# Mathematics and Statistics for Modelling and Prediction (MaP)

Our Thematic Doctoral Training Programme, MaP, offers a variety of interdisciplinary projects with the theme of Uncertainty Quantification.

## About MaP

There are three PhD studentships available, funded by the Engineering and Physical Sciences Council (EPSRC), and some further PhD studentships from other funding sources.

Reseaching at the point of contact between applied mathematics and statistics, we expect successful applicants to focus on modelling real-world problems under uncertainty, working collaboratively with fellow mathematicians, statisticians, researchers from other disciplines and industry contacts.

## How to apply

To apply for the programme for 2019 entry please:

- Identify three projects of interest
- Apply online using the University of Nottingham application page
- In the personal statement section indicate that you are applying to the '
**Mathematics and Statistics for Modelling and Prediction**' programme - Make sure to include a
**ranked list of your three preferred projects**, together with a CV of no more than two pages.

For more information, please contact Professor Andrew Wood.

## Eligibility and funding

All candidates should have, or expect to obtain, a First or 2:1 in mathematics, statistics or a related quantitative discipline, such as physics, engineering or computer science.

Fully funded studentships are available for UK applicants. EU applicants who are able to confirm that they have been resident in the UK for at least three years before October 2019 may also be eligible for a full award. EU students who are not able to prove that they meet the residency criteria may apply for a fees only award.

Successful applicants will receive a stipend (£14,777 per annum for 2018/19) for up to three-and-a-half years, tuition fees and a Research Training Support Grant.

## Projects

Please identify three projects of interest from this selection. If you have questions about a particular project, please contact the project supervisors directly.

**Bootstrap methods for hypothesis testing in non-linear models**

Dr Simon Preston and Prof Andrew Wood

Hypothesis testing is an important way to draw scientific conclusions from experimental data. However, models relevant in industrial settings (for example models that characterise manufacturing processes) are almost invariably non-linear in the model parameters, and hypotheses of interest often involve parameters that lie on the boundary of the parameter space; these are challenging to standard (asymptotic) approaches to hypothesis testing.

We will develop methods based on the "bootstrap" -- a powerful approach in computational statistics that involves computing null distributions using simulated data -- to address hypothesis testing in challenging non-linear settings.

**Controlling bacterial biofilm formation with shape**

**Developing machine learning and statistical techniques to analyse large ground motion datasets to determine the changes in the state of global peatlands**

**Bayesian inversion in resin transfer moulding**

Dr Marco Iglesias, Prof Michael Tretyakov

This project will be based at the University of Nottingham in the School of Mathematical Sciences and the Faculty of Engineering.

The use of fibre-reinforced composite materials in aerospace and automotive industries and other areas has seen a significant growth over the last two decades. One of the main manufacturing processes for producing advanced composites is resin transfer moulding (RTM). The crucial stage of RTM is injection of resin into the mould cavity to fill empty spaces between fibres; the corresponding process is described by an elliptic PDE with moving boundaries. Imperfections of the preform result in uncertainty of its permeability, which can lead to defects in the final product. Consequently, uncertainty quantification (UQ) of composites’ properties is essential for optimal RTM.

One of important UQ problems is quantification of the uncertain permeability. The objectives of this PhD project include (i) to construct, justify and test efficient algorithms for the Bayesian inverse problem within the moving boundary setting and (ii) to apply the algorithms to real data from composite laboratory experiments.

**Eligibility/Entry Requirements**: We require an enthusiastic graduate with a 1st class degree in Mathematics, preferably at MMath/MSc level (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). We are expecting that the successful applicant has a background in PDEs, Probability and Statistics and has exceptional computational skills.

For any enquiries please email: Marco.Iglesias@nottingham.ac.uk or Michael.Tretyakov@nottingham.ac.uk or Mikhail.Matveev@nottingham.ac.uk.

This project will be jointly supervised by Dr Mikhail Matveev in the Faculty of Engineering.

**Electromagnetic compatibility in complex environments – predicting the propagation of electromagnetic waves using wave-chaos theory**

Dr Stephen Creagh, Dr Gabriele Gradoni and Prof Gregor Tanner

Electromagnetic systems and devices are often complicated, irregular in their geometry and heterogeneous in their electrical characteristics. Such a system could be a PC, a mobile phone, or even an airplane cockpit. The prediction of the energy distribution becomes hard when using traditional analytical and numerical tools, especially if the wavelength is small compared to the size of the structure. Statistical methods are often more appropriate to describing the physical process under investigation in such cases. Appropriately chosen, such methods can lead to surprisingly simple and physically understandable characterization of the problem, which can be used to exploit complexity and turn collective behaviour into beneficial engineering technology.

This PhD project uses a phase-space representation of wave fields, the so-called Wigner distribution function (WDF), to unveiled transport properties of fields using tools of dynamical system theory. An exact evolution operator for the transport of these Wigner functions can be derived, and approximation schemes are obtained by using ray families that include reflections from irregular boundaries. The project will explore the possibility of linking the WDF operator to existing semiclassical approximations of quantum mechanics, used to transport densities of quantum particles. The challenge lies in constructing a phase space picture of those operators through the WDF before including the source operator.

Dr Gabriele Gradoni is also based at George Green Institute of Electromagnetics Research in the Faculty of Engineering.

**Energy storage bed dynamics –the ever-expanding magnesium bed conundrum**

In order to facilitate high penetration of renewable energy in to the grid, energy storage is needed to better manage the supply and demand for the grid. Hydrogen offers a high energy density solution and, rather than storing the hydrogen as a gas at high pressures, solid state storage of hydrogen in a metal like magnesium offers a low pressure and low cost technology. The hydrogenation of magnesium is very exothermic (74.5 kJ mol-1) and the material is also being investigated as a thermal energy store (i.e. using the exotherm of hydrogenation to liberate the stored thermal energy back as heat at 400°C).

A fear was that cycling a magnesium bed at high temperatures would lead to sintering and a loss of void space. However, the startling result was that the powdered magnesium bed when cycled at temperatures of 350-400°C, rather than losing porosity, gained porosity. The form of the bed had changed from a loose powder to a metal porous plug which had swelled in dimensions to fill the available head space in the vessel. Further cycling at temperature below 350°C results in the bed resorting back to a more densely packed loose powder.

The intriguing question is to uncover the fundamental mechanism(s) behind this process and to develop a predicative model based on the physical and chemical processes occurring. For the application, understanding these processes will enable optimisation of the porous structure for heat and mass flow; moreover, there is also concern the expanding bed may exert significant stress on the wall of the storage vessel eventually leading to failure of the vessel.

This challenging research project will develop new mathematical models based on the chemical and physical processes occurring in order to develop a model that simulates the expanding porous bed phenomenon. Some of these processes include: nucleation, growth of the metal hydride phase, crystal lattice expansion leading to defect formation, decrepitation, atomic diffusion and surface energy minimisation, annealing. The models developed will thus need to encompass a wide range of physical phenomena; the focus will be on partial-differential-equation/moving-boundary formulations, building on the established sintering literature but, for the reasons described above (specifically, to generate increased, rather than decreased, porosity), of necessity raising significant additional challenges. The project will accordingly equip the student with an unusually wide experience of experimental and modelling questions and of mathematical techniques, as applied in a context with clear energy and sustainability implications.

This project will be jointly supervised by Prof Gavin Walker in the Faculty of Engineering and Dr Richard Wheatley in the School of Chemistry.

**Form, function and utility in small community energy networks**

Dr Etienne Farcot and Dr Reuben O'Dea

This is a unique and exciting opportunity to undertake research that spans across the disciplines of energy engineering and mathematical sciences. Successful applicants will be joining a strong interdisciplinary team from academia and industry who are currently working on the delivery of the Energy Research Accelerator (ERA) Community Energy System (CES) demonstrator at the 15 acre Trent Basin site in Nottingham.

The project will investigate the energy challenges and complexity science issues associated with heat and electrical power generation, storage and use arising from the connections between micro-generation output, grid/heat loads, weather, and energy/power demands (including occupant behavior) combined with variable load energy storage devices in order to provide energy stability, a reduction of cost and associated carbon emissions from fossil fuel use.

The PhD research will develop new multi-vector CES models that utilise ‘big data’ obtained from a dedicated onsite monitoring platform at the housing development applied to a heterogeneous network of users. The work will ultimately help inform the design, implementation and operation of local community energy schemes in the UK.

Applicants should have a Bachelor Science or Engineering (at least 2i) and/or a Master of Science or Engineering in Mathematical Sciences, Engineering or Energy related disciplines.

This project will be jointly supervised by Prof Mark Gillott and Dr Parham Mirzaei Ahrnjani in the School of Architecture & Built Environment.

**Geometric statistical methods for curves and surfaces with applications in medical imaging**

What is the average shape of a brain tumour in a cohort of patients? How might this evolve over time? Is it possible to associate the shape and its evolution with certain genetic and clinical characteristics of a patient?

Advent of high-resolution imaging technologies have enabled meaningful answers to such questions based on continuous representations of tumours and organs as parametric curves and surfaces. These data objects typically reside on manifolds equipped with non-trivial geometries and symmetries (invariances).

The project will focus on developing statistical methods for such data using tools from stochastic processes, differential geometry and group theory.

**Inference for noisy ordinary differential equation models with application to "rust" modelling in sugar beet**

Dr Theodore Kypraios and Prof Andrew Wood

There are many areas of science and technology where ordinary differential equations provide suitable models but the complication is that empirical observations contain measurement errors. The challenge is such situations is to develop inference techniques, such as maximum likelihood or Bayesian approaches, for "noisy" ordinary differential equation models. Ideas and techniques from the topic Uncertainty Quantification are relevant here.

Various approaches will be developed and these will be applied to the modelling of "rust" in Sugar Beet, a form of disease in Sugar Beet which adversely affects growth and therefore yield. The methodology developed will be applied to a Sugar Beet dataset.

Applications to other datasets from plant biology will be considered too.

This project will be jointly supervised by Prof Neil Crout and Prof Debbie Sparkes in the School of Biosciences.

**Linking epidemiological and genomic data for infectious diseases**

Prof Philip O'Neill and Dr Theodore Kypraios

In the past few years, advances in sequencing technology and the reduction in associated costs have enabled scientists to obtain highly detailed genomic data on disease-causing pathogens on a scale never seen before. In addition to the inherent phylogenetic information contained in such data, combining genomic data with traditional epidemiological data (such as time series of case incidence) also provides an opportunity to perform microbial source attribution, i.e. determining the actual transmission pathway of the pathogen through a population.

These advances have seen a corresponding surge of activity in the modelling and statistical analysis community, so that now a number of methods and associated computer packages exist to carry out source-attribution, i.e. estimating who-infected-whom in a particular outbreak.

All the methods have their own limitations; a very common issue is that the models used to perform estimation are conditional upon the observed data, which can create estimation biases and lead to misleading results. In contrast, the method developed by Worby, Kypraios and O'Neill involves a model that can explain how the data arose, overcoming such problems.

This project is concerned with developing this approach to both (i) extend the idea to more complex model settings, relaxing certain technical assumptions and (ii) improve computational efficiency. A highly-detailed data set on MRSA provided by collaborators at Guy's and St Thomas' hospital trust, London, provides one opportunity for applying such methods.

**Relevant Publications**

Worby, C. J., O'Neill, P. D., Kypraios, T., Robotham, J. V., De Angelis, D., Cartwright, E. J. P., Peacock, S. J. and Cooper, B. S. (2016) Reconstructing transmission trees for communicable diseases using densely sampled genetic data. Annals of Applied Statistics 10(1), 395-417.

**Machine learning for first-principles calculation of physical properties**

The physical properties of all substances are determined by the interactions between the molecules that make up the substance. The energy surface corresponding to these interactions can be calculated from first-principles, in theory allowing physical properties to be derived ab-initio from a molecular simulation; that is by theory alone and without the need for any experiments.

Recently we have focussed on applying these techniques to model carbon dioxide properties, such as density and phase separation, for applications in Carbon Capture and Storage. However, there is enormous potential to exploit this approach in a huge range of applications. A significant barrier is the computational cost of calculating the energy surface quickly and repeatedly, as a simulation requires.

In collaboration with the School of Chemistry we have recently developed a machine-learning technique that, by using a small number of precomputed ab-initio calculations as training data, can efficiently calculate the entire energy surface.

This project will be jointly supervised by Dr Richard Wheatley in the School of Chemistry.

**Mathematical Modelling of Lubrication in Grinding Wheels**

Grinding wheels are used to machine surfaces in many different manufacturing processes. Two key components of the process are: the fluid lubricant, which is sprayed into the point where the wheel meets the workpiece; the surface of the grinding wheel, which may be either random or patterned. Between them, the lubricant and the grinding wheel surface must fulfil the, often competing, requirements to remove material from the workpiece in a controlled manner, remove cuttings from between the wheel and the workpiece, and cool the wheel and workpiece. This aggressive engineering environment is hard to investigate experimentally, and mathematical modelling is in its infancy, so trial and error is the usual means of improving the process. Recent work has considered the flow outside the grinding wheel, [1], and a current PhD student is developing a basic model for the lubrication flow.

There are many possible extensions and uses of the simple models we have at the moment that can form the basis of a project, including: optimization of grinding surface patterns to promote even flow and cuttings removal; the use of homogenization to investigate periodic, slowly-varying periodic and random grinding surfaces; the use of uncertainty quantification techniques to characterise random grinding surfaces; inclusion of heat and cuttings transport into the model. **

**Relevant Publications**

[1] Textured grinding wheels: A review, Li, H.N and Axinte, D.A. 2016, Int. J. Machine Tools and Manuf.

**Mathematical Modelling of Powder Snow Avalanches**

**Relevant Publications**

**Mechanical modelling of the stability of Earth's peatland carbon reservoirs**

**Modelling flow and crystallisation in polymers**

**Modelling the environmental and genomic interactions in maize under changing climatic conditions**

**Modelling wave propagation in meta-materials: a graph network approach**

**Molecule comparison using electrostatic fields and 3D shape representation**

**Opening the black-box: understanding the mechanisms and behaviours of data-driven water resource models**

**Opening the black-box: understanding the mechanisms and behaviours of data-driven water resource models**

**Quantum tomography for high dimensional systems**

**Relevant Publications**

**Stochastic Numerics**

**Relevant Publications**

**Text-Analytics for Major Event Detection**

**Thermal characterisation of the building fabric under uncertainty**

**Relevant Publications**