sub-diffusive regime. We study this dynamical regime in an Anderson model on random regular graphs, where the emergence of a sub-diffusive regime before the localization transition is highly debated. Looking at long-range spectral correlations, we found that the sub-diffusive regime may be extended over the whole thermal phase of the model. Moreover, when disorder is strong enough, quantum many-body systems can undergo an ergodicity breaking transition to a many-body localized (MBL) phase. These systems do not follow ETH, so they present a challenge for conventional statistical mechanics. In particular, we study how the structure of local operator eigenstate matrix elements (central assumption of ETH) change between the thermal and MBL phase. A complete characterization of matrix elements of correlation functions is achieved via strong disorder quasi-degenerate perturbation theory. Furthermore, we study the MBL transition mechanism, which is still an open question due to the limitations of the available techniques for addressing that regime. Focusing on the avalanche mechanism, we simulate MBL spin chains coupled to a finite and infinite thermal bath. We could estimate the thermalization rate, which behaves as an order parameter and provide bounds for the actual critical disorder in the thermodynamic limit. We propose the existence of an intermediate MBL ``regime' where the system is slowly de-localizing, but relevant time scales are out-of-reach for current experiments and numerical simulations

Topological Photonics

Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.Comment: 87 pages, 30 figures, published versio