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Optimization OR & Statistics

Applied Probability and Electricity Networks

Reference Number: 2017-HW-AMS-14

Abstract: The decarbonisation of the electricity network is leading to substantial changes in both generation, e.g. renewable generators and increased electricity storage, and demand, e.g. electric vehicles and dynamic demand response. This is leading to the development of novel mathematical tools to address the challenges arising from these changes.  Many of these tools are related to ideas from probability and queueing theory including stability analysis, large deviations and decentralised algorithms.  In this PhD project the student will consider a number of models arising from power systems and explore the application of ideas and methods from queuing theory.

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Astrophysical inference using the LIGO gravitational wave detectors

Abstract: In September 2015, the LIGO detectors observed gravitational waves from a binary black hole merger for the first time. LIGO has subsequently observed five further events. As LIGO makes further observations, the properties of the observed population will allow us to constrain the astrophysical properties and origin of the sources. This project will focus on developing techniques for such inference and applying them to the LIGO data set.

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Developing data analysis techniques for low frequency gravitational wave detectors

Abstract: There are ongoing efforts to detect gravitational waves at nanohertz frequencies using the accurate timing of millisecond pulsars with pulsar timing arrays. ESA plans to launch a space based gravitational wave detector, LISA, observing in the millihertz band, and it has recently been realised that gravitational waves at microhertz frequencies can be detected by monitoring the positions of stars in the sky using astrometric satellites such as GAIA. Techniques for analysing the data from these various detectors and extracting the science are not fully developed and this project will focus on that. The project could concentrate on one of the three detectors or on methods that could be applied to all of them.

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High-Dimensional Extremes

Abstract: Extreme values of high-dimensional random vectors appear naturally in risk management. This thesis develops dimension reduction methods for multivariate extreme values, with a principal focus on high-dimensional extremes. A main goal is on tracking directions which account for a larger proportion of risk, and to use corresponding lower-dimensional representations to quantify risk in the raw high-dimensional random vector. Applications are envisioned in high- dimensional portfolios.

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Mathematical and statistical modelling of energy systems

Abstract: I can supervise broadly across energy systems analysis, including in operational and planning optimization, uncertainty quantification in large scale energy system models, security of supply risk analysis, statistical modelling of renewable resource, and people/institutional/communication aspects of the use of modelling in public policy decisions. My research in these areas is strongly informed by industry and government links, and there is particular scope for linking basic research to important application in security of supply analysis and modelling of renewable resources (wind, solar etc) and their relationship with demand.

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PDE-constrained optimization in scientific processes

Reference Number:

Abstract: A vast number of important and challenging applications in mathematics and engineering are governed by inverse problems. One crucial class of these problems, which has significant applicability to real-world processes, including those of fluid flow, chemical and biological mechanisms, medical imaging, and others, is that of PDE-constrained optimization. However, whereas such problems can typically be written in a precise form, generating accurate numerical solutions on the discrete level is a highly non-trivial task, due to the dimension and complexity of the matrix systems involved. In order to tackle practical problems, it is essential to devise strategies for storing and working with systems of huge dimensions, which result from fine discretizations of the PDEs in space and time variables. In this project, "all-at-once" solvers coupled with appropriate preconditioning techniques will be derived for these systems, in such a way that one may achieve fast and robust convergence in theory and in practice. This project is related to the EPSRC Fellowship.

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Statistical analysis of writing style in literature and social media (with application to cybersecurity)

Abstract: Suppose that the manuscript of a newly discovered play purported to be written by Shakespeare is found in an antique shop.  Professors of English Literature begin to debate whether the author was indeed Shakespeare, or an imposter. To what extent can statistical analysis help answer this question? in fact, the use of statistical methods for tasks such as this is quite common. The technical name for this is "authorship attribution" - creating quantitative models to describe the writing styles of authors and using these to assess whether they are the true authors of written texts of interests. Until recently, this was a niche field restricted to solving academic literary disputes. However the rise of the internet and social media has made such tools directly relevant to cybersecurity. Suppose that an anonymous post is made on an extremist Internet forum. Can analysis of the writing style be used to identify (or exclude) potential authors as part of a deanonymisation analysis? Or suppose that a person's Twitter account is hijacked, and a hacker starts posting new content on it. Can statistical analysis identify the change in writing style when it occurs? This project will examine traditional methods for authorship attribution where there is only a small number of candidate authors (Shakespeare, etc) and extend them to the modern internet era where there may be millions of possible authors, and only small text fragments associated with each. Potential tools include statistical classification, clustering, and Bayesian hierarchal modelling.

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Statistical mechanics for reaction-diffusion systems

Reference Number: ACM

Abstract: Many systems in chemistry, biology and engineering can be modelled by reaction-diffusion equations in which the populations of a number of different species move and interact or react, leading to the interchange of population or mass between the species. When the populations are large and/or reside in a complex environment (such as particles suspended in a turbulent flow), a powerful approach is to use techniques from statistical mechanics, which describes the 'average' behaviour of such systems. Dynamical density functional theory is one such approach that has met with great success over the past decade or so. This project will extend existing models, which generally describe only the dynamics, to include the reaction terms. The topics covered can be tailored to the interest of the student, covering both rigorous analysis and numerics. Techniques include statistical mechanics, stochastic dynamics, mathematical modelling, homogenisation theory of PDEs, and computational methods such as pseudo-spectral methods and finite elements. This project also has strong links to the work of members of the School of Engineering. 

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Statistical methods for earthquake forecasting

Abstract: Statistical analysis of earthquake recurrence and its input into seismic hazard assessments is of interest to both the insurance and disaster management industries. This is typically based on probability models for the expected time between earthquakes in a particular region. Building these statistical models is complicated by the limited number of events in historical records. Many current approaches resolve this by combining earthquake histories of multiple regions into a single catalogue. However, more sophisticated methods for combining information across multiple regions and coping with limited data in a more principled and flexible manner are needed. This project will focus on Bayesian hierarchical modelling to allow natural pooling of information across related regions, along with machine learning/artificial intelligence approaches to gain insight into earthquake recurrence and the variability in seismic cycles. (No previous experience in these methods is required).

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Stochastic differential equations, sampling and big data

Reference Number: ACM

Abstract: For details on the range of potential topics in this area please contact Konstantinos Zygalakis.

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Stochastic hybrid modelling of chemical systems

Reference Number: ACM

Abstract: It is well known that stochasticity can play a fundamental role in various biochemical processes, such as cell regulatory networks and enzyme cascades. Isothermal, well-mixed systems can be adequately modelled by Markov processes and, for such systems, methods such as Gillespies algorithm are typically employed. While such schemes are easy to implement and are exact, the computational cost of simulating such systems can become prohibitive as the frequency of the reaction events increases. This has motivated numerous coarse grained schemes, where the fast reactions are approximated either using Langevin dynamics or deterministically. While such approaches provide a good approximation for systems where all reactants are present in large concentrations, the approximation breaks down when the fast chemical species exist in small concentrations, giving rise to signicant errors in the simulation. This project will be concerned with developing further and analysing a new hybrid approach (a stochastic dierential equation with jumps) capable of dealing with more general systems.

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Tests of fundamental physics with gravitational waves

Abstract: Gravitational waves are generated in a highly dynamical regime of gravity that has not been well constrained observationally. Observations with LIGO and future detectors such as LISA will provide a unique way to probe the properties of gravitational theory in this regime. This project will develop techniques for producing systematic and comprehensive constraints on deviations form general relativity in future observations.

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Very large scale optimization with Interior Point Methods

Abstract: This PhD project will develop theory and implementation of interior point methods (IPMs) for huge scale constrained optimization problems. IPMs are particularly well-suited to solving very large problems and indeed as demonstrated by excessive computational experience, they stay beyond any competition. They are proved to solve problems of dimension N in no more that the square root of N iterations. However, in practice, they never need more than log(N) iterations. This project will aim at closing the gap between the best to-date known worst-case complexity result and the observed excellent behaviour of IPMs. 

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Volatility-Models for Multivariate Extremes

Abstract: Statistical models for multivariate extremes are one of the most commonly used strategies to approach problems where it is necessary to understand the association of variables during extreme scenarios (say joint extreme losses in a portfolio). This thesis develops the angular volatility as a natural modeling tool for settings where the structure of dependence between extreme values may be changing over time. The target is on developing measures which can track the dynamics governing extremal dependence of losses in a portfolio over time.

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