Analysis of Self-Organization

The dissertation is devoted to the study of problems in calculus of variation, free boundary problems and gradient flows with respect to the Wasserstein metric. More concretely, we consider the problem of characterizing the regularity of minimizers to a certain interaction energy. Minimizers of the interaction energy have a somewhat surprising relationship with solutions to obstacle problems. Here we prove and exploit this relationship to obtain novel regularity results. Another problem we tackle is describing the asymptotic behavior of the Cahn-Hilliard equation with degenerate mobility. By framing the Cahn-Hilliard equation with degenerate mobility as a gradient flow in Wasserstein metric, in one space dimension, we prove its convergence to a degenerate parabolic equation under the framework recently developed by Sandier-Serfaty.

Spectral graph theory with applications to quantum adiabatic optimization

In this dissertation I draw a connection between quantum adiabatic optimization, spectral graph theory, heat-diffusion, and sub-stochastic processes through the operators that govern these processes and their associated spectra. In particular, we study Hamiltonians which have recently become known as ``stoquastic'' or, equivalently, the generators of sub-stochastic processes. The operators corresponding to these Hamiltonians are of interest in all of the settings mentioned above. I predominantly explore the connection between the spectral gap of an operator, or the difference between the two lowest energies of that operator, and certain equilibrium behavior. In the context of adiabatic optimization, this corresponds to the likelihood of solving the optimization problem of interest. I will provide an instance of an optimization problem that is easy to solve classically, but leaves open the possibility to being difficult adiabatically. Aside from this concrete example, the work in this dissertation is predominantly mathematical and we focus on bounding the spectral gap. Our primary tool for doing this is spectral graph theory, which provides the most natural approach to this task by simply considering Dirichlet eigenvalues of subgraphs of host graphs. I will derive tight bounds for the gap of one-dimensional, hypercube, and general convex subgraphs. The techniques used will also adapt methods recently used by Andrews and Clutterbuck to prove the long-standing ``Fundamental Gap Conjecture''.

DISPATCHING AND RELOCATION OF EMERGENCY VEHICLES ON FREEWAYS: THEORIES AND APPLICATIONS

Resource allocation decisions are made to serve the current emergency without knowing which future emergency will be occurring. Different ordered combinations of emergencies result in different performance outcomes. Even though future decisions can be anticipated with scenarios, previous models follow an assumption that events over a time interval are independent. This dissertation follows an assumption that events are interdependent, because speed reduction and rubbernecking due to an initial incident provoke secondary incidents. The misconception that secondary incidents are not common has resulted in overlooking a look-ahead concept. This dissertation is a pioneer in relaxing the structural assumptions of independency during the assignment of emergency vehicles. When an emergency is detected and a request arrives, an appropriate emergency vehicle is immediately dispatched. We provide tools for quantifying impacts based on fundamentals of incident occurrences through identification, prediction, and interpretation of secondary incidents. A proposed online dispatching model minimizes the cost of moving the next emergency unit, while making the response as close to optimal as possible. Using the look-ahead concept, the online model flexibly re-computes the solution, basing future decisions on present requests. We introduce various online dispatching strategies with visualization of the algorithms, and provide insights on their differences in behavior and solution quality. The experimental evidence indicates that the algorithm works well in practice. After having served a designated request, the available and/or remaining vehicles are relocated to a new base for the next emergency. System costs will be excessive if delay regarding dispatching decisions is ignored when relocating response units. This dissertation presents an integrated method with a principle of beginning with a location phase to manage initial incidents and progressing through a dispatching phase to manage the stochastic occurrence of next incidents. Previous studies used the frequency of independent incidents and ignored scenarios in which two incidents occurred within proximal regions and intervals. The proposed analytical model relaxes the structural assumptions of Poisson process (independent increments) and incorporates evolution of primary and secondary incident probabilities over time. The mathematical model overcomes several limiting assumptions of the previous models, such as no waiting-time, returning rule to original depot, and fixed depot. The temporal locations flexible with look-ahead are compared with current practice that locates units in depots based on Poisson theory. A linearization of the formulation is presented and an efficient heuristic algorithm is implemented to deal with a large-scale problem in real-time.

SOCIAL AND ECOLOGICAL FACTORS INFLUENCING COLLECTIVE BEHAVIOR IN GIANT DANIO

A fundamental problem in biology is understanding how and why things group together. Collective behavior is observed on all organismic levels - from cells and slime molds, to swarms of insects, flocks of birds, and schooling fish, and in mammals, including humans. The long-term goal of this research is to understand the functions and mechanisms underlying collective behavior in groups. This dissertation focuses on shoaling (aggregating) fish. Shoaling behaviors in fish confer foraging and anti-predator benefits through social cues from other individuals in the group. However, it is not fully understood what information individuals receive from one another or how this information is propagated throughout a group. It is also not fully understood how the environmental conditions and perturbations affect group behaviors. The specific research objective of this dissertation is to gain a better understanding of how certain social and environmental factors affect group behaviors in fish. I focus on two ecologically relevant decision-making behaviors: (i) rheotaxis, or orientation with respect to a flow, and (ii) startle response, a rapid response to a perceived threat. By integrating behavioral and engineering paradigms, I detail specifics of behavior in giant danio Devario aequipinnatus (McClelland 1893), and numerically analyze mathematical models that may be extended to group behavior for fish in general, and potentially other groups of animals as well. These models that predict behavior data, as well as generate additional, testable hypotheses. One of the primary goals of neuroethology is to study an organism's behavior in the context of evolution and ecology. Here, I focus on studying ecologically relevant behaviors in giant danio in order to better understand collective behavior in fish. The experiments in this dissertation provide contributions to fish ecology, collective behavior, and biologically-inspired robotics.