###### Stochastic Mathematical Biology in Ecology and Evolution

In this research we describe population dynamics in ecosystems and evolutionary processes through the lens of stochastic mathematical biology and develop methods to analyse phenomena that emerge from fluctuations and uncertainty. In particular, we model the dynamics of large random ecosystems in terms of randomly distributed interaction parameters and derive solutions for the power spectral density of this stochastic process based on statistical properties of the underlying interaction network.

Furthermore, we investigate the evolution of mating types in isogamous species, where the number of compatible mating types for sexual reproduction is not necessarily limited to two. Unlike in a model with neutral mutations, we find that fitness differences damp the growth of the average number of mating types and derive predictions independent of the underlying fitness distribution. This research opens up further questions on how fluctuations in ecosystems affect the evolutionary dynamics of embedded species.

###### On primal-dual algorithms for non-smooth large-scale machine learning

Modern abundance of data motivates automated data analysis methods, such as those provided by machine learning. Classification models such as Support Vector Machines (SVM) belong to an extensive class of optimisation problems which present non-smooth and large scale problems. To this purpose, Eric is studying a wide class of iterative algorithms with two main features:

first-order primal-dual algorithms, practical for non-smooth optimisation, and stochastic algorithms that can lower the computational workload and memory requirements for large scale models. The main focus of the project is to establish convergence for these algorithms, and it is expected that they can guarantee faster rates of convergence over state of the art methods.

###### On-line drill system parameter estimation and hazardous event detection

Dan’s research, in collaboration with Schlumberger, develops statistical methods for automatic detection of hazardous events in oil and gas drilling operations. Initially, only two particular hazardous events are considered. The first is called washout and it means the appearance of a hole in the drill pipe which may compromise the safety and efficiency of the operation as well as equipment integrity. The second event is called mud loss and it means the loss of drill due to a leakage in the drill system to the surrounding rock formation. As the project progresses, more complex scenarios will be considered, involving multiphase flow, influx of gas from the formation, accumulation of rock cuttings around the drill pipe, wear of the drill bit, plugged bit nozzles, or the degradation of the motor. The initially one dimensional model could also be extended to two or three dimensions for increased accuracy.

###### Mathematical modelling of formulation composition trade-offs for pesticides

Creating validated mathematical models that can inform the process of risk assessment during pesticide product development is an industry-wide aspiration. It is particularly challenging given the wide range of formulations that may be used to produce new pesticides and the complexity of developing products that have good foliar uptake but poor dermal absorption. Working with Syngenta, Jenny is developing and analysing a series of spatially explicit mathematical models for membrane penetration parameterised using existing data sets. The impact of formulation products is explored in relation to their physicochemical properties in an attempt to categorise formulation impact across the two membranes. The models will subsequently be combined and analysed within a novel optimisation framework which should highlight the key parameter groupings responsible for good foliar uptake and poor dermal absorption based on existing data sets.

###### Hybrid methods for modelling the cell-division cycle

Biological systems exhibit a tremendously wide variety of behaviours at many different spatial scales. While in theory, the behaviour of a system at any scale can be viewed as emergent from the behaviour of its smallest components, deriving these scale relationships is analytically intractable except in only very carefully constructed examples. Numerical methods based purely on a system’s microscopic behaviour can quickly become cost-prohibitive as the number of atomic components of a system increases. To achieve computational feasibility, different modelling regimes are used at different scales, though at the cost of information loss when using coarser representations. Josh’s work is concerned with hybrid methods, which combine multiple regimes to balance the advantages and disadvantages of each. In particular, he aims to construct a hybrid method based on a model of the cell-division cycle as a reaction-diffusion system, building upon previous hybrid methods such as the pseudo-compartment method.

###### Impact of climate change on cocoa farming

Cocoa is a delicate and sensitive crop whose production has been hugely affected by several different factors including climate change. These lead to uncertainties in planning for the future both in the short term (seeding and harvesting) and in the long term (allocation of land use). The planning process can be helped by the construction of an appropriate mathematical model and linking these to data on climate variation. Tosin’s project involves constructing such models and comparing them with data on both cocoa production and local climate variation. This model will be used to track the impact of time lags between the flowering and harvesting stage of the cocoa crop and further used to establish how cocoa trees will behave in the near future as atmospheric temperatures continue to rise due to climate change.

###### A priori bounds for fractional SPDEs

Salvador's research is in the field of singular SPDEs, more particularly in the fractional ϕ4 model which is an important model in Quantum Field Theory that arises as the natural Glauber dynamics in the Brydges-Mitter-Scoppola model. In recent years the theories of Regularity Structures and Paracontrolled Distributions have led to a lot of research for the non fractional ϕ4 model. The aim of the project is to extend some of this work to the fractional case by constructing a solution theory and a priori bounds to extend local in time existence of solutions.

###### Numerical and analytical approaches using complex ray theory and exponential asymptotics in 3D wave-structure interactions

Despite significant advances in computational hardware and numerical algorithms, the simulation of fully nonlinear three-dimensional free-surface flows around blunt-bodied objects remains particularly limited. On account of the processing power required, most modern desktop (and in some cases high-performance) computations still require the use of simplifying geometrical assumptions and coarse meshes on the order of a hundred points per spatial dimension. In contrast, numerical simulations of comparable two-dimensional flows can be routinely done with O(1000) grid points in the spatial dimension. There continues to be a need for the analytical theories that can provide explicit asymptotic descriptions of the flow properties, particularly for the use of efficient hybrid numerical-analytical approaches. Recently, there has been success in developing new asymptotic techniques for studying linear wave-structure flows in three-dimensions. These techniques are based on the use of exponential asymptotics applied to low-speed hydrodynamical flows. Yyanis’s research develops new analytical and numerical techniques related to the area of complex ray theory and asymptotic analysis, to extend these ideas to nonlinear problems.

###### Multivariate Regression on High-Dimensional Networks

A crucial feature of many modern data sets is their inherent graphical (network) structure embedded in them. Gaussian Graphical Models, known by various other names, are commonly used in Statistics to encode a graphical conditional relationship onto a multivariate Gaussian distribution. Many scientific applications manifest a high dimensionality combined with a sparse graph structure, which can hinder inference and high-level interpretation when fitting graphical models onto relevant data. However, as exemplified through geospatial crime data, the implementation of a regression-based transformation through a penalty term in the log-likelihood function can efficiently and informatively infer a covariate relationship between a high number of connected nodes (crime rates for different regions) with a smaller number of secondary variables (socioeconomic factors). Daniel's project aims to establish tractability conditions for parameter estimation under a generalised expression of this new Gaussian graphical model, through an iterative algorithmic procedure. Where a graph structure is not known, a Lasso-like penalty must be employed in tandem with the regression penalty. Further considerations include the extension of the single graph case to that of an evolving graph, mainly in a temporal sense.

###### Large scale differential geometric MCMC

Uncertainty Quantification (UQ) concerns both propagation of uncertainty through a physical model, known as the forward problem, and the inverse problem of inferring uncertain model parameters from noisy measurements. Markov Chain Monte Carlo (MCMC) methods 30are the most widely used tools for computing expectations in UQ and large statistical models in general. Conventional approaches to MCMC are often inefficient and must compute many samples for a high accuracy. Geometric ideas can be used to improve the methods’ statistical performance; two prominent algorithms in this line of thinking are Riemann Manifold Hamiltonian Monte Carlo (RMHMC) and Riemann Manifold Metropolis Adjusted Langevin Algorithm (RMMALA). Tom is interested in extending these ideas to exploit more general ideas from differential geometry, with a focus on developing methods that are suited to problems from UQ.