Mathematics & Statistics Seminars and colloquia

Title:   Deflation with POD vectors for porous media flow

Date: 2.00pm Wednesday 8th May 2019

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: We consider systems originating from the simulation of multi-phase flow through porous media.  The spatially discretized coupled nonlinear equations are solved with a two-stage preconditioner.  We consider the first step of this procedure, i.e. solving of the pressure equation.  We develop preconditioners based on deflation and a selection of deflation vectors motivated by Proper Orthogonal Decomposition (POD) to a number of pre-computed solutions.  We investigate alternatives using different varieties of reduced-order modelling.  Furthermore we explore the connection between POD-based preconditioning and deflation methods.  One of the difficulties for deflation methods is to find the right deflation vectors for general problems. The combination of deflation with the POD methods looks very promising in this respect.

 

Title:  How record-linkage has shed light on age-related increases in drugs-related deaths

Date: 3.30pm Wednesday 27th November 2019

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: Strongly age-related increases in drugs-related deaths (DRDs) in the 21^st century are the late sequelae of UK’s heroin-injector epidemics of the early 1980s. DRDs are mainly opioid-related deaths (ORDs); opioid-dependent clients’ DRD-rate not only increases as clients age beyond 35 and 45 years of age but females’ advantage is lost. Briefly, I explain how the before/after evaluation of Scotland’s National Naloxone Programme (NNP) was designed to circumvent the first of these epidemic forces. Secondly, I will show how record-linkage, an approach in which data from diverse records are combined, has been applied to Scotland’s methadone-client cohort (2009-2015) and has shed light on the effects of both age and gender as well as methadone-dose.

 

Title:   Riordan graphs and their properties

Date:   3pm Tuesday 8th October 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: In this talk, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The Riordan graphs are proved to have a number of interesting (fractal) properties, which can be useful in creating computer networks with certain desirable features, or in obtaining useful information when designing algorithms to compute values of graph invariants. The main focus in this talk is the study of structural properties which find certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle, diameter of io-decomposable Riordan graphs, and so on. Finally we pose several open questions and conjectures.

Title:   Port-Hamiltonian differential equations on infinite networks

Date:   3pm Tuesday 22nd October 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Given a graph G=(V,E) we associate with each edge an interval. On each interval we consider a port-Hamiltonian partial differential equation x_t = P_1*(Hx)_xi + P_0*Hx with matrices P_1, P_0 and Hamiltonian density function H. In the talk, we discuss which boundary conditions, imposed at the vertices, guarantee that solutions of the above equation are given by a (contraction) semigroup. Our results cover infinite graphs, graphs with edge length tending to zero and Hamiltonians H that can approach zero or be unbounded.

Title:   TBA

Date:   3pm Tuesday 29th October 2019

Venue: Livingstone Tower, 9th floor, room LT907

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Date:   3pm Tuesday 19th November 2019

Venue: Livingstone Tower, 9th floor, room LT907

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Title:   TBA

Date:   3pm Tuesday TBC

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: TBA

Title: Bubble propagation in modified Hele-Shaw channels

Date:  1.00pm Tuesday 15th October 2019

Venue: Livingstone Tower, 9th floor, room LT907

 

Title: Coating flow on a rotating horizontal cylinder subject to an airflow

Date:  1.00pm Tuesday 22nd October 2019

Venue: Livingstone Tower, 9th floor, room LT907

 

Title: Near-wall behaviour of a colloid particle in a liquid crystal fluid

Date:  1.00pm Tuesday 22nd October 2019

Venue: Livingstone Tower, 9th floor, room LT907

 

Title: Inverse Obstacle Scattering Problems: A Game of Hide and Seek

Date:  1.00pm Tuesday 29th October 2019

Venue: Livingstone Tower, 9th floor, room LT907

 

Title: Dropwise Condensation on Hierarchical Superhydrophobic Surfaces: Uncoated, Coated and Lubricant infused

Date:  1.00pm Tuesday 12th November 2019

Venue: Livingstone Tower, 9th floor, room LT907

 

Title: Topics in thick flows

Date:  1.00pm Tuesday 19th November 2019

Venue: Livingstone Tower, 9th floor, room LT907

 

Title: Using multiply scattered waves in heterogeneous media

Date:  1.00pm Tuesday 10th December 2019

Venue: Livingstone Tower, 9th floor, room LT907

 

Title: Algorithmics of the Student-Project Allocation problem 

Date:  4pm Tuesday 24th September 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:  Matching problems are all around us – they arise when we try to find the “best” allocation between two sets of entities. For example, matching kidney transplant patients with compatible donors, allocating junior doctors to hospitals, and assigning students to projects in a university department. 

My talk is centred around the Student-Project Allocation problem (SPA), which, in its natural form, involves finding a many-to-one matching of students to projects offered by lecturers, based on student preferences over projects and the maximum number of students that each project and lecturer can accommodate. Two models of SPA exist in the literature: one permits preferences from the students only, while the other permit preferences from both the students and the lecturers. In the latter case, three different variants have been studied, including lecturer prefereces over (i) students, (ii) projects, and (iii) (student,project) pairs. 

For my PhD, I have been exploring SPA with lecturer preferences over Projects (SPA-P) and SPA with lecturer preferences over Students with Ties (SPA-ST). The solution concept that we seek in this context is a “stable matching”, which ensures that no student and lecturer who are not matched together would rather be assigned to each other than remain with their current assigneess. In my talk, I will present the algorithmic results arising from my research on finding stable matchings in SPA-P and SPA-ST. Some of these results include integer programming models and exact polynomial-time algorithms. 

Sofiat Olaosebikan is a final year PhD in Computing Science at the University of Glasgow. Her interests lie at the intersection of mathematics and theoretical computer science. You can find out more about her work on her website, and you can follow her on twitter @soolaosebikan.

Title: Adaptive & Multilevel Stochastic Galerkin Approximation for Parameter-Dependent PDEs.

Date:  4pm Tuesday 29th October 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:  In this talk, we discuss aspects of numerical analysis associated with stochastic Galerkin approximation for performing forward UQ in PDE models with uncertain (parameter-dependent) inputs. Starting with the standard elliptic test problem, we first describe in general terms, a strategy for performing a posteriori error estimation and developing adaptive solution algorithms. We then discuss how this methodology can be extended to a more challenging linear elasticity problem with uncertain Young’s modulus. We introduce a three-field parametric PDE model and develop an adaptive stochastic Galerkin mixed finite element scheme. We estimate the error in the natural weighted norm with respect to which the weak formulation is stable. Exploiting the connection between this norm and the underlying PDE operator also leads to an efficient block-diagonal preconditioning scheme for the associated discrete problems. It can be shown that both the error estimator and the preconditioner are robust in the incompressible limit.

Title: TBA

Date:  4pm Tuesday 5th November 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:  TBA

Title: TBA

Date:  4pm Tuesday 10th December 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: TBA

Title: Host-Parasite Modelling Approaches as Applied to Sea Lice on Salmon Farms

Date: Wednesday 20th November, 1.00pm

Venue: Livingstone Tower, LT907

Abstract: 

Title: Patient-level probabilistic modelling to investigate the cost-effectiveness of six screening strategies for carbapenemase producingenterobacteriaceae (CPE) in the NHS.

Date: Wednesday 23rd October, 1.00pm

Venue: Livingstone Tower, LT907

Abstract:  Antimicrobial resistance has been recognised as a global threat to public health, with carbapenemase-producing-enterobacteriaceae (CPE) causing significant morbidity and mortality in hospital settings. There is evidence to suggest these pathogens propagate along hospital transmission routes making screening programmes important. At present, there is no gold standard available on which to base a CPE screening and testing programme.  The low incidence of CPE has led to the introduction of a clinical risk assessment (CRA) which identifies patients who may benefit from microbiological screening. In order to limit spread in clinical settings, isolation of suspected positives is being recommended putting a strain on resources. There are different screening technologies available which means screening can be done quickly or more slowly with differential impacts on costs. There is therefore a need for an evidence base for different approaches to screening in order to ensure that these programmes are effective. A patient level simulation approach is the most appropriate when dealing with infectious diseases since it allows modelling heterogeneous patients interacting with each other. Our individual based approach allows population-level costs and health benefits associated with infectious disease prevention and control to be captured. 

A decision-analytic model was developed to assess the cost per QALY gained for different NHS screening strategies. Patients can be colonised by CPE making them asymptomatic carriers, patient pathways change according to colonisation status and patient outcomes depend on characteristics. Patients can be discharged and re-admitted to hospital. Different scenarios are modelled to test how cost-effectiveness and patient outcomes change as CPE epidemiology changes. We use a combination of data to parameterise the model with information coming from literature sources and NHS Scotland patient level data. Health and economic benefits of screening are estimated taking into account parameter uncertainty. We report key events and probabilistic sensitivity analyses are also conducted. This is a highly detailed patient level simulation of a NHS hospital which provides estimates under uncertainty.

Screening has the potential to save lives and the recommendation is to continue screening for CPE in NHS hospitals. Screening results in a lower estimated number of infections, cross-infections and hospitalisations due to CPE. 

 

Title: Why Should we and How Can we Identify Structural Details in Size-Spectra of Marine Microplankton?

Date: Wednesday 25th September, 1.00pm

Venue: Livingstone Tower, LT907

Abstract:  Despite their microscopic size, unicellular marine plankton control how elements like carbon are transformed and distributed in the ocean. Phytoplankton are photosynthetic organisms that assimilate inorganic nutrients into their organic biomass, with cell sizes ranging from less than one up to several hundreds of micrometers. Phytoplankton live as single cells or in colonies, can form large sticky aggregates that may sink, or they become ingested by a variety of zooplankton.

Variations in size-spectra reveal some of the underlying dynamics involved within the unicellular plankton community. The size-abundance spectrum slope and the corresponding y-intercept are two parameters that are often thought to inform us about the biovolume and transfer efficiency of the microbial plankton community. This paradigm is challenged by measurements when looking at size-spectra precisely. Distinctive structural details are well resolved with the kernel density estimation method. I will present latest results from our analyses of elaborate, repeated microscopic measurements from the Fram Strait (Arctic Ocean). We could identify four ranges of size-selective grazing close to Equivalent Spherical Diameters (ESD) of 4, 9, 30 and 70 μm. These prominent and robust ESD ranges will be discussed in context with plankton size-spectra from other locations (e.g. Equatorial Pacific). Finally, I will discuss with you how size-spectral data could be used for calibrating size-based models.

Title: Phenotypic plasticity sustains modelled phytoplankton size diversity by flattening fitness gradients, but may confound observed relationships

Date: Friday 23rd August, 12.00pm

Venue: Livingstone Tower, LT907

Abstract:  Inducible phenotypic plasticity has long been known to impact the growth response of a wide variety of organisms, and more recently has been appreciated as a determinant of biodiversity, production, and ecosystem function. However, considerable uncertainty remains about how intra-specific trait variation may contribute to biodiversity. Photo-acclimation is a well known example of physiological flexibility for phytoplankton, and a variety of models have been developed to represent its effects on their growth, chlorophyll and nutrient content. I apply a sized-structured model accounting for the photo-acclimation response of phytoplankton (FlexPFT), as well as a control model lacking this response, to two contrasting time-series observation sites from the North Pacific ocean: a relatively calm subtropical site (stn. S1) and a more variable subarctic site (stn. K2). As previously reported (Smith et al. J. Plankton Res. 2016), compared to the control, the FlexPFT model reproduced better the available observations of size fractionated chlorophyll, nutrients, and primary production and predicted greater size diversity. Here I clarify that this is because phenotypic plasticity flattens fitness gradients, quantified here as specific growth rate vs. size. This effect contributed more to enhancing size diversity at the more variable subarctic site than at the calmer subtropical site. However, at both sites modelled size diversity differs substantially as calculated in terms of chlorophyll, carbon or nitrogen biomass, because the degree of flexibility differs with cell size, as a result of the size-scaling of traits. Modelled distributions of chlorophyll over size tend to be substantially less even (lower diversity) than those of either fitness or biomass. This suggests that, although much more widely available than observations of biomass or growth rate, chlorophyll-based size distributions should be interpreted with care. 

Title: Exploring the Drivers for the Sustainability of the Gooseneck Barnacle Fishery in Asturias, Northern Spain

Date: Wednesday 21st August, 1.00pm

Venue: Livingstone Tower, LT907

Abstract: The Asturian gooseneck barnacle fishery is a unique example of a complex social-ecological system that has been co-managed for over 20 years. As part of the co-management system fishers are allotted Territorial User Rights for Fishing and an active participation in management strategies in return for detailed data gathering. Here, we used this extensive time-series to assess the sustainability of the Asturian gooseneck barnacle fishery through time and to disentangle the key socio-ecologic drivers for its success. The fishery has succeeded in maintaining or increasing catch per unit effort in all management areas. Additionally, despite the national economic crisis, mean gooseneck barnacle market prices have remained stable in Asturias. Furthermore, the system has received vast public approval, where 73% of the stakeholders have indicated that the only way to maintain a sustainable gooseneck barnacle fishery in Asturias is through the current management regime. The co-management system has primarily achieved the sustainability of the fishery through 4 key characteristics: (1) the continuous incorporation of scientific information and fishers’ knowledge in management frameworks, (2) a matching of management scales with the main life-history traits of the species, (3) empowerment of the resource users and (4) embracing adaptive capacity through flexible management guidelines, resource diversification and selectivity. The Asturian gooseneck barnacle co-management system provides a set of basic principles for TURFs, which may be conducive to sustainable fisheries.

Title: Towards Constructing a Mathematically Rigorous Framework for Modelling Evolutionary Fitness

Date: Tuesday 11th June, 12.00pm

Venue: Livingstone Tower, LT907

Abstract: In modelling biological evolution, a major mathematical challenge consists in an adequate quantification of selective advantages of species. Current approaches to modelling natural section are often based on the idea of maximization of a certain prescribed criterion - evolutionary fitness. This paradigm was inspired by the seminal Darwin's idea of the 'survival of the fittest'. However, the concept of evolutionary fitness is still somewhat vague, intuitive and is often subjective. On the other hand, by using different definitions of fitness one can predict conflicting evolutionary outcomes, which is obviously unfortunate. In this talk, I present a novel axiomatic approach to model natural selection in dynamical systems with inheritance in an arbitrary function space. For a generic self-replication system, I introduce a ranking order of inherited units following the underlying measure density dynamics. Using such ranking, it becomes possible to derive a generalized fitness function which maximization will predict long-term evolutionary outcome. The approach justifies the variational principle of determining evolutionarily stable behavioural strategies. I demonstrate a new technique allowing to derive evolutionary fitness for population models with structuring (e.g. in models with time delay) which was so far a mathematical challenge. Finally, I show how the method can be applied to a von Foerster continuous stage population model.

Title: Gone with the Wind: Linking Phytoplankton Biomass and Wind Patterns using Satellite Data in the California Current System

Date: Wednesday 22nd May, 11.00am

Venue: Livingstone Tower, LT907

Abstract: 

Title: Modelling the Spread of HCV Amongst People Who Inject Drugs

Date: Wednesday 27th March, 1.00pm

Venue: Livingstone Tower, LT907

Abstract:  Mathematical modelling techniques are now being used by health organizations worldwide to help understand the likely impact that intervention strategies treatment options and combinations of these have on the prevalence and incidence of hepatitis C virus (HCV) in the people who inject drugs (PWID) population. In this talk, we develop a deterministic, compartmental mathematical model to approximate the spread of the HCV in a PWID population. The model assumes that after injection needles adopt the most infectious state of their previous state or that of the PWID who last injected with them. Using analytical techniques, we find that the model behaviour is determined by the basic reproductive number R0, where R0 = 1 is a critical threshold separating two different outcomes. If R0  1 there is only the disease-free equilibrium whereas if R0 > 1 there is the disease-free equilibrium and a unique endemic equilibrium. The disease-free equilibrium is globally stable if R0  1 and unstable if R0 > 1: Then we look at an approximate model obtained by using the fact that the timescale on which
injections take place is much faster than the timescale of epidemiological change. The approximate model has the same equilibria as the full model. For the approximate model we showed that if R0 > 1 the endemic equilibrium is locally stable. Simulations with realistic parameter values are used to illustrate the analytical results. A brief summary concludes the talk.

 

Title: Predicting Severe Complications Following Cardiac Surgery: An Evaluation of Predictive Modelling Methods for an Imbalanced Classification Problem

Date: Wednesday 27th March, 1.00pm

Venue: Livingstone Tower, LT907

Abstract: With the ageing population, the epidemic of obesity and patients having multiple morbidities, cardiac surgery is becoming more complicated and therefore postoperative complications are becoming more common. Patients with severe postoperative complications, such as kidney failure, stroke and sepsis, face significantly reduced quality of life after surgery, stay in the hospital for longer and increase the burden on healthcare resources. Hence, a robust and reliable predictive model for severe postoperative complications would prove extremely useful for managing patient flows and clinical resources in surgical care.

In this presentation I am going to talk about my most recent results from my Phd project that is in co-operation with the Department of Computer and Information Sciences, Mathematics and Statistics and the Department of Anaesthesia at the Golden Jubilee National Hospital. 

I am going to discuss my most recent results using predictive modelling methods, such as logistic regression, random forest, adaboost, gradient boosting model and stacking to predict severe postoperative complications after cardiac surgery. 

Title:  Data science in healthcare research: Challenges and Opportunities

Date: Thursday 14th March, 1.00pm

Venue: Livingstone Tower, LT907

Abstract:  Modern-day healthcare record keeping in NHS has evolved into electronic/digital format. This has created huge opportunities for “big data” research in medical science. Traditional and modern statistical methods have been applied to construct prediction models for a range of healthcare outcomes. This field of research has the potential to help with reducing prescription errors in medicine and implement more preventative strategies by improving precision in predicting adverse healthcare outcomes.

However, application of modern data science methods in healthcare research needs careful considerations. Lack of reproducibility is a major challenge, where the research results may be invalid when examined in another dataset or in real world. Majority of general population and clinicians remain sceptical about “big data” research with concerns around privacy and accuracy. Throughout the seminar, I will use illustrative case studies to discuss the challenges as well as the opportunities with data science in healthcare research.

Title: Bayesian models for safety outcomes in clinical trials and observational studies.

Date: Wednesday 27th February, 1.00pm

Venue: Livingstone Tower, LT907

Abstract: Bayesian models are increasingly appearing in the literature as methods to analyse clinical trial safety outcomes simultaneously, with the possibility of both modelling relationships between outcomes and handling multiple comparison issues which may occur when using multiple single-variate approaches. Adapting methods from a clinical trial setting for use with observational data presents some challenges, for example biased treatment allocation, lack of balance between comparator groups, and data which may require clean up.

Title: Individual behaviours and population response to environmental change in a context of informed dispersal.

Date: Wednesday 6th February, 1.00pm

Venue: Livingstone Tower, LT907

Abstract: The ability of individuals to select a future breeding area is particularly influential during their lifetime, because such decisions can have multiple consequences on individual life history traits and fitness, but also on metapopulation dynamics, structure and long-term persistence through non-random dispersal patterns. In the wild, many colonial and territorial species are known to gather personal and social information during prospecting to assess the local quality of different breeding areas and optimally decide whether and where to disperse. Under current climatic and anthropogenic pressures, understanding these informed dispersal strategies is essential for relating individual behaviour to subsequent movements and then determining how emigration and settlement decisions affects individual fitness and demography. Through empirical and theoretical examples, I will show how individuals acquire and use information for dispersal and what are the consequences at the individual and population level.

Title: Modelling the Effect of Auto-dissemination Traps on the Spread of Dengue in Malaysia.

Date: Wednesday 30th January, 1.00pm

Venue: Livingstone Tower, LT907

Abstract:  In this paper, we use the classical Ross-Macdonald model to analyse the effect of the Mosquito Home System (MHS), which is an example of an auto-dissemination trap, in controlling the spread of dengue in Malaysia in a high-rise condominium environment. By using the national dengue data from Malaysia, we are able to estimate lambda which represents the initial growth rate of the dengue epidemic and thus allows us to estimate the number of mosquitoes in Malaysia. The basic reproduction number R_0 is also obtained. We have constructed a mathematical expression which allows us to estimate the potential number of breeding sites for  Aedes mosquitoes. Later on, by using the data available from the eleven months trials carried out in three blocks of flats in Selangor, we improved on our dengue model by including the effect of the MHS and thus modelling the impact it has on the spread of dengue within the flats. Numerical simulations and tables are also produced to illustrate our results.

 

Title:  A New High-Accuracy Click Classifier to Study Harbour Porpoises in the Wild

Date: Wednesday 16th January, 1.00pm

Venue: Livingstone Tower, LT907

Abstract:  Harbour porpoises are difficult to observe at sea, even with good weather conditions, due to their small size and cryptic behaviour. However, they are highly vocal, producing narrow-band high-frequency (NBHF) echolocation clicks, which are produced as a series of clicks or ‘click trains’, making them well suited for passive acoustic monitoring (PAM). The first part of my PhD project includes the development of a new porpoise classifier (PorCC) that can be coupled to PAM systems to identify the likely porpoise signals among transient sounds. PorCC works on full-waveform signals and shows an improved performance over current classifiers. It was developed in MATLAB and uses the coefficients of two logistic regression models in a decision-making pathway to assign each signal to one of three categories: high-quality click (HQ), low-quality click (LQ), or high-frequency noise (N). PorCC achieved hit rates > 90% for HQ clicks while keeping false alarm levels < 1%. Moreover, PorCC could be applied for real time monitoring, as well as to study harbour porpoises, and potentially other NBHF species from data collected by towed hydrophones or static recorders. The following step is to separate overlapping click trains to then assess the likely behaviour the animals are engaged in, as evidence suggest the patterns of the click trains are indicative of at least some behaviours (e.g., feeding). The objective of the second part of my project is to study the acoustic behaviour of Kylie, a solitary female common dolphin who inhabits in the Firth of Clyde and interacts with harbour porpoises since at least 2004. Preliminary results suggest Kylie produces signals similar to those of the harbour porpoise when interacts with them. This would be the first case of vocal interaction between two species in the wild.

 

Title: Investigating the Variation of Antibiotic Prescribing Rates between NHS Scotland Health-boards and GP Practices Through the use of Maps, Principal Components and Spatial Analysis.

Date: Wednesday 16th January, 1.00pm

Venue: Livingstone Tower, LT907

Abstract:  

Background: In Scotland, there is a goal to raise awareness of unnecessary prescribing and inappropriate use of antibiotics, advocated by the Scottish Antibiotics Prescribing Group (SAPG), with the aim to lower the persistence of resistant organisms and control infections such as Clostridium-difficile infection (CDI). The aim of this analysis is to investigate the variation of antibiotic prescribing rates between NHS Scotland health boards and GP practices through the use of maps, principal components and spatial analysis.

Methods: This analysis began with multiple open source ISD prescription data files which were converted into a linked dataset. Other GP descriptive information and locations of GP practices were also merged to this dataset and were used to investigate how the rates of antibiotic prescribing vary across Scotland. Through the use of maps, it was possible to visualise total antibiotic prescribing rates, and individual antibiotic drug group rates, between NHS health boards in Scotland. Creating a spatial point data frame allowed for visualisation of the difference between antibiotic prescribing rates of individual GP practices, within the same health boards, assessing close proximity rates.

Principal Component Analysis (PCA) on the 13 Antibiotic groups was used to indicate GP practices that follow similar prescribing trends of antibiotic drug groups. A Poisson generalised linear regression model of total antibiotic prescribing was constructed, investigating potential explanatory variables. The residuals from this model were then assessed for spatial association, this was achieved using Monte Carlo Envelopes to test for spatial association.

Results: This analysis has shown that the variation in antibiotic prescribing rate between Scottish GP practices can be partly explained by practice demographics. Higher proportions of practice populations Over 74, Under 15 and in the Most Deprived quintile suggests higher prescribing at GP-level, whereas increasing proportion of Least Deprived suggests lower prescribing rates. The PCA has shown that 25% of the total variation can be explained, approximately, by an average across all antibiotic drug groups therefore total antibiotics is a reasonable representation of prescribing at GP-level. After adjusting for GP practice characteristics, there is no evidence of spatial association between GP practices.

 

Title: Stabilisation of Highly Nonlinear Hybrid Systems by Feedback Control Based on Discrete-Time State Observations

Date: 4-5.00pm, Wednesday 9th October 2019, LT907

Abstract:  Given an unstable hybrid stochastic differential equation (SDE), can we design a feedback control, based on the discrete-time observations of the state at times $0, \tau, 2\tau, \cdots$, so that the controlled hybrid SDE becomes asymptotically stable? It has been proved that this is possible if the drift and diffusion coefficients of the given hybrid SDE satisfy the linear growth condition.

However, many hybrid SDEs in the real world do not satisfy this condition (namely, they are highly nonlinear) and there is no answer to the question yet if the given SDE is highly nonlinear.  The aim of this paper is to tackle the stabilization problem for a class of highly nonlinear hybrid SDEs.  Under some reasonable conditions on the drift and diffusion coefficients, we show how to design the feedback control function and give an explicit bound on $\tau$ (the time duration between two consecutive state observations), whence the new theory established in this paper is implementable.

Title: The truncated Euler-Maruyama method for stochastic differential equations with H\"{o}lder diffusion coefficients

Date: 5-6.00pm, Friday 10th May 2019, LT907

Abstract:  In stochastic financial and biological models, the diffusion coefficients often involve the term \sqrt{x}, or more general |x|^r for r\in(0,1). In this paper, we study the strong convergence of the truncated Euler-Maruyama (EM) approximation first proposed by Mao for one-dimensional stochastic differential equations (SDEs) with superlinearly growing drifts and the H\"{o}lder continuous diffusion coefficients.

Title: Asymptotic Properties and Numerical Analysis of the Solution to Hybrid Stochastic Differential Equations with Jumps

Date: Friday 7th December, LT908, 4-5.00pm

Abstract:  In this paper, we are concerned with the asymptotic properties and numerical analysis of the solution to hybrid stochastic differential equations with jumps. By applying the theory of M-matrices, we study the pth moment asymptotic boundedness and stability of the solution. Under the non-linear growth condition, we also show the convergence in probability of the Euler-Maruyama approximate solution to the true solution. Finally, some examples are provided to illustrate our new results.

Title: On strong convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations

Date: 3pm Wednesday 10th October 2018, LT907

Abstract:  We prove that some time discretization schemes, such as the splitting, fully and semi-implicit ones, of the 2D Navier-Stokes equations subject to a random perturbation converge in $L^2(\Omega)$. The speed of convergence depends on the viscosity. The argument is based on convergence of a localized scheme, and on exponential moments of the solution to the stochastic 2D Navier-Stokes equations. This joint work with H.~Bessaih  improves previous results which only described the speed of convergence in probability of these numerical schemes.

Title: Stability in distribution of stochastic functional differential equations

Date: 4pm Wednesday 26th September 2018, LT907

Abstract:  The theory of stochastic functional differential equations (SFDEs) has been developing very quickly.  In particular, many research papers have been devoted to the stability analysis of SFDEs. However, most of these papers are concerned with the stability of the trivial solution in probability or moment and such stability is somehow too strong. In many practical situations it would be more useful to know whether or not the probability distribution of the solution will converge to some distribution). This convergence is called the stability in distribution and the limit distribution is known as the stationary distribution. The talk will review the current research on the stability in distribution of SFDEs and show our new results.

Title: Randomized Numerical Schemes for (S)ODEs/SPDEs

Date: 4.00pm Tuesday 12th June 2018

Venue: Livingstone Tower, LT9.07

Abstract:   A wide range of applications, for instance, in the engineering and physical sciences as well as in computational finance is still spurring the demand for the development of more efficient algorithms and their theoretical justification. In particular, the current focus lies on the approximation of ODEs/S(P)DEs which cannot be treated by standard methods found in textbook.  We, therefore, first developed two randomized explicit Runge–Kutta schemes for ordinary differential equations (ODEs) with time-irregular coeffcient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coeffcient functions are only integrable with respect to the time variable but are not assumed to be continuous.  An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule.  It is demanding to approximate numerical solutions of non-autonomous SDEs where the standard smoothness and growth requirements of standard Milstein-type methods are not fulfilled. In the case of a non-differentiable drift coefficient function f, we proposed a drift-randomized Milstein method to achieve a higher order approximation and discussed the optimality of our convergence rates.  We also pushed the idea to the numerical solution of non-autonomous semilinear stochastic evolution equations (SEEs) driven by an additive Wiener noise. Usually quite restrictive smoothness requirements are imposed in order to achieve high order of convergence rate. It turns out that the resulting method converges with a higher rate with respect to the temporal discretization parameter without requiring any differentiability of the nonlinearity. Our approach also relaxes the smoothness requirements of the coefficients with respect to the time variable considerably.

Title: Non-local Schrodinger Operators and Related Jump Processes

Date: 3pm Wednesday 15th March 2017

Abstract:  Classical Schrödinger operators have been the object of much research involving functional analysis, probability and mathematical physics in the past decades. The recent interest in non-local Schrödinger operators consisting of the sum of a pseudo-differential operator and a multiplication operator greatly extended the range of applications, and inspired much new research in pure mathematics too. I will discuss how Feynman-Kac-type representations can be derived for the non-local cases and which random processes they give rise to. Then I will consider various sample path properties of these jump processes in terms of spectral properties of the generating non-local operators, and will contrast them with diffusions and classical Schrödinger operators.