**Department Colloquia**

19 September 2018: Dr Sascha Trostorff, (TU Dresden)
Title: ** Well-posedness for a general class of differential inclusions**

Date: 3.30pm Wednesday 19th September 2018

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: The abstract is attached.

Title: ** **How long does it take to get there?

Date: 3.30pm Wednesday 7th November 2018

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: There are a huge number of nonlinear partial differential equations that do not have analytic solutions. Often one can find similarity solutions, which reduce the number of independent variables, but still leads, generally, to a nonlinear equation. This can, only sometimes, be solved analytically. But always the solution is independent of the initial conditions. What role do they play? It is generally stated that the similarity solution agrees with the (not determined) exact solution when (for some variable say t) obeys t >>t_1. But what is t_1? How does it depend on the initial conditions? How large must t be for the similarity solution to be within 15, 10, 5, 1, 0.1, ….. percent of the real solution? And how does this depend on the parameters and initial conditions of the problem? I will explain how a number such typical, but somewhat different, fundamental problems can be solved, both analytically and numerically, and compare some of the results with small scale laboratory experiments, performed during the talk. It will be suggested that many members of the audience could take away the ideas and apply them in their own special areas.

Title: ** **Variational Multiscale (VMS) methods for the simulation of turbulent incompressible flows

Date: 3.30pm Wednesday 14th November 2018

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: VMS methods are a comparably new class of methods for simulating convection-dominated problems, like turbulent incompressible flows. The talk will start with an introduction to the difficulties of turbulent flow simulations. Next, principal ideas of the classical Large Eddy Simulation (LES) approach will be presented. Then, the basic ideas of VMS methods will be discussed, together with highlighting the differences to LES methods.

Various realizations of VMS methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different.

In this talk, the derivation of the different VMS methods is presented in some detail, their relation among each other and their behavior in numerical simulations are discussed.

Title: ** **Combining livestock movement network data with a risk map to inform Rift Valley fever control in Northern Tanzania

Date: 3.30pm Wednesday 5th December 2018

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: The potential of modelling to inform infectious disease control in low-income countries, where need is generally greatest, is hampered by a lack of data that is routinely available in many high-income countries, in particular livestock movement data. I will present recent progress from a project that aims to simulate the spread of zoonoses (animal infectious diseases that infect humans) such as Rift valley fever (RVF) across three regions of northern Tanzania (Arusha, Manyara and Kilimanjaro, covering an area slightly larger than Portugal, and home to 5 million people and 4 million cattle). The simulation model combines three components: a model of livestock movement among the 398 wards (administrative subunits) in the study regions, fitted to data from archived livestock movement permits; a map of habitat suitability for the mosquito vectors of RVF; and a stochastic SEIR model of RVF transmission within wards. I will illustrate the simulation model’s potential to predict the effects of interventions (vaccination, market closure) on disease transmission, the potential to use the movement network to target interventions, and the model's main limitations, including the general lack of availability of good quality data on relevant parameters.

### Applied Analysis

9th October: Grant Silver (Department of Mathematics and Statistics)Title: Parameterised communicability metrics in networks: The case of alternative routes for urban traffic

Date: 3pm Tuesday 9th October

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Many systems in the real world are represented as networks and methods of network theory make it possible to analyse their properties. We demonstrate how communicability functions can be used to define node centrality indices, which gives a ranking of nodes based on their importance to the network. We then introduce a novel communicability function which makes greater use of longer walks in the network. We introduced a parameter that can be interpreted as the temperature in the network. We use the parameterised function to define a communicability distance between each pair of adjacent nodes. We show in Relative Neighbourhood Graphs how varying the parameter significantly changes the shortest communicability path between pairs of nodes. As the parameter increases, the length of the shortest communicability path increases, while the nodes themselves have, on average, decreased centrality values. We then apply the parameterised communicability function to the street network of Isfahan, where the parameter is interpreted as the level of traffic within the network. We show that for a particular parameter choice, the shortest communicability path becomes the preferred way to travel between nodes. This new path will be longer; however it will contain fewer nodes of high centrality, which corresponds to the avoidance of street intersections which are more likely to be congested with traffic.

Title: Applications of Centrality Measures to Simplicial Complexes

Date: 3pm Tuesday 23rd October

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Networks are a well-established method of representing complex systems mathematically. By representing individual entities by nodes and the connections between them as edges, we are able to model systems such as the interactions of proteins in a cell and the communication between wireless sensors. One limitation of networks is that there is no way to model three (or more) way interactions between nodes. Simplicial Complexes are one of the ways to remedy this situation. Centrality measures are ways of measuring the “importance” of a node, edge, triangle or higher order simplex to a simplicial complex in various situations. We shall focus on two applications of these centrality measures, the first to protein protein interaction networks and the second to random geometric simplicial complexes.

Title: Compressible Fluids Interacting with a Linear-Elastic Shell

Date: ** 4pm** Tuesday 6th November

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: We study the Navier–Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell of Koiter type. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter’s elastic energy. We show the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies γ > 12/7 (γ > 1 in two dimensions). The solution exists until the moving boundary approaches a self-intersection. This provides a compressible counterpart of the results in Lengeler and R°užiˇcka (Arch RationMech Anal 211(1):205–255, 2014) on incompressible Navier–Stokes equations.

Title: TBA

Date: 3pm Tuesday 13th November

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: TBA

Title: An introduction to the theory of word-representable graphs

Date: 3pm Tuesday 12th February 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word $xyxy\cdots$ (of even or odd length) or a word $yxyx\cdots$ (of even or odd length). A graph $G=(V,E)$ is word-representable if and only if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$.

Word-representable graphs generalize several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. In this talk, I will give a comprehensive introduction to the theory of word-representable graphs. In particular, I will discuss the characterization of word-representable graphs in terms of semi-transitive orientations.

### Continuum mechanics & industrial mathematics

19th February 2019 - Paul Grassia (Strathclyde)Title: Motion of an oil droplet through a capillary with charged surfaces

Date: 2.00pm Tuesday 19th February 2019

Venue: Livingstone Tower, 9th floor, room LT908

Abstract:

A model developed by Breward et al. (2018) for the advance of a charged oil droplet along a charged capillary pore is considered. The oil droplet is surrounded by an aqueous phase filling the pore, and the model considers a uniformly curved capillary static droplet front plus an aqueous thin film separating the body of the oil droplet from the capillary wall, with these two regions being joined by a transition region. The methodology follows a classical asymptotic approach proposed by Bretherton (1961) but incorporates additional electro-osmotic effects (specifically an electro-osmotic disjoining

tension) due to the charged surfaces. A number of dimensionless parameters control the model's behaviour, of which the most important is denoted $\chi'$ and represents the ratio between the ``nominal''

thickness of the aqueous film (as determined neglecting any electrostatic effects) and the Debye length within the film, which is sensitive to ion concentrations and hence to salinity. When $\chi'$ is large, electro-osmotic effects are screened and Bretherton's classical results are recovered. However as $\chi'$ decreases, electro-osmotic effects come into play and the film becomes much thicker than Bretherton's prediction to ensure that screening effects are not altogether lost, and also there is a noticeable increase in the pressure needed to drive the droplet front along. These results apply with minor variations in the case of singly charged surfaces (charge on either oil or on the capillary wall), oil and wall surfaces with like charges, or oil and wall surfaces with opposite but unequal charges. However in the case of opposite and equal charges, the system's behaviour changes dramatically. There is now a conjoining electro-osmotic pressure rather than a disjoining tension, the film becomes thinner than the analogous Bretherton film, and the pressure needed to drive the droplet front along decreases. Surprisingly in this case, for sufficiently small $\chi'$, the work done by the conjoining pressure can exceed the work done against viscous dissipation, meaning the pressure required to drive the droplet front is not just smaller than in Bretherton's predictions but also slightly less than would be estimated based on capillary forces alone.

Although the main effect of reducing salinity is to increase Debye length and hence reduce $\chi'$, salinity also affects surface charges. A situation is explored whereby reducing salinity affects charges producing a switch from disjoining tensions to conjoining pressures and back again: this leads to a non-monotonic response in film thickness and pressure required to drive the droplet front along.

Title: Viscous froth model applied to the motion of a two-dimensional bubble train

Date: 2.00pm Tuesday 26th February 2019

Venue: Livingstone Tower, 9th floor, room LT908

Abstract:

The viscous froth model is used to predict the rheological behaviour of a two-dimensional liquid-foam system contained between two glass plates. This model incorporates three physical phenomena: the viscous drag force, the pressure difference across the foam films and the surface tension acting along the films. The viscous froth model converts any mismatch between the pressure forces and the film curvature to film motion, leading to viscous drag forces. As the foam system flows, topological transformation or bubble rearrangement may occur, implying loss of stability. These topological transformations have an effect on system’s energy, quantified in terms of the film lengths, the energy relaxing after such events. It is known that, in the so-called infinite staircase structure, the system does not undergo any topological transformations, for any imposed driving back pressure, meaning that the bubbles flow out of the system in the same order in which they entered it. In contrast, for higher imposed back pressures, systems composed of a finite number of bubbles tend to undergo topological transformations: a train of bubbles can however exhibit distinct types of topological transformations in distinct locations in the train. To investigate under which physical and geometric conditions those topological changes take place, a simple three bubble symmetric case is studied, so as to understand the behaviour of the system as higher imposed back pressures are applied. An analytical solution is obtained for weak driving velocities, whilst for an arbitrary imposed back pressure, steady and unsteady state solution are obtained for a wide range of initial conditions.

Title: Poro-Hyperelasticity: The Mechanics of Fluid-Saturated Soft Materials

Date: 1.00pm Tuesday 5th March

Venue: Livingstone Tower, 9th floor, room LT908

Abstract:

The lecture presents the formulation of the mechanics of a fluid-saturated porous medium where the porous skeleton can undergo hyper-elastic deformations. The modelling has potential applications in the study of highly deformable biological tissues including brain matter, synthetic materials impregnated with fluids and highly deformable porous solids used as tactile sensors, where the fluid can be the air present in the void space. Conventional treatments of soft biological materials assume the applicability of classical hyperelasticity. The presence of the saturating fluid, however, completely changes the character of the modelling approach, in that the partitioning of stresses between the fluid and the porous skeleton needs to be addressed. The flow of the saturating fluid, induced by hydraulic gradients, is an added consideration. The presentation summarizes recent analytical results for canonical problems involving one-dimensional strains, pure shear and expansion of annuli. The role of these developments in the validation of computational schemes that can ultimately be used in the solution of problems with complex geometries is also discussed.

________________________________________________

*A.P.S Selvadurai is *William Scott Professor* and *James McGill Professor* in the Department of Civil Engineering and Applied Mechanics at McGill University. He received his PhD in *Theoretical Mechanics* from The University of Nottingham under the tutelage of the late Professor A.J.M. Spencer FRS and his DSc in *Theoretical Mechanics* also from the University of Nottingham. His research interests include continuum mechanics, geomechanics and computational mechanics. He is Fellow of the *Royal Society of Canada*, *The Institute of Mathematics and its Applications* and the *American Academy of Mechanics*. He is in the Editorial Board of several Journals including the *Journal of Engineering Mathematics*.

Title: *Pattern formation & complex bifurcations in a class of nonlocal hyperbolic models*

Date: 2.00pm Tuesday 12th March

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

*Bifurcation theory has been used for decades to investigate transitions between different dynamical states displayed by various fluid dynamics models. This also led to the identification of novel dynamical phenomena: from rotating waves to heteroclinic cycles, and even snake-and-ladder structures. *

*Here I will focus on the application of bifurcation theory to biological aggregations. In particular, I will discuss pattern formation and transitions between different patterns as a result of symmetry breaking in a class of nonlocal hyperbolic models developed to describe self-organised biological aggregations (from schools of fish and flocks of birds, to swarms of insects or bacteria). I will discuss solutions arising near codimension-1 and codimension-2 bifurcation points. I will also mention briefly some more complex phenomena that could be exhibited by this class of models: from heteroclinic cycles to snake-and-ladder localized structures. *

Title: From dry quicksand to unsinkable suspensions--exploring the underlying links between dry granular flow and wet suspensions

Date: 2.00pm Tuesday 19th March

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

The flow behaviour of dry granular materials, e.g. free flowing sand, is commonly observed to be very different from wet suspensions, e.g. sticky cornstarch-water mixtures. We show, through particle simulations and rheology experiments, that the different behaviour can be put in a unifying rheological framework, where the interplay between particle-size scaling and the observation window gives rise to the perception. More interestingly, the underlying microstructure and force networks bear surprising similarities, which can be used to understand many sometimes puzzling non-linear flow phenomena.

A particular example is shear thickening, where suspension viscosity increases with shear stress or shear rate. It is a ubiquitous feature of many different particle systems and flow processes in nature and industry, particularly for particles of intermediate sizes (diameter 1 μm ≤ *d* ≤ 50 μm) at high concentrations. This phenomenon has been traditionally explained as being driven purely by hydrodynamic interactions. However, recent theoretical, modelling and experimental work has shown the inadequacy of this mechanism, by elucidating the important role of frictional particle contact.

In this talk, I will present experimental and simulation evidence for shear thickening as a transition from a typical colloidal to a granular behaviour, in which the formation of frictional *non-hydrodynamic* contacts is key. We have directly quantified the contact contribution to the suspension viscosity during shear thickening by means of shear-reversal rheological measurements, providing new and unambiguous evidence. I will finally describe the ‘tuning’ of shear thickening and other rheological properties based on the understanding of particle contacts, by means of active control or suspension formulation.

**Bio-sketch**: Dr Jin Sun is a Reader in the School of Engineering at the University of Edinburgh. He graduated from Iowa State University in 2007 with a PhD in Mechanical and Chemical Engineering and did postdoctoral research in Chemical Engineering at Princeton University until 2010. His main research interests are in rheology and flow of dense particle systems in industrial and natural environments, such as manufacturing of ceramics, food stuff, battery electrodes and pharmaceuticals and debris and pyroclastic flows. He has been principal investigator of three UK EPSRC projects (total value ~£2.4M), currently leading the “Predictive formulation of high-solid-content complex dispersions” project; and co-investigator on three EU projects (~€11M), working on particle-fluid simulation method development. He has been awarded a Royal Academy of Engineering/ Leverhulme Trust Senior Research Fellowship to explore the novel application of suspension rheology in 4D printing technology.

Title:

Date: 2.00pm Tuesday 26th March

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: The origin of fluid manipulation using acoustic waves, or 'acoustofluidics', can be traced back to the early experiments of Chladni and Faraday, which lead to the discovery of the eponymous Faraday waves. Since then, the science of acoustic-wave propagation in fluids has advanced considerably; however, numerical modelling has remained under-utilised in this field, and the majority of work tends to be theoretical or empirical in nature. This seminar will review the history of acoustofluidic modelling, summarising the strengths and limitations of the different approaches, and highlighting the essential multiscale nature of the underlying physics, which makes it a formidable problem to simulate.

In recent years this classic field of research has found new life when applied to smaller length scales, due to the development of acoustofluidic lab-on-a-chip devices, which have applications across chemistry, biology, and medicine. These devices typically use small-amplitude, high-frequency sound waves to manipulate micro/nano droplets and films on surfaces of varying wettability. The physics of micro/nano-scale acoustofluidics is distinct from its larger scale counterpart, in ways that simplify the consequent numerics. This lecture will conclude with discussing nanoscale acoustofluidics, and presenting original results involving molecular dynamics simulations of acoustofluidic evaporation/boiling of nanodroplets and nanofilms.

Title: Modelling Phase Separation In Amorphous Solid Dispersions

Date: 2.00pm Wednesday 1st May 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Much work has been devoted to analysing thermodynamic models for solid dispersions with a view to identifying regions in the phase diagram where amorphous phase separation or drug recrystallization can occur. However, detailed partial differential equation non-equilibrium models that track the evolution of solid dispersions in time and space are lacking. Hence theoretical predictions for the timescale over which phase separation occurs in a solid dispersion are not available. In this talk, we address some of these deficiencies by (i) constructing a general multicomponent diffusion model for a dissolving solid dispersion; (ii) specializing the model to a binary drug/polymer system in storage; (iii) deriving an effective concentration dependent drug diffusion coefficient for the binary system, thereby obtaining a theoretical prediction for the timescale over which phase separation occurs; and (iv) presenting a detailed numerical investigation of the HPMCAS/Felodipine system assuming a Flory-Huggins activity coefficient. The numerical simulations exhibit numerous interesting phenomena, such as the formation of polymer droplets and strings, Ostwald ripening/coarsening, phase inversion, and droplet-to- string transitions.

### Numerical Analysis and Scientific Computing

2nd October: Dr Dante Kalise (Imperial College London)Title: TBA

Date: 4.00pm Tuesday 2ndOctober

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

Title: TBA

Date: 4.00pm Tuesday 23rd October

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: TBA

Title: Reducing Reachability in Temporal Networks

Date: **2.00pm** Tuesday 6th November

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: The concept of reachability sets (i.e. which vertices in a network can be reached by travelling along edges from a given starting vertex) is central to many network-based processes, including the dissemination of information or the spread of disease through a network. Depending on the application, it might be desirable to increase or decrease the number of vertices that are reachable from any one starting vertex. In most applications, time plays a crucial role: each contact between individuals, represented by an edge, will only occur at certain time(s), when the corresponding edge is "active". The relative timing of edges is clearly crucial in determining the reachability set of any vertex in the network.

In this talk, I will address the problems of reducing the maximum reachability of any vertex in a given temporal network by two different means:

(1) we can remove a limited number of time-edges (times at which a single edge is active) from the network, or

(2) the number of timesteps at which each edge is active is fixed, but we can change the relative order in which different edges are active (perhaps subject to constraints on which edges must be active simultaneously, or restrictions on the timesteps available for each edge).

Mostly, we find that these problems are computationally intractable even when very strong restrictions are placed on the input, but we identify a small number of special cases which admit polynomial-time algorithms, as well as some general upper and lower bounds on what can be achieved.

Everything in this talk is based on joint work with Jessica Enright (University of Edinburgh); I will also mention some joint results with George B. Mertzios and Viktor Zamaraev (University of Durham) and Fiona Skerman (Uppsala University).

Title: TBA

Date: 4.00pm Tuesday 13th November

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: TBA

Title: Matrix Structures arising in Fractional Differential Equations: Spectral Features and Fast Solvers

Date: 1pm Thursday 31st January 2019

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: Fractional partial diﬀerential equations (FDEs) are a generalization of classical partial diﬀerential equations, used to model anomalous diﬀusion phenomena. Several discretization schemes (ﬁnite diﬀerences, ﬁnite volumes, etc.) combined with (semi)-implicit methods lead to a Toeplitz-like matrix-sequence. In the constant diﬀusion coeﬃcients case such a matrix-sequence reduces to a Toeplitz one, then exploiting well-known results on Toeplitz sequences, we are able to describe its asymptotic eigenvalue behavior. In the case of nonconstant diﬀusion coeﬃcients, we show that the resulting matrix-sequence is a Generalized Locally Toeplitz (GLT) sequence and then we use the GLT machinery to study its singular value/eigenvalue distribution as the matrix size diverges. The new spectral information is employed for analyzing preconditioned Krylov and multigrid methods recently appeared in the literature, with both positive and negative results. Moreover, such spectral analysis guides the design of new preconditioning and multigrid strategies. We propose new structure preserving preconditioners with minimal bandwidth (and thus with eﬃcient computational cost) and multigrid methods for related 1D and 2D problems. Numerical results conﬁrm the theoretical analysis and the eﬀectiveness of the new proposals

Title: Dynamic primal-dual regularization in interior point methods

Date: 2pm Thursday 31st January 2019

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: Interior point methods (IPMs) for linear and quadratic programming problems need to solve a large saddle point system at each iteration. Such systems are inherently ill-conditioned as the method approaches optimality. Moreover, they sometimes involve rank deficient matrices which further complicates the task. The aforementioned difficulties can be addressed by applying a primal-dual proximal point method to the problem; the added proximal terms in the objective improve the spec- tral properties of the saddle point system while allowing convergence of the interior point method to an optimal solution of the initial problem.

In this talk we propose a non-diagonal regularization for interior point methods which acts in primal and dual spaces and dynamically chooses the right level of perturbation; in the context of generalized proximal point methods, the regularization matrices are in fact the penalty matrices corresponding to the added proximal terms. Depending on a measure of infeasibility, we construct regularization matrices that improve the sparsity of the KKT system matrices, without affecting the convergence of the method. The ultimate objective of such a regularization is to deliver better conditioning in the KKT systems and allow for their solution with less computational effort.

Title: Limited-memory approximation of the inverse Hessian in 4D-Var

Date: 2.30pm Thursday 31st January 2019

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: Networks are versatile representations of the interactions between entities in complex systems. Cycles on such networks represent feedback processes which play a central role in system dynamics. In this seminar we present a measure of the importance of any individual cycle as the fraction of the total information flow of the network passing through the cycle. This measure stems from a number theoretic sieve which asymptotically counts the number of walk multiples of any given cycle, in a manner analogous to how integers are the multiples of primes. We will show that this centrality measure is computationally cheap, numerically well-conditioned, induces a centrality measure on arbitrary subgraphs and reduces to the eigenvector centrality on vertices. We demonstrate that in the protein-interaction network of the plant Arabidopsis thaliana, a model based on cycle-centrality better accounts for pathogen activity than the state-of-art one. This translates into pathogen-targeted-proteins being concentrated in a small number of triads with high cycle-centrality. In the protein-interaction network of the yeast, we will show that cycle-centrality is capable of finding biological complexes by itself, a feat which is not reproduced by other notions of group-centrality. Finally, we will mention the latest developments from sieve theory, which yield a formula for the centrality that can be evaluated in a time independent from the network size under certain conditions.

Title: Networks analysis from number theoretic sieves : biological applications

Date: 1pm Thursday 7th February 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Networks are versatile representations of the interactions between entities in complex systems. Cycles on such networks represent feedback processes which play a central role in system dynamics. In this seminar we present a measure of the importance of any individual cycle as the fraction of the total information flow of the network passing through the cycle. This measure stems from a number theoretic sieve which asymptotically counts the number of walk multiples of any given cycle, in a manner analogous to how integers are the multiples of primes. We will show that this centrality measure is computationally cheap, numerically well-conditioned, induces a centrality measure on arbitrary subgraphs and reduces to the eigenvector centrality on vertices. We demonstrate that in the protein-interaction network of the plant Arabidopsis thaliana, a model based on cycle-centrality better accounts for pathogen activity than the state-of-art one. This translates into pathogen-targeted-proteins being concentrated in a small number of triads with high cycle-centrality. In the protein-interaction network of the yeast, we will show that cycle-centrality is capable of finding biological complexes by itself, a feat which is not reproduced by other notions of group-centrality. Finally, we will mention the latest developments from sieve theory, which yield a formula for the centrality that can be evaluated in a time independent from the network size under certain conditions.

Title: Stability of network indexes defined by means of matrix functions

Date: 1pm Thursday 14th February 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Given a network G, a matrix function of the adjacency matrix (e.g., the exponential or the resolvent) defines a centrality index I(k) measuring the importance of the node k in the network. We address the problem of estimating the changes in I(k) with respect to changes in the edge structure of G. We propose several bounds showing that the magnitude of the variation of I(k) decays exponentially with the shortest-path distance in G that separates k from the set of nodes touched by the perturbed edges.

Title: Krylov-Tikhonov methods for large-scale inverse problems

Date: 1pm Thursday 14th February 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Inverse problems are ubiquitous in many areas of Science and Engineering and, once discretised, they lead to ill-conditioned linear systems, often of huge dimensions: regularisation consists in replacing the original system by a nearby problem with better numerical properties, in order to find a meaningful approximation of its solution. After briefly surveying some standard regularisation methods, both iterative (such as many Krylov methods) and direct (such as Tikhonov method), this talk will introduce the recent class of the Krylov-Tikhonov methods, which merge an iterative and a direct approach to regularisation. In particular, strategies for choosing the regularization parameter and the regularization matrix will be emphasised.

Title: TBA

Date: 1pm Thursday 7th March 2019

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

### Population Modelling and Epidemiology

14th March: Dr Bhautesh Jani (University of Glasgow)**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: **Optimal-detail circulation models for fjords and sea lochs: an application to the Puget Sound **

Date: Wednesday 28th November, 1.00pm

Venue: Livingstone Tower, LT907

Abstract: The Puget Sound is a fjord on the west coast of the USA near Seattle. It is an important habitat for marine life, but in the last 30 years three species of wild salmon have declined dramatically. A possible explanation for this decline is the impact of water circulation and salinity variations on plankton production. This context motivates the investigation of long-term changes in circulation due to environmental dynamic conditions and their impacts on primary production. Our long-term goal is to create a new circulation model that runs over decades and to couple it with a bio-chemical model. Here we focus on the development of the physical model. We use a tidally and laterally averaged framework to describe the estuarine circulation over a long time-scale. However, this requires us to define tidally averaged eddy coefficients. We present a new tidally averaged mixing parametrization for the fjord of the Puget Sound. We show that different physical processes drive the mixing in the bottom and surface layers. For the bottom layer the eddy viscosity relies only on the tidal velocity. However, for the surface layer, wind stress also needs to be considered.

Title: **Assessing the Impact of Offshore Windfarm Developments on Marine Mammal Populations**

Date: Wednesday 28th November, 1.00pm

Venue: Livingstone Tower, LT907

Abstract: The construction of offshore windfarms can impact marine mammals via the underwater noise generated by the installation of foundations. Potential consequences for extended or repeated disturbance include behavioural and physiological changes which can affect the health and vital rates of individual animals. This, in turn, can translate into population effects.

Consents have been granted for a number of major windfarm developments on the East Coast of Scotland, with further applications anticipated. These major developments each have the potential to input construction noise over tens of kilometres and, taken cumulatively, will input construction noise over a substantial number of years. In this talk we use the iPCoD (Interim Population Consequences of Disturbance) model to assess the impact of proposed developments on the East Coast of Scotland on the long term population dynamics of several key species of marine mammals: bottlenose dolphin, grey seal, harbour porpoise and minke whale.

Title: Effect of Viral Plasticity on the Survival of Host-Virus Systems

Date: Wednesday 14th November, 1.00pm

Venue: Livingstone Tower, LT907

Abstract: Phenotypic plasticity is the ability of an individual to vary their features when environmental conditions change. It can be found in organisms at any trophic level and many studies pointed out the importance of plasticity in the survival of individuals. However, the effects of plasticity on the ecological interactions are not well characterized. Here, we focus on viral plasticity (i.e. viruses are more virulent when the host growth rate increases) and we use models to investigate how plastic viruses affect the interaction with bacteria under a diversity of hosts size and under different environmental conditions. We investigate under which conditions plasticity translates into an increase of coexistence between host and virus and how plasticity influences oscillations as typically observed for antagonistic. We will discuss the stability regions that we compare with a non-plastic system (i.e. viral traits are fixed). The self-regulation involved by the plasticity can have important general applications such as for predator-prey systems.

Title: A Statistical Assessment of the Utility of TOXBASE in the Emergency Department

Date: Wednesday 14th November, 1.00pm

Venue: Livingstone Tower, LT907

Abstract: The National Poisons Information Service maintain TOXBASE, a system which provides information on the treatment of poisoned patients. It contains information on over 17,000 different toxicants. In this talk, I will focus on work linking TOXBASE usage data with routinely collected attendances and admissions data from England and Wales in order to assess whether usage of TOXBASE in the emergency department has a positive impact on admissions due to poisoning.

This analysis began by comparing temporal patterns in accesses to TOXBASE with emergency admissions due to poisoning. It was then of interest to determine whether there were hospitals which were unusual with respect to their admissions or their use of TOXBASE. An informal assessment of these “unusual” values was carried out in order to assess whether there were any commonalities which may explain why the locations are unusual.

Finally, regression models were fit in order to assess the impact of TOXBASE use. An interaction model was first used to assess the overall effect of TOXBASE use on emergency admissions. This showed that, while admissions will always increase with attendances, locations which use TOXBASE more will see a smaller increase than those which use TOXBASE less. The final piece of analysis used simple models to assess whether accesses to six different drug sub-groups had different effects on admissions.

Date: 3pm Wednesday 3rd October 2018

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** The main revenue source for beef farmers worldwide is carcass value (carcass weight, conformation and fat score). In general, heavier, better conformed (muscular), lean carcasses are awarded premium grades at the abattoirs. The grade a carcass receives is related to a number of factors, such as the genetics (breed) of the animal and the environment in which it is reared. In order to improve the profitability of the herd, breeders need an accurate means of comparing and selecting the best animals to breed from. Beef producers also need information on how their farm management (environmental) practices affect the performance of their animals. For predictive purposes, both Ireland and the UK routinely assess the productivity of animals for carcass traits, to estimate the improvement in these traits by the use of particular animals in breeding programs. The primary purpose of these evaluations is to distinguish the elite breeding stock in the population. These evaluations may be enhanced through employing alternative methods of analysis and by incorporating more information on animals or relatives of animals. This increase in information can be achieved through countries pooling their data. Pooling data also means that farmers in each country will have an accurate means of comparing foreign and domestic bulls, therefore getting access to the best selection candidates.

Using information on beef carcasses from abattoirs in Ireland and the UK, this study makes use of random regression analyses to generate tools for the enhancement of selection and management decisions at a national (within each country) and international (between countries) level. The results from this study show that the influence of the genetic make-up of an animal on carcass traits varies across age and that there is variation between breeding bulls in their growth profiles for carcass traits. This means that the progeny of some bulls develop at different rates compared to the progeny of other bulls, therefore they will be ready for slaughter at different ages. Knowledge of individual sire growth profiles for carcass traits could help farmers identify the most profitable time at which to slaughter the progeny of particular bulls, leading to a more efficient use of farm resources.

This study also showed how data collected for the purpose of genetic evaluations for carcass traits can yield useful information for consideration in farm management practices. The herd environment plays a significant role in carcass trait performance across ages at slaughter and years of slaughter and this information is a useful indicator of management practices across time.

In addition to the enhancement of within country evaluations, this study shows the potential benefits of an international evaluation for carcass traits. Access to international evaluations would allow Irish and UK beef farmers to make more informed decisions on the selection of breeding stock needed to increase genetic gain in carcass traits and subsequently increase herd profitability.

Date: 3pm Wednesday 3rd October 2018

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:**Understanding how different hydrographic processes drive long-term changes in plankton is crucial to unveil the marine ecosystem functioning and project the consequences of future climate scenarios. To address this question in the Cantabrian Sea (southern Bay of Biscay), we combined a monthly time-series (1993-2010) of physicochemical and biological data collected in situ and satellite observations. Zooplankton biomass increased during the study period, especially for larger size classes. These results contrast with the decrease in biomass and size expected under global warming, which is shown in the region by the rise in summer sea surface temperature, suggesting the operation of other processes. Indeed, winter mixing and coastal upwelling were key drivers of zooplankton dynamics in spring and autumn, respectively, when zooplankton inter-annual increases were stronger. In particular, winter-mixing control occurs through the spring phytoplankton bloom: deeper and later mixing in winter was followed by later, larger and more productive blooms. We found that winters with weaker mixing (that led to weaker spring blooms) were associated with warmer surface temperatures. Consequently, global warming may lead in the future to smaller and less productive spring blooms in the Bay of Biscay, reversing the observed positive trends in zooplankton.

Date: 1pm Wednesday 19th September 2018

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** The warming of high latitude marine environments presents a challenge to ecosystem models which aim to predict future change in primary production magnitude and phenology. A major difficulty in predicting these changes and their impact on higher trophic levels is that the strong seasonal variations in physiology and strategy of phytoplankton are poorly understood. Further, seasonality is usually not accounted for in ecosystem models. Here, a mechanistic explanation for observed seasonality in light response (the photoparameters µ_{0} and α) is proposed using a trade-off between photosynthetic efficiency and respiration costs. This seasonality is integrated into an NPZD ecosystem model applied to the eastern Bering Sea, which captures the timing and magnitude of the spring bloom in a cold year more accurately than without seasonality. The final aim of this project is a comparison of the model's ability to reproduce spring blooms in both timing and magnitude for both cold and warm years, using cruise data collected in cold years 2007-2009, and warm years 2014-2015.

Title: A pilot project of real world outcomes in patients with metastatic breast cancer treated with taxanes

Date: 1pm Wednesday 19th September 2018

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** Aim To compare the outcome of two taxane treatments (docetaxel and paclitaxel) in women with metastatic breast cancer.

Design Cohort study via linking existing health datasets (Chemocare, hospitalisation and cancer registry).

Cohort Patients with metastatic breast cancer in GGC who are treated with taxane regimen during Jan 2010 - Dec 2015 - all patients are followed from first treatment date until their death or the censor date March 2017.

Statistical methods Adjusted hazard ratio comparing the survival of two taxane treatments is derived using Cox proportional-hazards regression accounting for the baseline characteristics (eg. Age, social deprivation and comorbidity). Proportionality assumption for the adjusted model is tested using Schoenfeld residuals. Proportionality assumption is failed which means the hazard ratio for taxane treatments is time dependent in the adjusted model. We therefore truncate the data set into three time intervals and derive a step function for the hazard ratio of taxane treatment i.e. a different hazard ratio in each time interval.

Results Compared to docetaxel, patient with paclitaxel are 1.7 times (95% CI 1.03-2.87) more likely to die during first year after treatment, but among survivors, no significant difference in survival after one year.

Title: Zooplankton Diapause in a Warmer World: Modelling the Impact of 21st Century Climate Change on Calanus Finmarchicus

Date: 1pm Wednesday 30th March 2016

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** To avoid starving in winter, many zooplankton species spend over six months dormant in deep waters. The time animals can remain dormant will likely be reduced by global warming. We therefore modelled changes in potential dormancy duration in the key species Calanus finmarchicus under 21st century climate change. Climate change impacts varied markedly. Western Atlantic populations see large reductions in potential dormancy duration, but the Norwegian Sea experiences only marginal change. The reductions in the Western Atlantic will likely cause important changes to the populations of C. finmarchicus and species that prey on it.

**Title:** The Risk of Dengue for Non-Immune Foreign Visitors to the 2016 Summer Olympic Games in Rio de Janeiro, Brazil

**Date:** 1.00pm, Wednesday 25th May 2016

**Venue:** Livingstone Tower, 9th Floor, LT9.07

**Abstract:** Dengue is a viral infection caused by 4 dengue serotypes transmitted by mosquitoes that is an increasing problem in Brazil and other countries in the tropics and subtropics. As Brazil is the country with the highest number of dengue cases worldwide. Rio de Janeiro, the venue for the 2016 Olympic Games, has been of major importance for the epidemiology of dengue in Brazil. After the DENV 1–4 introductions in 1986, 1990, 2000 and 2011, respectively, the city has suffered explosive outbreaks. Properly quantifying the risk of dengue for foreign visitors to the Olympics is important. A mathematical model to calculate the risk of developing dengue for foreign tourists attending the Olympic Games in Rio de Janeiro in 2016 is proposed. A system of differential equation models the spread of dengue amongst the resident population and a stochastic approximation is used to assess the risk to tourists.

Title: Dancing in the Moonlight: Vertical Migration of Arctic Zooplankton during the Polar Night

Date: 1pm Wednesday 16th November 2016

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** This talk will focus on the results from my PhD, which I completed this year at the Scottish Association for Marine Science before starting here at Strathclyde. In recent years, evidence has been found of Diel Vertical Migration (DVM) in zooplankton during the
Polar Night in the Arctic Ocean. However, the drivers of this light mediated behaviour during an apparent lack of
illumination and food are poorly understood. A novel
dataset comprising 58 deployments of moored Acoustic Doppler Current Profilers is used in this study
to observe the vertical migratory behaviour of zooplankton on a pan-Arctic scale. Methods of circadian rhythm analysis are applied to detect synchronous activity. During the Polar Night, the moon is seen to control the vertical positioning of zooplankton, and a new type of migratory behaviour is described: Lunar Vertical Migration (LVM). This exists as LVM-day (24.8 hour periodicity) and LVM-month (29.5 day periodicity), and is observed throughout the Arctic Ocean. The results presented here show continuous activity throughout winter, and
challenge assumptions of a quiescent Polar Night.

Title: Including biology in spatial statistical models. Examples from vector-borne disease studies.

Date: 12.30pm, Thursday 1st June 2017

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:**

Vector borne diseases (e.g. Malaria, Dengue, Leishmaniasis) account for 20% of all infectious diseases, causing several million of infections and more than 1 million deaths annually. The majority of the vectors are insects (e.g. mosquitoes, midges and flies) and ticks, which biology and epidemiology are not often fully understood.

Biological and statistical models are used for mapping and modelling vector-borne diseases, however, rarely these methods are combined to produce maps and tools for disease surveillance and control (e.g. vector hot spots). In this talk I will present some techniques that can make data biologically meaningful; and the use of geo-bio-statistical models for tsetse flies (sleeping sickness) surveillance and control in Zambia. We show how mapping tsetse flies immigration, emigration, mortality and fertility can be the key element for successful disease eradication.

Title: Disease Mapping and Visualization using Data from Spatio-Temporally Referenced Prevalence Surveys

Date: 1pm Wednesday 3rd May 2017

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** We set out general principles and develop statistical tools for the analysis of data from spatio-temporally referenced prevalence surveys. Our objective is to provide a tutorial guide that can be used in order to identify parsimonious geostatistical models for prevalence mapping. A general variogram-based Monte Carlo procedure is proposed to check the validity of the modelling assumptions. We describe and contrast likelihood-based and Bayesian methods of inference, showing how to account for parameter uncertainty under each of the two paradigms. We also describe extensions of the standard model for disease prevalence that can be used when stationarity of the spatio-temporal covariance function is not supported by the data. We discuss how to define predictive targets and argue that exceedance probabilities provide one of the most effective ways to convey uncertainty in prevalence estimates. We describe statistical software for the visualization of spatio-temporal predictive summaries of prevalence through interactive animations. Finally, we illustrate an application to historical malaria prevalence data from 1334 surveys conducted in Senegal between 1905 and 2014.

Title: Why Primary Production Peaks at Surface During Summer in the Subtropica, Oligotrophic Open Ocean?

Date: 1pm Tuesday 20th March 2018

Venue: Livingstone Tower, 9th floor, room LT908

**Abstract:** Classic understanding on environmental controls on primary production (PP) in the subtropical, oligotrophic ocean gyre is that phytoplankton growth and PP are limited by the upward supply of inorganic nutrients delivered largely by diffusion. Since nutrient supply mostly comes from below and light attenuates from surface to the depth, phytoplankton growth rate and PP should peak at some intermediate depth, coinciding with the deep chlorophyll maximum (DCM) layer. However, examination on the PP data measured at three stations (ALOHA, S1, and BATS) in the subtropical North Pacific and Atlantic reveals that PP peaks within the surface mixed layer despite the negligible nutrient concentration at surface and a pronounced deep chlorophyll maximum (DCM) around 100 m. While the formation of DCM can be largely explained by phytoplankton photo-acclimation (adjustments of chlorophyll-to-carbon ratios), the surface peak of phytoplankton growth rate is difficult to explain. I evaluate several hypotheses that try to explain the surface peak of phytoplankton growth rate. The preliminary finding is that the coexistence of high- and low-light adapted ecotypes can best explain the observed patterns of nutrient, chlorophyll, and PP. This is highly consistent with many biodiversity–ecosystem functioning (BEF) studies that biodiversity can enhance productivity and nutrient utilization via niche complementarity. Thus, evaluation of the effects of climate change on ocean productivity is hard to be reliable without considering biodiversity.

### Stochastic Analysis

7th December 2018: Professor Wei Mao (Donghua University, China)**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: 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.

Title: Stochastic Lotka-Volterra Food Chains

Date: 3.30pm Wednesday 19th April 2017

**Abstract: **We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka-Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka-Volterra systems in the literature but few for their stochastic counterparts. The food chain we analyze consists of one prey and $n-1$ predators for $n\in\{2,3,4,\dots\}$. The $j$th predator eats the $j-1$th species and is eaten by the $j+1$th predator; this way each species only interacts with at most two other species - the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on an explicit quantity depending on the interaction coefficients of the system, which species go extinct and which converge to their unique invariant probability measure. Our work can be seen as a natural extension of the deterministic results of Gard and Hallam '79 to a stochastic setting. A novelty of our analysis is the fact that we can describe the behavior the system when the noise is degenerate. This is relevant because of the possibility of strong correlations between the effects of the environment on the different species. This is joint work with Dang H. Nguyen.

Title: Time-Varying Feedback and its Control Ability

Date: 3.00pm Friday 19th May 2017

**Abstract: **Comparison to pure feedback control, time-varying feedback control has distinct advantages, e.g., in handling system nonlinearities, counteracting system uncertainties and achieving prescribed performance. But due to the time-variations, time-varying feedback always keeps most people away, and its potential has been investigated far from enough. Here I shall illustrate some good and ability of time-varying feedback, and introduce some applications in SDEs, as well as several problems to be further investigated.

Title: Last Passage Percolation Models in a Bernoulli Environment

Date: 3.00pm Wednesday 14th June 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **We will discuss two different last passage percolation models in an i.i.d. Bernoulli random environment. In particular, I will show explicit laws of large numbers and order of fluctuations for the last passage time - the maximum number of Bernoulli points one can collect by following a sequence of admissible steps that ends in a predetermined lattice site. I will show how the behaviour of these models change depending on the set of admissible steps (e.g. the LLN changes, directions that belong in a "percolation cluster” change) and also show how the order of fluctuations change if the direction of the path endpoint changes. This is joint work with Janosch Ortmann and Federico Ciech (Univ. of Sussex).

Title: Quantized Feedback Control for Control Systems with Saturation Nonlinearity

Date: 3.30pm Friday 16th June 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **In control systems, every physical actuator or sensor is subject to saturation owing to its maximum and minimum limits. Common examples of such limits are the deflection limits in aircraft actuators, the voltage limits in electrical actuators. Saturation nonlinearities are also purposely introduced into engineering systems such as control systems and neural network systems. In addition, one of the most important research areas in control theory is quantized control. Quantized feedback is found in many engineering systems including mechanical systems and networked systems. Since communication that need to transmit the feedback information from

the sensor to the controller may become less reliable as the bandwidth is limited. Here, I shall investigate quantized feedback control problems for systems subject to saturation nonlinearity.

Title: Stability of Two Kinds of Stochastic Runge-Kutta Methods for Stochastic Differential Equations

Date: 3.30pm Wednesday 5th July 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **We present two kinds of explicit Runge–Kutta methods for solving stochastic differential equations by using the three–term recurrence relations of Chebyshev and Legendre polynomials. The almost sure stability and mean-square stability of the numerical solutions generated by the two kinds of methods are investigated respectively. Numerical examples are provided to confirm theoretical results.

Title: Bayes' Rule and the Law

Date: 3.00pm Thursday 24th August 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **Bayesian inference is an approach in mathematical statistics where the probability of a hypothesis is updated as more evidence and data become available. It has wide applications in many areas such as machine learning, evolutionary biology, medicine and even in the judicial system. This talk will explore how Bayesian inference can be used in a specific court case to assist jurors in the process of legal decision making, demonstrating the power of mathematics in the court room.

Title: MLMC for Value-At-Risk

Date: 4.00pm Tuesday 19th September 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **In this talk, I explore Monte Carlo methods to estimate the Value-At-Risk (VaR) of a portfolio, which is a measure of the risk of the portfolio in some short time horizon. It turns out that estimating VaR involves approximating a nested expectation where the outer expectation is taken with respect to stock values at the risk horizon and the inner expectation is taken with respect to the option index and stock values at some final time. Following (Giles, 2015), our approach is to use MLMC to approximate the outer expectation where deeper levels use more samples in the Monte Carlo estimate of the inner expectation. We look at various control variates to reduce the variance of such an estimate. We also explore using an adaptive strategy (Broadie et al, 2011) to determine the number of samples used in estimating the inner expectation. Finally, we discuss using unbiased MLMC (Rhee et al., 2015) when simulating stocks requires time discretization. Our results show that using MLMC to approximate a probability of large-loss with an error tolerance of order $\epsilon$, we are able to get an optimal complexity of order $\epsilon^{-2}(\log(\epsilon^{-1})^2$ that is independent of the number of options, for a large enough number of options.

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: On Uniqueness and Blowup Properties for a Class of Second Order SDEs

Date: 2.30pm Wednesday 18th October 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (X_t; Y_t) of the equations dX_t = Y_tdt, dY_t =|X_t|^\alpha dB_t, (X_0; Y_0) = (x_0; y_0). In particular, we prove that solutions are nonunique if 0 < \apha < 1 and (x_0; y_0) = (0; 0) and unique if 1=2 < \alpha and(x_0; y_0) \not= (0; 0). We also show that blowup in finite time holds if \alpha > 1 and (x_0; y_0) \not= (0; 0).