Network Optimization

Mathematics & Statistics Available PhD projects

Dr Sarah Barry

University of Strathclyde Diversity in Data Linkage Doctoral Training Centre (STRADDLE DTC) – Molecular, Clinical and Population Health (Funding Available)

Summary

1 fully funded, 3.5 year PhD studentship is available to a highly motivated UK or EU student to work within The STRADDLE DTC, a centre of excellence in the linkage and analysis of data across disciplines. The successful applicant will be trained to be part of the next generation of data scientists and develop skills and know how to deliver on the integration of diverse clinical data types – from molecules to man to populations.

Open date for applications: Accepting applications all year round

Duration of Project: 42 months

Funding Details: Funding will cover full tuition fees for UK/EU candidates, and provide a tax-free PhD stipend in the region of £14,000 per annum. Please note that candidates outside the EU will not be eligible for funding under this studentship.

Project Description

In recent years, the health sector has seen a significant increase in the volume and diversity of data that is gathered and stored on a daily basis. Ranging from administrative data, clinical observations and patient level attributes, through to epidemiological, omics data and images, the data is frequently siloed, often in different organisations, and analysed in the absence of the other data streams relating to that individual. To take advantage of the growing availability of these data types, a more comprehensive, systems modelling approach needs to be applied to account for multiple dimensions, integration of diverse data types, and changes over time. The STRADDLE DTC brings together experts in basic science, image analysis, statistics and health economics that are focused on deriving clinically relevant hypotheses from fusing a diverse range of datasets generated from molecular up to population level data. This collaborative and highly interdisciplinary approach has the potential to achieve new insights in to disease and to maximize the value of health research data, for example addressing problems such as antimicrobial resistance. 

We are looking for an applicant to join our ground breaking PhD programme, to help explore and advance new methodology, develop new visualisation and data analysis tools that are capable of harnessing the full potential of these diverse data sets. As a PhD candidate, you will sit within the Mathematics and Statistics department of the University of Strathclyde but will have equal supervision from the Business School. As a Strathclyde University PhD candidate, you will have access to the Strathclyde Researcher Development programme (PG Cert) offering you a competitive advantage as a research professional.

Supervisor Webpages

https://www.strath.ac.uk/staff/barrysarahdr/

https://www.strath.ac.uk/staff/megiddoitamarmr/

Keywords: big data, machine learning, statistics, health informatics, epidemiology, population health

Further Information: 

How to Apply: 
Applicants must have obtained, or expect to obtain, a first or 2.1 UK honours degree, or equivalent for degrees obtained outside the UK, in a quantitative or scientific discipline.

Please direct all enquiries and applications to sarah.barry@strath.ac.uk or itamar.megiddo@strath.ac.uk.  Applications will be reviewed when received, shortlisted candidates will be invited to interview on a rolling basis and it is anticipated that PhD Studentships will start in October 2019. The application process will remain open until the position is filled.

All applications must be submitted via email (subject line: PhD applicant – University of Strathclyde STRADDLE DTC) as a single pdf file and include the following:

  • A cover letter (max 1 page) explaining your interest and fit to the DTC programme
  • A CV (maximum three pages)

Dr Bingzhang Chen

Protecting our seas using computers - Modelling and forecasting the expansion of a highly invasive seaweed (Funding Available)

This fully-funded 3.5 year studentship will remain open only until filled. We recommend applying immediately.

In this project, you will meet with the challenge of developing an individual based model to simulate the growth and dispersal of a highly invasive seaweed (Japanese eelgrass) driven by coastal ocean currents. This alien seaweed has been identified as one of the top invasive species threatening UK coastal environments. The project mainly consists of two model components: a biological part (life history model) and a physical part (particle tracking model). The ocean currents will be derived from a state-of-art unstructured grid Finite Volume Community Ocean Model (FVCOM) specifically targeted to UK coastal seas. The model outputs will be fitted against field observational data and the parameters will be optimised by Bayesian inference such as Metropolis-Hastings Monte Carlo sampling.

It is anticipated that your model codes and results will provide invaluable information for prediction of this notorious invasive seaweed. Your mathematic, statistical, and programming skills are expected to be substantially sharpened during the PhD training. You will also have the invaluable opportunity to work on the high-performance computing system in Strathclyde. These skills will be very useful for securing some of the most popular jobs in this Big Data era.

You will mainly work within the Marine Population Modelling group, Department of Mathematics and Statistics, University of Strathclyde. You will also be co-supervised by Dr. Andrew Blight and Prof. David M. Paterson in the School of Biology, University of St Andrews and will also have the opportunity to work with the physical ocean modelers in Marine Scotland.

Qualifications

Applicants should have or expect to obtain a good honours degree (1, 2.1, or equivalent) in applied mathematics, statistics, ecology, or a highly quantitative science. Experience of numerical modelling and programming in Fortran or R would be beneficial.

How to apply

To apply, send 1) a complete CV, 2) a 1 page personal statement explaining your interests and skills for this project, and 3) names and contact information of three references to the lead supervisor, Dr Bingzhang Chen, Department of Mathematics and Statistics, University of Strathclyde, Glasgow at bingzhang.chen@strath.ac.uk.

The preferred starting date is 30 September 2019.

We value diversity and welcome applications from all sections of the community.

The University currently holds a Bronze Athena SWAN award, recognising our commitment to advancing women’s careers in science, technology, engineering, maths and medicine (STEMM) employment in academia.

Funding Notes

This studentship is funded by the University of Strathclyde and is open to all nationalities. However, it is expected that non-EU/UK students should bring their own funding to match up with the extra international fee.

Background reading

Sigman, D. M., & Boyle, E. A. (2000). Glacial/interglacial variations in atmospheric carbon dioxide. Nature, 407, 859.

Laws, E. A., Falkowski, P. G., Smith Jr, W. O., Ducklow, H., & McCarthy, J. J. (2000). Temperature effects on export production in the open ocean. Global Biogeochemical Cycles, 14, 1231-1246.

Cael, B.B. and Follows, M.J., 2016. On the temperature dependence of oceanic export efficiency. Geophysical Research Letters, 43, 5170-5175.

Chen, B. and Laws, E.A., 2017. Is there a difference of temperature sensitivity between marine phytoplankton and heterotrophs?. Limnology and Oceanography, 62, pp.806-817.



Dr Young-Ho Eom

Mathematical modelling and analysis of network cascades

Many complex systems ranging from brain and ecosystems to social organisations and financial systems can be represented as networks of interacting entities (e.g., neurons, species, people, and banks). Network science is the study of connectivity among different entities and its impact on (and coupling with) dynamics and function [1].

Cascades in networks are self-amplifying processes by which a relatively small event may precipitate a change across a substantial part of a system [2]. For examples, the failure of few banks can trigger the failure of the global financial system as the banks are interdependent through the network of obligations. Other examples of network cascades can include (i) information diffusion in social networks (some news or videos become ‘viral’ and millions of people share them while most of them have only limited influence) and (ii) blackouts in power grids (a local failure of the transmission lines can cause the failure of the grid because the redistributed load from the failed lines make the other lines overloaded), to name a few. Therefore, how cascades arise in networked systems, what is the role of network structure in cascades, and how to predict and control cascades are important questions in network science.

To address the above questions, this project will develop mathematical models for network cascades based on data analysis of real-world systems, analyse the models, and apply the outcomes to real-world problems (e.g., developing design principles for stable financial networks and reducing the impact of sudden shocks in infrastructures). In particular we will focus on the role of modular network structure in network cascades. Both mathematical and computational skills are required. Furthermore domain knowledge on specific systems (e.g., finance, physics, or engineering) will be necessary. Hence the student needs to be willing to learn these topics and open to interdisciplinary research.

If you are interested in the project, please feel free to contact. Please note that early contact is better for your PhD studentship application.

References and Suggested reading

[1] M. A. Porter and S. D. Howison. The Role of Network Analysis in Industrial and Applied Mathematics. arXiv:1703.06843. (https://arxiv.org/pdf/1703.06843)

[2] A. E. Motter and Y. Yang. The unfolding and control of network cascades. arXiv:1701.00578. (https://arxiv.org/pdf/1701.00578)

 

Dr Alison Gray

Digital image analysis for the classification of tea

Classification of teas (types, quality grades, region of origin etc) has been examined by many researchers, using both chemical composition and more recently digital image analysis techniques to extract features from the image that are useful for classification.

Factors affecting the success of classification include the choice of features, the classifier and the imaging modality. Building on previous work at Strathclyde in collaboration with the EEE department, this project will allow the student to examine the choice of any of these to achieve optimal results. Intending students should have a strong statistical background and excellent computer skills, and be competent/be able to quickly become competent in the use of both R and Matlab. 

Honey bee colony losses and associated risk factors

Second supervisors: George Gettinby, Magnus Peterson

Worldwide losses of honey bee colonies have attracted considerable media attention in recent years and a huge amount of research. Researchers at Strathclyde have experience since 2006 of carrying out a series of surveys of beekeepers in Scotland (http://personal.strath.ac.uk/a.j.gray/) and now have 5 years of data arising from these surveys. These data have been used to estimate colony loss rates in Scotland and to provide a picture of beekeepers’ experience and management practices.

This project will examine the data in more detail than has been done so far, and is likely to involve data modelling and multivariate methods to identify risk factors. Part of the project will involve establishing the spatial distribution of various bee diseases.

This project will build on links with the Scottish Beekeepers’ Association and membership of COLOSS, a network linking honey bee researchers in Europe and beyond. Intending students should have a strong statistical background and excellent computer skills, and be competent/be able to quickly become competent in the use of R.

Prof David Greenhalgh

Stochastic Models in Epidemiology and Population Dynamics

Introduction
In this project we shall look at how stochastic models can be used to describe how infectious diseases spread. We will start off by looking at one of the simplest epidemic models, the SIS (susceptible-infected-susceptible) model. In this model a typical individual starts off susceptible, at some stage catches the disease and after a short infectious period becomes susceptible again.

These models are used for diseases such as pneumococcus amongst children and sexually transmitted diseases such as gonorrhea amongst adults (Bailey, 1975). Previous work has already looked at introducing stochastic noise into this model via the disease transmission term (Gray et al. 2011). This is called environmental stochasticity which means introducing the random effects of the environment into how the disease spreads. This results in a stochastic differential equation (SDE) model which we have analysed. We have derived an expression for a key epidemiological parameter, the basic reproduction number.

In the deterministic model this is defined as the expected number of secondary cases caused by a single newly-infected individual entering the disease-free population at equilibrium. The basic reproduction number is different in the stochastic model than the deterministic one, but in both cases it determines whether the disease dies out or persists. In the stochastic SIS SDE model we have shown the existence of a stationary distribution and that the disease will persist if the basic reproduction number exceeds one and die out if it is less than one.

SDE Models with Environmental Stochasticity
We have also looked at other SDE models for environmental stochasticity. One of these took a simple deterministic model for the effect of condom use on the spread of HIV amongst a homosexual population and introduced environmental stochasticity into the disease transmission term. Again we found that a key parameter was the basic reproduction number which determined the behaviour of the system.

As before this was different in the deterministic model than the stochastic one. Indeed it was possible for stochastic noise to stabilise the system and cause an epidemic which would have taken off in the deterministic model to die out in the stochastic model (Dalal et al., 2007). Similar effects were observed in a model for the internal viral dynamics of HIV within an HIV-infected individual (Dalal et al., 2008).

Demographic Stochasticity
The real world is stochastic, not deterministic, and it is difficult to predict with certainty what will happen. Another way to introduce stochasticity into epidemic models is demographic stochasticity. If we take the simple homogeneously mixing SIS epidemic model with births and deaths in the population we can derive a stochastic model to describe this situation by defining p(i, j, t) to be the probability that at time t there are exactly i susceptible and j infected individuals and deriving the differential equations satisfied by these probabilities.

Then we shall look at how stochastic differential equations can be used to approximate the above set of equations for p(i, j, t). This is called demographic stochasticity and arises from the fact that we are trying to approximate a deterministic process by a stochastic one (Allen, 2007). Although the reasons for demographic and environmental stochasticity are quite different the SDEs which describe the progress of the disease are similar. The first project which we shall look at is analysis of the SIS epidemic model with demographic stochasticity along the lines of our analysis of the SIS epidemic model with environmental stochasticity.

Further Work
After this we intend to look at other classical epidemiological models, in particular the SIR (susceptible-infected-removed) model in which an individual starts off susceptible, at some stage he or she catches the disease and after a short infectious period he or she becomes permanently immune. These models are used for common childhood diseases such as measles, mumps and rubella (Anderson and May, 1991). We would look at introducing both environmental and demographic stochasticity into this model.

Other epidemiological models which could be analysed include the SIRS (susceptible-infected-removed-susceptible) epidemic model, which is similar to the SIR epidemic model, except that immunity is not permanent, the SEIS (susceptible-exposed-infected-susceptible) model which is similar to the SIS model, but includes an exposed or latent class, and the SEIR (susceptible-exposedinfected- removed) model, which similarly extends the SIR model.

We would also aim to look at other population dynamic models such as the Lotka-Volterra predator-prey model. There is also the possibility of developing methods for parameter estimation in all of these epidemiological and population dynamic models, and we have started work on this with another Ph. D. student (J. Pan).

References
1. E. Allen, Modelling with Itˆo Stochastic Differential Equations, Springer-Verlag, 2007.
2. R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
3. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Second Edition, Griffin, 1975.
4. A.J. Gray, D. Greenhalgh, L. Hu, X.
5. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for AIDS and condom-use. J. Math. Anal. Appl. 325, 36-53, 2007.
6. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics. J. Math. Anal. Appl. 341, 1084-1101, 2008.

Mathematical Modelling of Disease Awareness Programs

Differential equation models are commonly used to model infectious diseases. The population is divided up into compartments and the flow of individuals through the various classes such as susceptible, infected and removed is modelled using a set of ordinary differential equations (Anderson and May, 1991, Bailey, 1975). A basic epidemiological parameter is the basic reproduction number. This is defined as the expected number of secondary cases produced by a single newly infected case entering a disease-free population at equilibrium (Diekman and Heesterbeek). Typically the disease takes off if R0 > 1 and dies out if R0≤1.

However media awareness campaigns are often used to influence behaviour and if successful can alter the behaviour of the population. This is an area which has not been studied much until recently. The student would survey the existing literature on media awareness models in the literature and with the supervisor formulate mathematical models using differential equations for the effect of behavioural change on disease incidence. These would be examined using both analytical methods and computer simulation with parameters drawn from real data where appropriate. The mathematical techniques used would be differential equations, equilibrium and stability analyses and computer simulation.

References
1. R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
2. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Second Edition, Griffin, 1975.
3. O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction number R0 for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365-382.
4. A. K. Misra, A. Sharma and J. B. Shukla, Modelling and analysis of effects of awareness programs by media on the spread of infectious diseases. Math. Comp. Modelling 53, 1221- 1228.
5. A. K. Misra, A. Sharma, V. Singh, Effect of awareness programs in controlling the prevalence of an epidemic with time delay, J. Biol. Systems, 19(2), 389-402,

Dr Michael Grinfeld

Dynamics of reaction-diffusion systems with non-Fickian diffusion

In recent years there has been much work on reaction-diffusion equations in which the diffusion mechanism is not the usual Fickian one. Examples are integro-differential equations, porous media type equations, pseudodifferential equations, p-Laplacian type equations and prescribed curvature type (saturating flux) equations.

The motivation for this work comes from material science and mathematical ecology. However, there are applied contexts where these diffusion mechanisms have never been considered. One is in the area of combustion and the other is in the area of regularised conservation laws and shock propagation. This project, which would build on the work I did through the years on integrodifferential models and recently with M. Burns on the prescribed curvature equations, will use PDE, asymptotic, and topological methods to explore the dynamics of blowup and of shock propagation in canonical examples of reaction equations and nonlinear scalar conservation laws regularised by non-Fickian diffusion terms.

References:

[1] M. Burns and M. Grinfeld, Steady state solutions of a bistable quasilinear equation with saturating flux, European J. Appl. Math. 22 (2011), 317-331.

[2] M. Burns and M. Grinfeld, Steady state solutions of a bistable quasilinear equation with saturating flux, European J. Appl. Math. 22 (2011), 317-331.

Coarsening in integro-differential models of materials science

Recently, a new class of model has been developed to describe, for example, phase separation in materials such as binary alloys. These take the form of integrodifferential equations. Coarsening, that is, creation of large scale patterns in such models is poorly understood.

There are partial results [1, 2] that use the maximum principle, while for most interesting problems such a tool is not available. This will be a mixture of analytic and numerical work and will need tools of functional analysis and semigroup theory.

References:

[1] D. B. Duncan, M. Grinfeld, and I. Stoleriu, Coarsening in an integro-differential model of phase transitions, Euro. J. Appl. Math. 11 (2000), 561-572.

[2]  V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math. 17 (2006), 221-232.

Professor Mike Heath

Mathematical ecology of ocean food webs (Funding available)

Project Description

Marine sonar records have long documented  the existence of a layer of acoustic scattering at around 400m depth throughout the world’s oceans, which can undertake diel (night-time) vertical migrations to the near-surface waters. For many years the origins this phenomenon were uncertain, but now we know that it is due to concentrations of small fish and invertebrates which make up what we call the “mesopelagic community” of marine life. We also know that this community plays key role in the ocean food web, providing food for top predators such as tuna, seabirds and whales. Mesopelagic organisms also affect the vertical flux of carbon into the interior of the ocean as a result of eating at the surface and respiring at depth, and so are involved in global climate regulation [1,2].

The fish component of the mesopelagic community (e.g. lantern fishes) is attracting particular attention. Recent global assessments suggest that these fishes represent a huge unexploited resource that could be used to meet human feed demands [1]. However the technological and economic challenges of harvesting, and the biodiversity and climate change consequences, raise doubts about the viability of such a venture. These issues will be addressed in a newly funded EU Horizon2020 project “Sustainable management of mesopelagic resources (SUMMER)” due to start in September 2019 [3]. This studentship will form part of the UK contribution to SUMMER.

The studentship will be part of a Work-package on “Food-web structure and resilience”, specifically on the development and application of mathematical models of ocean food webs incorporating mesopelagic fish and invertebrates, and top predators, to investigate the consequences of mesopelagic exploitation. The student will be based in the Marine Population Modelling group, Department of Mathematics and Statistics, University of Strathclyde [4] and focus on extending an existing food web model of shelf seas to represent ocean food webs [5,6]. The work will involve close collaboration with other modelling teams in Denmark, Portugal and Turkey, and also with sea-going researchers who will be gathering new data to support the models. In the first instance the model developments will focus on legacy field data archives from the Irminger Sea region east of Greenland, and extend to the Mediterranean and equatorial regions of the Atlantic later in the project.

The project will require the student to become an expert in ocean ecology and the application of mathematics and computer programming skills to ecological problems. Realistically, the applicant should have, or expect to obtain, a good honours degree (1, 2.1, or equivalent) in applied mathematics, statistics, earth science, ecology, or a highly quantitative science, and be able to demonstrate  experience of programming in C and/or R.  It is anticipated that you will receive trainings on mathematical and statistical modelling including but not limited to numerical implementation and solving of differential equations and optimisation methods. Your mathematical, statistical, and programming skills are expected to be substantially enhanced during the PhD training. These skills will be very useful for securing some of the most popular jobs in this Big Data era.

To apply, send 1) a complete CV, 2) a 1 page personal statement explaining your interests and skills for this project, and 3) names and contact information of three references to the lead supervisor, Prof. Michael Heath, Department of Mathematics and Statistics, University of Strathclyde, Glasgow at m.heath@strath.ac.uk

There is no specific closing date for this project – we will appoint when we find the most appropriate candidate, but the earliest starting date is 30 September 2019.

We value diversity and welcome applications from all sections of the community.
The University currently holds a Bronze Athena SWAN award, recognising our commitment to advancing women’s careers in science, technology, engineering, maths and medicine (STEMM) employment in academia.

Funding Notes

This studentship is funded by the EU Horizon 2020 Programme and is open to all nationalities. However, it is expected that non-EU/UK students should bring their own funding to match up with the extra international fee.  The participation of the University of Strathclyde in the H2020 SUMMER project is not expected to be dependent on the outcome of Brexit.

References

[1] Irigoien, X., et al. (2014). Large mesopelagic fishes biomass and trophic efficiency in the open ocean. Nat. Communicat. 5, 3271. doi: 10.1038/ncomms4271

[2] St John et al. (2016). A Dark Hole in Our Understanding of Marine Ecosystems and Their Services: Perspectives from the Mesopelagic Community. Front. Mar. Sci. 17 March 2016 | https://doi.org/10.3389/fmars.2016.00031

[3] EU CORDIS website – Horizon 2020 Sustainable management of mesopelagic resources. project. https://cordis.europa.eu/project/rcn/223251/factsheet/en?WT.mc_id=RSS-Feed&WT.rss_f=project&WT.rss_a=223251&WT.rss_ev=a

[4] Strathclyde Marine Population Modelling Group: https://www.strath.ac.uk/science/mathematicsstatistics/smart/marineresourcemodelling/

[5] Heath, M.R. ( 2012). Ecosystem limits to food web fluxes and fisheries yields in the North Sea simulated with an end‐to‐end food web model. Prog. Oceanogr. 102, 42– 66.

[6] Heath M.R. et al. (2014). Understanding patterns and processes in models of trophic cascades. Ecology Letters 17, 101-114.

 

Professor Adam Kleczkowski 

Modelling plant trade networks and their response to plant diseases (Funding available)

The dispersal of plant diseases is among the most important side effects of a closely integrated global economy. There are many examples of plant diseases that were introduced to the UK and other countries by trade, including Phytophthora ramorum, Dutch elm disease and ash dieback. The number of new plant diseases appearing in Europe each year more than quadrupled during the 20th and the early 21st century.

The plant trading network of the UK is likely to undergo significant modifications due to policy changes, both Brexit-related and due to potential reorientation of the British agriculture towards more emphasis on conserving the environment. This might result in an increase in long-range trade with more diverse partners, resulting potentially in the rise of the risk of importing pests, or an increase in domestic production where non-compliant businesses create larger risks. Thus, introducing stricter regulation might have an unintended consequence of a rearrangement of trading networks which may lead to a shift in the risk pathways rather than the overall reduction. The diversification and increased short-term dynamics of the trading networks is also associated with the rise of e-commerce. The rise of internet trading affects both the traditional supplier (e.g. plant nurseries) and the individual customer, resulting in a potential loss of traceability and quality.

By combining data analysis with bioeconomic modelling, this project aims to develop a framework to analyse the risks associated with plant trade, to highlight the pathways associated with high risk, and to assess strategies that can be used to minimise the risks at different border points (international and domestic “borders”). Firstly, the research will look to assess the gains from combining different data sources as well as improved data. This might include studying public data such as that held for England & Wales at FERA, as well as a consideration of statutory plant movement data that is currently privately held and unlinked. Secondly, we will use the novel bioeconomic framework to study influence of changes in disease risk on the change of trade decisions on the way buyers and sellers are connected. Finally, we will use the cutting-edge mathematical models of dynamic and adaptive networks to capture the traders behaviour whereby the individuals alter the pattern of with whom to trade and how much to trade in response to the epidemic.

The studentship is jointly funded by Defra and the University of Strathclyde for a 3.5 years study period. Applicants should have a first-class or a 2.1 honours degree in a relevant subject; Masters degree will be an advantage. A more detailed plan of the studentship is available to candidates upon application. Funding is available for UK and European applications.

Professor Xuerong Mao

Numerical Methods for SDEs under the Local Lipschitz Condition

Up to 2002, most of the existing strong convergence theory for numerical methods requires the coefficients of the SDEs to be globally Lipschitz continuous [1]. However, most SDE models in real life do not obey the global Lipschitz condition. It was in this spirit that Higham, Mao and Stuart in 2002 published a very influential paper [2] (Google citation 319) which opened a new chapter in the study of numerical solutions of SDEs---to study the strong convergence question for numerical approximations under the local Lipschitz condition.

Since the classical explicit Euler-Maruyama (EM) method has its simple algebraic structure, cheap computational cost and acceptable convergence rate under the global Lipschitz condition, it has been attracting lots of attention.

Although it was showed that the strong divergence in finite time of the EM method for SDEs under the local Lipschitz condition, some modified EM methods have recently been developed these SDEs. For example, the tamed EM method was developed in 2012 to approximate SDEs with one-sided Lipschitz drift coefficient and the linear growth diffusion coefficient. The stopped EM method was developed in 2013. Recently, Mao [3] initiated a significantly new method, called the truncated EM method, for the nonlinear SDEs. The aim of this PhD is to develop the truncated EM method. The detailed objectives are:

(1) To study the strong convergence of the truncated EM method in finite-time for SDEs under the generalised Khasminskii condition and its convergence rate.

(2) To use the truncated EM method to investigate the stability of the nonlinear SDEs. Namely to study if the numerical method is stochastically stable when the underlying SDE is stochastically stable and to study if we can infer that the underlying SDE is stochastically stable when the numerical method is stochastically stable for small stepsize.

A PhD studentship might be available for the project.

References:

[1] Mao X., Stochastic Differential Equations and Applications, 2nd Edtion, Elsevier, 2007.

[2] Higham D., Mao X., Stuart A., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40(3) (2003), 1041--1063.

[3] Mao X., The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), 370--384.

Stochastic Modelling of Saving Accounts Linked to Stock Market

Income needs seem to be a top priority at the moment and with low savings rates and top UK equity income funds yielding less than 4\%, it is perhaps easy to understand why. With savings rates continuing at their record lows, some savers are turning to alternatives. It is in this spirit that many financial institutions are offering stock market linked savings plans to those looking to combine a high yield opportunity with some protection against a falling stock market. That is why the stock market linked savings accounts have recently become very popular. The returns of these accounts are random so the returns, even the initial capital, are not guaranteed. They are very much different from the familiar fixed-term-fixed-rate savings accounts.

The aim of this PhD project is to perform the stochastic and numerical analysis on the stock market linked savings accounts in order to establish the theory on the mean percentage of return (MPR). The MPR depends mainly on two factors:

(1) The structure of the saving accounts, namely the terms of the portfolios (plans);

(2) The behaviour of the stock market linked.

There are various portfolios in the market. This PhD project will analyze a number of typical plans. On the stochastic modelling of the stock market, there are various stochastic differential equation (SDE) models. In this project, some of these SDE models will be used. The key objective here is to establish the explicit formulas for the MPRs of the underlying stock marker linked saving accounts if possible; otherwise develop some new techniques to establish better approximation schemes.

The project will then develop the techniques of the stochastic and numerical analysis to deal with other more complicated financial derivatives including bond, fund.

A PhD studentship might be available for the project.

Professor Nigel Mottram

Mathematical modelling of active fluids

The area of active fluids is currently a “hot topic” in biological, physical and mathematical research circles. Such fluids contain active organisms which can be influenced by the flow of fluid around them but, crucially, also influence the flow themselves, i.e. by swimming. When the organisms are anisotropic (as is often the case) a model of such a system must include these inherent symmetries. Models of bacteria and even larger organisms such as fish have started to be developed over the last few years in order to examine the order, self-organisation and pattern formation within these systems, although direct correlation and comparison to real-world situations has been limited.

This project will use the theories and modelling techniques of liquid crystal systems and apply such modelling techniques to the area of anisotropy and self-organisation derived from active agents. The research will involve a continuum description of the fluid, using equations similar to the classical Navier-Stokes equations, as well as both the analytical and numerical solution of ordinary and partial differential equations.

Flow of groundwater in soils with vegetation and variable surface influx

Flow of groundwater in soils with vegetation and variable surface influx Groundwater is the water underneath the surface of the earth which fills the small spaces in the soil and rock and is extremely important as a water supply in many areas of the world. In the UK, groundwater sources, or aquifers, make up over 30% of the water used, and a single borehole can provide up to 10 million litres of water every day (enough for 70,000 people).

The flow of water into and out of these aquifers is clearly an important issue, more so since current extraction rates are using up this groundwater at a faster rate than it is being replenished. In any specific location the fluxes of water occur from precipitation infiltrating from the surface, evaporation from the surface, influx from surrounding areas under the surface, the flow of surface water (e.g. rivers) into the area, and the transpiration of water from underground directly into the atmosphere by the action of rooted plants.

This complicated system can be modelled using various models and combined into a single system of differential equations. This project will consider single site depth-only models where, even for systems which include complicated rooting profiles, analytical solutions are possible, but also two- and three-dimensional models in which the relatively shallow depth compared to the plan area of the aquifer can be utilised to make certain "thin-film" approximations to the governing equations.

Prof Tony Mulholland

Mathematical Modelling to aid the Detection of Cracks in Safety Critical Structures (Funding available)

Short Abstract: Artificial Intelligence is set to revolutionise every aspect of our lives and this PhD will enable you to start to make your contribution to this exciting field.  You will combine mathematical modelling with experimental data to study a nonlinear dynamical system involving colliding particles suspended in a fluid flow.  The project is co-funded by our industry partners and so this PhD will act as the perfect platform for you to have a career as a mathematician in industry.

 

Number of Places:    1

Opens:                        25th Feb 2019

Deadline:                    31st Mar 2019

Funding:                     UK/EU (Home Fees and Stipend)

Mode of Study:          Full-time

 

Eligibility

Qualifications: You should have, or expect to obtain, a first or upper second class honours degree or equivalent in mathematics, physics, engineering or a discipline with a high mathematical content.

Funding:  This studentship covers tuition fees and provides an annual tax-free stipend of approx. £14,000 in the first year (typically rising with inflation for subsequent years) - additional support for conference travel is provided. The award is for 3.5 years, subject to satisfactory performance. To qualify applicants should be UK or EU nationals, or should have been “ordinarily resident” in the UK for the last 3 years.  Overseas students can apply but this award only covers the UK/EU fees and so applicants would need to have sufficient additional funds to cover the difference between UK/EU and overseas fees (details can be provided upon request). 

 

Project Details

Background and Motivation

Approximately 70% of the world’s oil and gas reserves are in reservoirs where the fluids that travel along the pipelines contain sand.  This is a real problem for the engineers; for example, 14% of leaks in the UK Continental Shelf oil and gas production are caused by pipe erosion by the sand.  Acoustic emission sensors are used to monitor this sand in a bid to prevent such disasters. Ultrasound sensors record the impact of the sand on the pipe walls but are currently limited to simply detecting the presence of the sand and, with regular calibration, the sand rate.

 

The Project

The technical objective of this PhD is to use mathematical techniques and machine learning to extract more information from these complex data sets; for example, to quantify the sand size distribution, volume, shape, and velocity.  Armed with this information, UK industry would be able to realise a step-change in its capability to reduce the effects of sand erosion and this would lead to increased production rates, extended well lifetimes, and optimised pipe inspection regimes. 

 

You

So this is where you come in.  You will work in a multi-disciplinary team across two universities and several industry partners.  You will develop new mathematical methods and algorithms to tackle this problem. The project is ambitious and transformative and will require mathematical skills to be brought together from topics such as nonlinear dynamical systems, differential equations, and artificial intelligence.  Don’t worry however as we will provide all the necessary training.  What we need from you is an enquiring mind, a thirst for knowledge, and an excitement about this project.

There will be lots of opportunities to travel to for training and to meet with other PhD students in the different centres, industries and universities involved.  Your interactions with the industry partners will provide you with an appreciation of the industrial and commercial opportunities surrounding this research project.  This PhD will act as a perfect springboard for a career as a mathematician in industry.

 

Funding Details

This studentship is co-funded by industry (TUV-SUD-NEL) and Strathclyde University.

 

Supervisors

You will be supervised by Prof Tony Mulholland who is the Head of the Department of Mathematics and Statistics at Strathclyde.  He has published papers on the mathematics of ultrasonic systems for more than 20 years.  He also has published extensively on nonlinear dynamical systems and on combining experimental data with mathematical models (so called inverse problems). 

Prof Tony Mulholland

https://www.strath.ac.uk/staff/mulhollandanthonyprof/

The supervisory team will also include

Prof Philip Aston from the Department of Mathematics at the University of Surrey who has published extensively in areas such as bifurcation theory, symmetry, computation of Lyapunov exponents using spatial integration, and attractor reconstruction methods for analysing time series data.

Prof Philip Aston

https://www.surrey.ac.uk/people/philip-aston

Prof Tony Gachagan who is the Director of the Centre for Ultrasonic Engineering; a vibrant, cross-faculty and multi-disciplinary research centre which has world class facilities and a vibrant research environment, including around 70 researchers with backgrounds in engineering, chemistry, biology, mathematics and statistics, physics, material science and computing.

Prof Tony Gachagan

https://www.strath.ac.uk/staff/gachagananthonyprof/

The techniques developed will be tested on data from experimental test facilities both at Strathclyde (Dr Bill Dempster, Weir Advanced Research Centre) and industry, alongside data from computer simulations (Dr Mark Haw, Chemical and Process Engineering, Strathclyde).

Dr Mark Haw

https://www.strath.ac.uk/staff/hawmarkdr/

 

You will work with industry!

Craig Marshall is a Flow Measurement Consultant with 10 years’ experience of USM’s, at TUV-SUD-NEL in East Kilbride, the National Measurement Institute for Flow in the UK. 

Craig Marshall

https://www.tuv-sud.co.uk

Prof Alistair Forbes is Science Area Leader for Data Analysis and Uncertainty Evaluation at the National Physical Laboratory (NPL) in London. 

Prof Alistair Forbes

http://www.npl.co.uk/people/alistair-forbes

Dr Daniel Csimszi is a Software Development Engineer with SMS Oilfield in Aberdeen; they supply, install and monitor the ultrasound measurement systems to the oil and gas industry.

Dr Daniel Csimszi

https://www.smsoilfield.com/

You will have opportunities to visit and interact with these industries on a regular basis throughout your PhD. 

Contact us

Prof Tony Mulholland

https://www.strath.ac.uk/staff/mulhollandanthonyprof/

 

Mathematical Modelling to aid the Detection of Cracks in Safety Critical Structures

Safety critical structures, such as those found in nuclear plants, aircraft engines, oil pipelines, and railway tracks, require to be routinely checked for the presence of cracks. This testing has to be performed non-destructively and often in-situ. To look inside these structures, one of the dominant imaging technologies that is deployed is ultrasound. In the last few years a new non-destructive testing (NDT) technology has emerged that has the potential to transform the capability of ultrasound transducer systems in detecting cracks in steel welds.

This technology consists of a spatial array of individual ultrasound transducers with each being capable of independently transmitting and receiving an ultrasound signal. The challenge is now for mathematicians to develop fast, real-time algorithms that optimise the use of this data in flaw detection and characterisation. This PhD project will use ideas from stochastic calculus (as used in the finance industry) to address this problem.

The project will involve close collaboration with engineers and other disciplines within the Centre for Ultrasonic Engineering (www.cue.ac.uk) and with industry partners.

Mathematical Modelling to Improve Drug Manufacturing

Many high added value products and processes (such as those found in the manufacture of drugs) utilise crystalline solids whose manufacture typically involves several discrete batch processes to achieve the final desired product.

The aim of the Centre for Continuous Manufacturing and Crystallisation (CMAC) is to enable conversion of current batch manufacturing processes to a continuous basis to provide a substantial competitive advantage to UK manufacturing. However, advances are required in the fundamental understanding of the processes and in the use of sensors to measure the crystal shapes as they evolve. This PhD project will look at developing this understanding and in developing mathematical methods for extracting key properties of these crystals from sensor measurements.

The project will involve close collaboration with engineers and other disciplines within the Centre for Ultrasonic Engineering (www.cue.ac.uk) and with industry partners.

Mathematical Analysis of Insect Hearing Systems

The aim of this project is to develop a mathematical understanding of insect tympanal membrane vibrations. Hearing airborne vibrations, or sound, is a sense particularly important in many insects, and plays a key role in both predator and prey detection and mate attraction.

One of the mechanical systems used by insects to detect sound waves is the air pressure receiver. Such a membrane, or tympanum, is a structurally heterogeneous structure and the way it is actuated by incident sound energy has been investigated in only a few systems. Recent work has examined tympanal vibrations in response to sound in the ears of several insects, including the locust, the moth and the cicada. These investigations have unearthed several un-expected phenomena.

The challenge now is to combine the dynamic measurements of the insect tympana vibrations with the sparse structural data in order to explain how the dynamic phenomena occur. This PhD project will involve the mathematical modelling of these systems with the aim of furthering the basic understanding of how these systems operate.

The project will involve close collaboration with engineers and biophysicists within the Centre for Ultrasonic Engineering (www.cue.ac.uk).

Dr Jiazhu Pan

Modelling multivariate time series

We wish to develop innovative methods for modelling high-dimensional time series. Practical time series data, including both continuous-valued and discrete-valued data such as climate record data, medical data, and financial and economic data, are used for empirical analysis.

Models for forecasting multivariate conditional mean and multivariate conditional variance (volatility) are concerned. Techniques for dimension reduction, such as dynamic factor analysis, are used.

The estimation of models for panel data analysis and the option valuation with co-integrated asset prices is discussed.

Functional time series models: estimation and applications

Nowadays people often meet problems in forecasting a functional. A functional may be a curve, a spatial process, or a graph/image. In contrast to conventional time series analysis, in which observations are scalars or vectors, we observe a functional at each time point; for example, daily mean-variance efficient frontiers of portfolios, yield curves, annual production charts and annual weather record charts.

Our goal is to develop new models, methodology and associated theory under a general framework of Functional Time Series Analysis for modelling complex dynamic phenomena. We intend to build functional time series models and to do forecasting.

Modelling large economic systems and high frequency data

When the true economic system consists of many equations, or our economic observations have a very high dimension, one may meet the ``curse of dimensionality" problem. We try to impose a common factor structure to reduce dimension for the parametric and nonparametric stability analysis of a large system. Replacing unobservable common factors by principle components in parametric and nonparametric estimation will be justified.

In contrast to conventional factor models which focuses on reducing dimensions and modeling conditional first moment, the proposed project devotes attention to dimensional reduction and statistical inference for conditional second moments (covariance matrices). The direct motivation lies in the increasing need to model and explain risk and uncertainty of a large economic system.

The other distinctive point is that the proposed project considers factor models for high frequency data. A key application is the analysis of high dimensional and high-frequency financial time series, although the potential uses are much wider.

Dr Douglas Speirs

Surfing the Size Spectrum using Length-Structured Models

Marine ecosystems are generally composed of large numbers of species of widely varying sizes, ranging from unicellular species, through zooplankton and up to large fish and whales. The distribution of the total biomass of all species by size is known as the biomass spectrum.

In addition to the size variation due to differing characteristic sizes of the different species, and unlike the case in terrestrial systems, individuals themselves often undergo increases in body size of several orders of magnitude, from small eggs and larvae at a few millimetres in length up to large adults at the metre length scale.

At different parts of its life cycle an individual will be present at different parts of the biomass spectrum. Despite this apparent complexity it has long been known from field observations that biomass spectra show many regularities. In particular the logarithm of biomass density is approximately linearly related to the logarithm of body length with negative slope. Moreover, the slope of the spectrum is potentially sensitive to environmental and anthropogenic perturbations, for example the removal of large fish due to commercial fishing.

For these reasons biomass spectra have gained currency in recent years as a tool for studying the integrated ecosystem impacts of climate change and human exploitation of the seas.

The relative simplicity of representing the entire ecosystem as a size-structured spectrum has also permitted the development of mathematical representations in terms of partial differential equations, and these can be used to make predictions about ecosystem level responses to fishing and environmental change.

To date most modelling efforts have focused on steady state analyses of biomass spectra, which may be compared to annually averaged size spectra from field observations. In temperate shelf seas such as the North Sea, however, the biomass spectrum is not static but subject to seasonal impulses caused by increased primary production from phytoplankton in the spring bloom, and by seasonal reproduction by zooplankton and fish. This is known from both observational and mathematical models to induce annual ripples in the biomass distribution that propagate up the spectrum, gradually attenuating at larger sizes.

Fish larvae can exploit this by growing in size at a rate that allows them to feed near the peak of the wave, a phenomenon that has been dubbed by John Pope and his co-workers “surf-riding the biomass spectrum”. This project will focus on developing existing models to better represent these processes. In order to test the models the parameters will be estimated using Bayesian inference methods by fitting to a variety of data sets, including a long term (multi-annual) high temporal resolution (weekly) data from the North Sea.

Modelling fisheries-induced evolution

Many fish populations worldwide have been heavily exploited and there is accumulating evidence from both observational and theoretical studies that this harvesting can induce evolutionary changes. Such responses can affect the stock sustainability and catch quality, and so there is a recognized need for new management strategies that minimise these risks. Most results suggest that high mortality on larger fish favours early maturation.

However, recent theoretical work has shown that trade-offs between growth and maturation can lead to more complex evolutionary responses. Surprisingly, harvesting large fish can select for either late or early maturation depending on the effect of maturation on growth rate. To date most theoretical studies have used evolutionary invasion analyses on simple age-based discrete-time models or on continuous-time coupled ODE representations of size structure.

In common with many generic models of fish population dynamics, population control occurs by unspecified density-dependence at settlement. While these simplifications carry the advantage of analytical tractability, the analysis assumes steady state populations. This, together with the stylised life-histories precludes comparing model results with field data on secular changes in size-distributions an sexual maturity.

The work in this project will develop a new generation of testable model for fisheries-induced adaptive changes with the potential to inform future management decisions. This will involve developing a consumer-resource model which a length-structured fish population feeding on a dynamic biomass spectrum.

Differently-sized fish will compete for food by exploiting overlapping parts of the food size spectrum. The population will be partitioned by length at maturity, and this will be the heritable trait under selection. The model will be used to explore how changes in mortality and food abundance affect the evolutionarily stable distribution of maturation lengths.

Comparisons with survey data on North Sea demersal fish will be used to assess whether the historical harvest rates are sufficient to explain growth rate changes as an evolutionary response. Finally the evolutionarily stable optimal harvesting strategies will be identified.

Professor Stephen Wilson

Mathematical Modelling and Analysis of Sessile Droplets

The behaviour of sessile droplets is an area of very active international research, with new publications appearing on an almost daily basis and entire conferences now dedicated to the topic. Over the last decade, Professor Stephen Wilson has collaborated very successfully with Professor Khellil Sefiane from the School of Engineering at the University of Edinburgh (and a Visiting Professor at the University of Strathclyde) of a variety of practically important fluid-dynamical problems, including evaporating droplets, bubble dynamics, self-rewetting fluids, and anti-surfactants.

The aim of the present project is to build on the proposed supervisors’ previous work on evaporating sessile droplets to explore two exciting new aspects of this scientifically and practically important problem.

Very recent work by Sefiane and his collaborators have provided the first comprehensive experimental investigation of vapour absorption by sessile droplets of a desiccant liquid (i.e. one which draws moisture from the air). The mathematical modelling and analysis of such systems is an exciting and challenging open problem.

In practice, droplets almost never occur singly, but, due to the inherent complexity of the multiple-droplet problem, so far very little work has been done on the interactions between evaporating droplets. The aim of the proposed work is to investigate the fascinating but virtually unexplored subject of the collective behaviour of large arrays of small sessile droplets. Interactions between the droplets (in particular, so-called “shielding” effects) are expected to lead to very different collective behaviour compared to that of isolated droplets.

The student will join a lively and mutually supportive cohort of fellow PhD students within the Continuum Mechanics and Industrial Mathematics (CMIM) research group.

You should have (or expect to have) a UK Honours Degree (or equivalent) at 2.1 or above in Mathematics, Mathematics and Physics, Physics or a closely related discipline with a high mathematical content. Knowledge of Continuum Mechanics and mathematical methods (such as asymptotic methods) for the solution of partial differential equations is desirable (but not essential, particularly for overseas applicants).

Informal enquiries can be made to the lead supervisor, Professor Stephen K. Wilson, Department of Mathematics and Statistics, University of Strathclyde, Glasgow at s.k.wilson@strath.ac.uk and/or +44(0)141 548 3820. However, formal applications must be made via the online application procedure which can be found at

https://www.strath.ac.uk/courses/research/mathematicsstatistics/

remembering to list the title of the project as “Mathematical Modelling and Analysis of Sessile Droplets” and Professor Stephen K. Wilson as the first supervisor. There is no need to include a detailed research plan, but a brief outline of your relevant experience (if any) and your motivation for choosing this project would assist with the selection procedure.

This project will be supported by a Research Excellence Award (REA) Studentship funded by the University of Strathclyde with the support of the University of Edinburgh.

Unfortunately, funding rules mean that this studentship is only open to UK and EU students, and not to EEA or International students. Moreover, EU students are only eligible for the stipend element of the studentship if they have been resident in the UK for 3 years, including for study purposes, immediately prior to starting their PhD.

Microfluidics

In recent years there has been an explosive growth of interest in the behaviour and control of fluids at small (typically sub-millimetre) scales motivated by a range of novel applications including ink-jet printing and lab-on-a-chip technologies.

Much of the most exciting current research concerns the interaction between fluids and both rigid and flexible structures at small scales, and so the aim of the present project is to use a judicious combination of asymptotic methods and judiciously chosen numerical calculations to bring new insight into the behaviour of a variety of novel fluid-structure interaction problems in microfluidics.

Dynamics of Self-Rewetting Droplets

In the last decade or so there has been an explosion of interest in droplet evaporation, driven by new technological applications as diverse as crop spraying, printing, cooling technologies such as heat pipes, and DNA micro-array analysis.

One particularly interesting aspect of this problem which has thus far received relatively little attention is that of fluids whose surface tension exhibits a local minimum with temperature, known as self-rewetting fluids, a property that can have a profound effect on the dynamics of droplets on heated substrates.

The aim of the project is to build on the existing literature on conventional surface-tension-gradient driven spreading and droplet drying (see, for example, the references to some of our recent work on these problems given below) to bring new physical insight into this challenging scientific problem, and hence to harness the novel properties of self-rewetting droplets in a range of applications.

The project will be a collaboration colleagues at the University of Edinburgh who will be undertaking a parallel series of experimental investigations on this problem which will be key to the successful outcome of the project.

Dunn, G.J., Wilson, S.K., Duffy, B.R., David, S., Sefiane, K. “The strong influence of substrate conductivity on droplet evaporation” J. Fluid Mech. 623 329-351 (2009)

Dunn, G.J., Duffy, B.R., Wilson, S.K., Holland, D. “Quasi-steady spreading of a thin ridge of fluid with temperature-dependent surface tension on a heated or cooled substrate” Q. Jl. Mech. appl. Math. 62 (4) 365-402 (2009)

MSP group

The MSP group engages in research at the frontier of Mathematics, Physics and Theoretical Computer Science with particular strengths in category theory, type theory, logic, functional programming and quantum computation. Fundamentally, we want to change the world around us and believe we have the mathematical ideas to do so. If you want to help us achieve this, look us up at http://msp.cis.strath.ac.uk.

If you would like to discuss opportunities to do graduate work with the MSP group then please contact Dr. Anders Claesson at anders.claesson@strath.ac.uk