Network Optimization

Mathematics & Statistics Available PhD projects

Dr Neil Banas

Satellite- and model-based prediction of harmful algae dynamics in the US Pacific Northwest

Certain species of the phytoplankton Pseudo-nitzschia (PN) produce a potent neurotoxin, domoic acid (DA), that bioaccumulates in shellfish, and can cause severe illness and death in humans. Past outbreaks have caused fisheries closures and millions of dollars of losses to coastal communities, and contributed to the death of many species of marine mammals. A partnership among US/UK academic, government, and tribal scientists has recently implemented an early-warning system, the "Pacific Northwest HAB Bulletin," that helps government agencies in Washington and Oregon protect health and livelihoods. This PhD project will synthesize state-of-the-art numerical models, satellite-based remote sensing, and existing bio-physical field observations in order to improve the spatial analyses that go into the production of that Bulletin.

The arrival of a PN harmful algal bloom (HAB) on a Pacific Northwest beach is the result of a chain of ocean processes: i) the buildup of PN species relative to other phytoplankton in one of a few known offshore hotspots; ii) transport by wind-driven ocean currents from those hotspots to the beaches; and iii) the maintenance of the PN cell population and their toxin levels during transport. Currently, the HAB Bulletin makes use of a high-resolution ocean simulation, LiveOcean (faculty.washington.edu/pmacc/LO/LiveOcean.html), in addition to a variety of direct observations, but the predictive skill of the forecast relies on the physics of the transport step (ii), not the biology (i,iii). This is the area for potential improvement that this studentship targets: adding the ability to predict PN population dynamics and DA production, by a combination of statistical and mechanistic/first-principles approaches.

The project thus combines 1) statistical analysis of large datasets, 2) dynamical modelling, 3) spatial ecology in the coastal ocean, and 4) applied ecoforecasting serving both industry and deprived coastal communities. An excellent quantitative background (programming, statistics, calculus) is required; a background in ocean science and ecology is preferred but not required. The student will be based at University of Strathclyde, Glasgow, UK, and co-supervised by Dr Neil Banas (Strathclyde Maths and Stats: http://neilbanas.com/projects/), Dr David McKee (Strathclyde Physics: http://www.strath.ac.uk/staff/mckeedaviddr/), and Prof Raphael Kudela (Ocean Sciences, University of California Santa Cruz: http://people.ucsc.edu/~kudela/). The student will join the Marine Alliance for Science and Technology Scotland (MASTS); be encouraged to attend international conferences; attend and help organise annual project/stakeholder meetings in Seattle, USA; and have the opportunity for visits to UC Santa Cruz and University of Washington, both of which are leading centres for marine science in the US.

The position is open to all UK and EU applicants and comes with three years of full support, including fees and an annual living stipend of approximately £14,000, as well as support for conference travel and other expenses. Start date is flexible, with summer-autumn 2018 preferred. Review of applications will begin 15 Apr 2018 and continue until the position is filled. To apply, send 1) a complete CV, 2) a 1-2 page personal statement explaining your specific interest in this position and the skills you bring to it, and 3) names and contact info for three references. Please send applications and other inquiries to Dr Neil Banas, neil.banas@strath.ac.uk.

Dr Robin Cook

Evaluation of Maximum Sustainable Yield (MSY) in an ecosystem context

MSY has become the standard basis on which fishery management reference points, such a Fishing mortality rate (F) and Spawning Stock Biomass(SSB), are set.

In almost all stocks these reference points are estimated using single species steady state models that assume invariance of life history parameters such as growth reproduction and natural mortality. There are good reasons to suppose that these life history parameters are not constant and it would be expected that natural mortality would change as a result of species interactions. Thus while it might be possible to fish at a value of F based in single species models, there is not good reason to believe that by so doing, SSB and catch will be achieved. Despite this major difficulty many analyses of the status of global fish stock are based on the assumption that single species MSY is observable and has been used to estimate foregone catch and estimate “safe” limits for SSB.

The purpose of this project will be to use current ecosystem models to assess the implications for both realised SSB and catch when stocks in a multispecies complex are all fished at single species Fmsy. The project would also examine methods that have been used to classify the exploitation status of stocks based on catch data alone when it has been assume that single species MSY is observable. Based on this analysis alternative reference points will be proposed that take into account species interactions and hence provide a more robust basis for fishery management.

Development of Bayesian statistical methods for the assessment of data poor fish stocks

MSY has become the standard basis on which fishery management reference points, such a Fishing mortality rate (F) and Spawning Stock Biomass(SSB), are set.

In almost all stocks these reference points are estimated using single species steady state models that assume invariance of life history parameters such as growth reproduction and natural mortality. There are good reasons to suppose that these life history parameters are not constant and it would be expected that natural mortality would change as a result of species interactions. Thus while it might be possible to fish at a value of F based in single species models, there is not good reason to believe that by so doing, SSB and catch will be achieved.

Despite this major difficulty many analyses of the status of global fish stock are based on the assumption that single species MSY is observable and has been used to estimate foregone catch and estimate “safe” limits for SSB. The purpose of this project will be to use current ecosystem models to assess the implications for both realised SSB and catch when stocks in a multispecies complex are all fished at single species Fmsy.

The project would also examine methods that have been used to classify the exploitation status of stocks based on catch data alone when it has been assume that single species MSY is observable. Based on this analysis alternative reference points will be proposed that take into account species interactions and hence provide a more robust basis for fishery management.

Dr Penny Davies

Time-dependent scattering problems

Time dependent wave propagation and scattering is important in acoustics, electromagnetics and seismology. These problems involve transient wave fields (often short pulses) and give rise to systems of hyperbolic PDEs, which can be approximated directly or first reformulated as boundary integral equations (BIEs) posed on the surface of the scatterer.

The BIE approach is computationally attractive because it only requires quantities to be approximated on a two-dimensional surface, but standard methods are complicated to implement and/or have numerical stability problems which need to be overcome. Recently progress has been made by using time-stepping approximations based on convolution quadrature or other methods which share its "backwards-in-time" framework, but several interesting open problems remain, such as coupling these time-stepping methods to spatial approximations based on B-splines.

The overall aim is to develop reliable and efficient approximation schemes.

Dr Victorita Dolean 

Iterative preconditioners for time-harmonic heterogeneous electromagnetic problems

Mathematical and numerical modelling is unavoidable nowadays when one tries to investigate and understand complex physical phenomena such as electromagnetic wave propagation. The underlying mathematical model in this case is Maxwell’s equations - according to applications it can take two different forms (time-domain and time-harmonic). 

There is currently a large international research effort dedicated to the efficient numerical solution frequency-domain or time-harmonic PDEs, driven by the fact that in many applications (including electromagnetic scattering), the frequency-domain formulation is a viable alternative to the time domain, provided suitably-efficient methods are available for solving the large linear systems that arise. Solving wave propagation problems in time-harmonic regime and heterogeneous media is challenging and requires sophisticated methods. This project seeks to design, analyse, and implement fast, highly-parallel DD preconditioners for heterogeneous frequency-domain wave problems involving electromagnetic waves. We will consider coarse grids based on oscillatory basis functions, and introducing another ingredients as absorption will allow us to design methods that are provably efficient, even for general geometries and general decompositions/meshes.

Suggested reading (on possible applications of Maxwells equations and the need for efficient solvers):

https://sinews.siam.org/Details-Page/high-performance-computing-for-the-detection-of-strokes-2

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.

Dr 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,

Nonintrusive management of complex systems

This studentship will be of 3 years duration with stipend and fees for a UK/EU student, funded by the University (funding application submitted outcome pending). This project is part of a collaboration with the Department of Mechanical and Aerospace Engineering.

Our immune system, the movement of people in congested environments, and the economy, are all examples of complex systems that are key to our well-being. These systems normally operate in a predictable way, that is, when the immune system is healthy, crowds move homogeneously, and economic business cycles oscillate steadily about an increasing mean.

However, catastrophe in these systems can occur, in these cases, through disease, crowd panic and economic shocks such as acts of terrorism, that lead to undesirable and unpredictable behaviour. This unpredictable behaviour is seemingly impossible to manage effectively.

For example government bodies may introduce constrictive legislation to regulate the decision maker but this is often at the expense of efficiency and increased bureaucracy. This studentship will investigate a mathematically informed method to introduce a robustness in these systems that will not require aggressive short term fixes but uses continuous non-intrusive management instead.

Moreover, current strategies and control methods fail to cope with the complexities of these systems and there is an opportunity here to test the possibility of a new management philosophy through fundamental research.

Our methodology will be informed by a technique used in physics to control chaos in low-dimensional systems of ordinary differential equations (ODEs). This technique is coined time-delayed auto-synchronization (T-DAS) [1,2]. Analytical and numerical techniques will be used to investigate models in the areas of disease management, economic management and crowd control.

References
1. Pyragas, K., Continuous control of chaos by self-controlling feedback, Physics Letters A, vol. 170, pp. 421-428, 1992.
2. Scholl, E. and Schuster, H.G. (Editors): Handbook of Chaos Control, Wiley-VCH, Weinheim, 2008.

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 Des Higham

Centrality Measures for Dynamic Networks

The digital revolution is generating novel large scale examples of connectivity patterns that change over time. This scenario may be formalized as a graph with a fixed set of nodes whose edges switch on and off. For example, we may have networks of interacting mobile phone users, emailers, Facebookers or Tweeters.

To understand and quantify the key properties of such evolving networks, we can extend classical graph theoretical notions like degree, pathlength and centrality. This project will focus on the development of linear algebra-based algorithms that can capture various aspects of information flow around an evolving network, with the aim of identifying important players. Real data sets from the world of on-line social interaction will be used to test ideas.

Suggested reading:

Collaboration Blooms from SIAM News Article, SIAM News, 2012

Peter Grindrod, Desmond J. Higham, and Peter Laflin http://www.siam.org/pdf/news/2039.pdf

Stochastic Models for Online Interaction

Social scientists have developed empirical rules, including homophily and triadic closure, that aim to describe how human social interactions evolve over time. The emergence of large scale digital data sets opens up the possibility of testing these hypotheses.

This project will involve the development of mathematical models, in the language of stochastic processes, and the validation of these models via computer simulation and Bayesian inference.

Suggested reading:

Bistability through triadic closure, P. Grindrod, D. J. Higham and M. C. Parsons, Internet Mathematics, Internet Math. Volume 8, Number 4 (2012), 402-423.

http://personal.strath.ac.uk/d.j.higham/Plist/P111.pdf

Dynamic Communities

This project will look at the development and validation of computatonal algorithms that aim to summarize the community structure in a dynamically evolving network. The work will involve linear algebra, matrix computation, applied statistics and high performance computing. Applications include the analysis of on-line behaviour, e.g., through Twitter (who tweeted who) and email (who emailed who). The project will also look at methods for real-time summaries of other network properties.

Suggested reading:

A matrix iteration for dynamic network summaries, P. Grindrod and D. J. Higham, SIAM Review, 55(1), 118–128.

http://personal.strath.ac.uk/d.j.higham/Plist/P109.pdf

Dr Philip Knight

How and Why of Matrix Balancing

Matrix balancing aims to transform a nonnegative matrix A by a diagonal scaling by matrices D and E so that P = DAE has prescribed row and column sums. Historical motivation for achieving the balance has included interpreting economic data, preconditioning sparse matrices and understanding traffic circulation. More recently, it has been acknowledged that there is a role for matrix balancing in network analysis, inferring phylogenies (via hierarchical clustering) and increasing fairness in elections.

Application of balancing has been limited by a lack of analysis of existing algorithms in order to assess their efficacy for large scale data sets. We have recently filled in some of the holes in this analysis and have had some success in implementing existing and new algorithms on very large problems. The new algorithms we have developed for balancing have been shown to work an order of magnitude faster than the standard algorithm. We would like to develop these algorithms to find implementations suitable for parallel computations on large data sets. And we would like to develop more full the theory of balancing by drawing from the theory of linear programming as well as linear algebra.

In tandem with algorithmic and theoretical work, applications in disciplines such as electoral policy and phylogenic reconstruction are a central feature of the project. The student will be able to influence the direction of the project based on her/his interests.

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.

Dr Tony Mulholland

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 Jennifer Pestana

Novel Preconditioned Iterative Methods for Radial Basis Linear Systems (Funding Available)

For more information please contact Dr Pestana (jennifer.pestana@strath.ac.uk).

Applications are invited for a fully funded 3 year PhD studentship in numerical linear algebra, under the supervision of Dr Jennifer Pestana and Dr Alison Ramage, at the Department of Mathematics and Statistics at the University of Strathclyde, Glasgow, Scotland.

This project will develop effective solvers for linear systems in these RBF methods. In particular, we will focus on certain iterative methods (Krylov subspace methods) that start with an initial guess of the solution that is improved at each step. For these ill-conditioned RBF problems, finding matrices known as preconditioners that accelerate the solution process are essential. Thus, at the core of this project will be the proposal, and analysis, of new solvers for RBF-based PDE solvers. The preconditioners will be tested on real-world applications.
Radial basis functions (RBFs) have several advantages when used to numerically approximate the solution of partial differential equations (PDEs) in applications. However, solving linear systems that arise within RBF-based solvers is challenging. This project will develop and mathematically analyse new preconditioned iterative solvers of these linear systems, and will test their performance on interesting applications.
Mathematical modelling is increasingly used to investigate and understand phenomena and forecast future events, particularly when experimentation is prohibitive or costly. However, real-world problems are often posed on complicated domains and involve scattered data, e.g. in geophysical and biological applications, or are inherently high-dimensional, e.g. in quantum physics, finance and systems biology.
Complex geometry, scattered data and high dimensionality can be difficult for some numerical methods for PDEs to deal with. However, these problem features are handled relatively easily by radial basis function approaches. RBF methods represent the solution of PDEs as a combination of radial basis functions that can be placed anywhere in the computational domain. The suitability of RBFs for complex problems is evidenced by their use in applications, including fluid flow, geophysics, plasma physics, finance and biology.  Despite their advantages for dealing with complex, real-world problems, RBF methods can be difficult to implement. This is because obtaining the combination of radial basis functions that describes the PDE solution requires the solution of one or more challenging (i.e. ill-conditioned) systems of linear equations.

This project will develop effective solvers for linear systems in these RBF methods. In particular, we will focus on certain iterative methods (Krylov subspace methods) that start with an initial guess of the solution that is improved at each step. For these ill-conditioned RBF problems, finding matrices known as preconditioners that accelerate the solution process are essential. Thus, at the core of this project will be the proposal, and analysis, of new solvers for RBF-based PDE solvers. The preconditioners will be tested on real-world applications.

Applicants should have or expect to obtain a good (I or II(i)) honours degree in mathematics or in a related discipline. This project would suit students with an interest in linear algebra and/or numerical analysis. Experience of numerical mathematics and/or programming would be beneficial, but is not essential.   The successful applicant will be part of a vibrant postgraduate community, and will have access to a range of training opportunities, including the Scottish Mathematical Sciences Training Centre (https://smstc.ac.uk/).

The studentship covers UK/EU tuition fees and comes with an annual tax-free stipend at the standard UK rate. International students who can fund the difference between UK/EU and International fee rates are also encouraged to apply.

For more information, please contact Dr Pestana (jennifer.pestana@strath.ac.uk). Formal application is via the University of Strathclyde postgraduate research application process at
https://www.strath.ac.uk/courses/research/mathematicsstatistics/.Please ensure that you clearly state your interest in this project with these supervisors when makinga formal application.For full consideration, please apply by February 12, 2018.

Dr Dougie 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.

Dr Marcus Waurick

Ultrasonic transducers in high-temperature regime with boundary dynamics (Funding available)
For more information please contact Dr Waurick.

Applications are invited for a fully funded 3 year PhD studentship in mathematical analysis, under the supervision of Dr Marcus Waurick and Prof Anthony Mulholland, at the Department of Mathematics and Statistics at the University of Strathclyde, Glasgow, Scotland.

Summary: The aim of the project is to describe and develop models for ultrasonic transducers taking into account both the high-temperature regime as well as possible boundary dynamics. For this one needs to transfer existing models with fixed boundary conditions to the regime of impedance or Leontovich type boundary conditions. After establishing the model, we aim for proving its well-posedness in the sense of Hadamard in order to more accurately describe ultrasonic transducers and to more effectively being able to address the inverse problem of identifying the model parameters by a full understanding of the forward problem.

Description: The full time-dependent equations for ultrasonic transducers ([5]) taking into account the elastic, the electro-magnetic as well as thermal effects are discussed in [4]. In this reference, the authors prove the well-posedness in the sense of Hadamard for the resulting equations. The main technique is to formulate the problem in question as an operator equation in a suitable Hilbert space modelling space-time. With this formulation it is possible to apply the theory of evolutionary equations developed by Picard [6].

The description outlined in [4], however, does not include boundary condition of impedance or Leontovich type, see e.g. [2, 3]. These type of boundary conditions are used to describe the influence of the boundary by means of a separate partial differential equation posed on the boundary as the underlying domain. In the recent preprint [7] these boundary conditions have been discussed using abstract boundary data spaces and for the equations for ultrasonic transducers disregarding any thermal effects.

For applications, it is important to take temperature effects into account, as well. In particular when smart ceramic materials are designed that adjust their mechanical properties corresponding to the fluctuation of the temperature of the environment. Most prominently, for non-destructive testing, there is the need for ultrasonic transducers that withstand high temperatures.

After having introduced a space-time framework for the piezo-electric equations under consid- eration, the author of [7] used the results of [8] to prove Hadamard well-posedness. The abstract implementation of the boundary dynamics roots in results developed in [9]. These results make it possible to have a well-defined theory valid for any boundary regularity, because the Hilbert space describing the boundary can be replaced by an auxiliary Hilbert space of functions living on the whole underlying domain and not only on its boundary. For possible numerical implementation of these equations it is, however, desirable to reduce the complexity of the problem at hand and, thus, to have a formulation using classical trace spaces (see e.g. [1]), as well.

In order to accurately describe ultrasonic transducers in a high-temperature regime taking into the finite size of the device, the aim of the project is to develop a mathematical model combining elastic, electromagnetic as well as thermal effects of ultrasonic transducers subject to boundary conditions of Leontovich type. This will lead to an operator theoretical description of the problem at hand such that the theory of evolutionary equations applies. This in turn yields well-posedness of the present model, that is, existence and uniqueness of solutions as well as continuous dependence of the solution on the data for the full time-dependent equations.

The understanding of the forward problem can then be employed for the inverse problem of deriving material parameters based on the knowledge of outcomes of experiments, which measure the solution of the model to be developed in this project.

The aimed for operator theoretical description shall not use any quasi-static or time-dependent approximation, but will also lead to the possible consideration of non-local coefficients of convo- lution type. Moreover, the materials can be fully anisotropic. For the special case of Maxwell’s equation there is already an effective description of so-called metamaterials at hand, see [6]. Thus, the final theoretical description for ultrasonic transducers will also cover the description of metame- terials and their electromagnetic properties subject to bending and temperature.

Applicants should have or expect to obtain a good (I or II(i)) honours degree in mathematics or in a related discipline. This project would suit students with an interest in functional analysis and partial differential equations. Prior experience and knowledge on unbounded operators in Hilbert space would be beneficial, but is not essential. The successful applicant will be part of a vibrant postgraduate community, and will have access to a range of training opportunities, including the Scottish Mathematical Sciences Training Centre (https://smstc.ac.uk/).

The studentship covers UK/EU tuition fees and comes with an annual tax-free stipend at the standard UK rate. International students who can fund the difference between UK/EU and International fee rates are also encouraged to apply.

For more information, please contact Dr Waurick (marcus.waurick@strath.ac.uk).

Formal application is via the University of Strathclyde postgraduate research application process at https://www.strath.ac.uk/courses/research/mathematicsstatistics/. Please ensure that you clearly state your interest in this project with these supervisors when making a formal application.

For full consideration, please apply by April 12, 2018.

Professor Stephen Wilson

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.

Mathematical Modelling and Analysis of Thin-Film Flows

Not only are thin-film flows ubiquitous in nature, industry and biology (where they appear as, for example, lava and mud flows, coating flows, microfluidics, and biofilms and other mucus linings within the human body, respectively), but they also provide an endlessly fascinating "playground" in which to investigate a wide range of nonlinear and dynamical phenomena. In particular, the flow of rivulets and droplets and the interaction between thin films of fluid and the external environment (such as, for example, the airflow over a car windscreen on a rainy day or the evaporation of a fluid droplet) display complex and unexpected phenomena who understanding often necessitates an insightful combination of asymptotic and numerical methods. The proposed project will use a variety of analytical and numerical methods to bring new understanding into a range of real-world problems involving thin films of both simple and complex fluids. In particular, while the details are open to discussion with the student, we plan to build on the supervisor’s previous work in rivulet flow [see, for example, Al Mukahal et al. Proc. Roy. Soc. A 473 (2207) 20170524 (2017)] and droplet evaporation [see, for example, Stauber et al. Phys. Fluids 27 122101 (2015)] and will hopefully involve collaboration with experimentalists.

Prerequisites
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).
Application Procedure

Informal enquiries can be made to the 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 Thin-Film Flows” 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.

There is currently no founding allocated to this project but there are various funding schemes (both internal and external to the University) to which we can apply for support for the right candidate. The project is, of course, also available to self-funded students who already have their own source of funding.

Dr Alex Wray

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)

Computer and Information Science

Strathclyde Combinatorics Group

There is an opportunity to do graduate work with the Strathclyde Combinatorics Group in enumerative combinatorics, algebraic combinatorics and graph theory, some of this with connections to physics and theoretical computer science. More information about the group and its research can be found at http://combinatorics.cis.strath.ac.uk.

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

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