Research
Current research interests
My major current research interests are in the area of mathematical optimization and control. This field lies at the fascinating interface between mathematics and engineering, marrying the practical applications of engineering science with the rigorous theory of mathematical analysis. Optimization problems are of fundamental interest in many practical applications today, and the study of various schemes by which these may be solved is thus of significant importance. The study of mathematical optimization and control has strong links with a great variety of other fields of research, for example convex analysis, network theory, and control theory. These connections can often be drawn upon in the analysis of schemes for solving optimization problems to aid in the derivation of strong theoretical results. In todays massively connected technological world, however, many classical techniques can become problematic either due to the large amounts of data involved or as a result of restrictions and prohibitions on information exchange. These problems motivate the idea of decentralized optimization - where the overall optimization problem is divided into smaller subproblems which are solved locally. These local solutions are then combined to reproduce a global solution.
At present, I am investigating ideas of decentralized optimization in the context of the Optimal Power Flow (OPF) problem. In the OPF problem, we have a known network topology, corresponding to a power grid, and known power loads at all of the nodes. The unknowns in the system are the generated powers at the given generation nodes and the voltages across the network. The goal of the OPF problem is then to determine the optimal operational choice for the system such that a convex generation cost function is minimized subject to the satisfaction of the power loads and various other system constraints. We are currently investigating properties of stability and convergence for an important class of algorithms for solving such problems in a decentralized manner, reducing both the data processing cost and the quantity of information that must be shared between rival operators.
Previous work
In the first year of my PhD, I undertook two short research projects, which are briefly described below.
1. "Decentralized control of large-scale networks", supervised by Dr. Ioannis Lestas, Department of Engineering
In this first project, we studied the general theory of decentralized control, and then applied this to a general class of distributed algorithms for the control of power allocations in wireless networks. We considered both undelayed networks and those in which the system nonlinearity incorporates a delay, meaning that stability theory for functional differential equations must be invoked. We proved uniform asymptotic stability for these algorithms whenever they produce bounded power allocations, and derived a condition that can be used to confirm that such bounded power allocations exist.
Downloads: Project description, Final report, Presentation slides.
2. "Wave singularities in ideal fluids", supervised by Dr. Gordon Ogilvie, DAMTP
Ideal fluids are those that lack dissipative properties. They can be modeled by a variety of systems of equations, depending on the precise situation under consideration. If we linearise these systems about particular solutions and assume harmonic time-dependence in the perturbation, a local dispersion relation can be obtained. In certain cases, this relation is observed to admit solutions for which the oscillation frequency remains finite in the limit as the wavelength vanishes. This corresponds to a solution oscillating at a finite frequency but confined within an infinitessimal region in space, i.e. a singular asymptotic solution that is localized about some particular surface in space. The aim of this project is to classify the situations in which such solutions can arise and to investigate their mathematical structure and significance.
Downloads: Project description, Final report, Presentation slides.