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Statistics, Probability, Analysis and Applied Mathematics

Below is an outline of the planned events we have for the forthcoming academic year! This page is updated on the fly, so do check back periodically for updates.

SPAAM Seminar Series

The Statistics, Probability, Analysis and Applied Mathematics (SPAAM) seminar series will take place virtually at 3pm on Thursdays during term time on the SPAAM Microsoft Teams channel. It will host a variety of talks from PhD students involved in applied mathematics research at Warwick (see the bottom of this page for the talk abstracts!).

Each seminar will usually host two speakers (unless otherwise stated) with each talk taking around 15-20 minutes with 5-10 minutes of questions afterwards. Speakers and committee members will hang around for some time after the talks for coffee and further questions. Please do contact one of the committee if you would like to join and be added to the MS Teams channel. Note that these talks may be recorded for later viewing on our Youtube channel so do join with audio and video off if you don't wish to feature!

If you would like to give a talk this term, please contact Diogo Caetano (Diogo.Caetano@warwick.ac.uk) or Haoran Ni (Haoran.Ni@warwick.ac.uk) and we will find you a slot!

Term 2
Date Talk 1 Talk 2
14th January 2020

Helicity for volume-preserving Anosov flows - Solly Coles (Maths)

Stochastic parareal: a novel application of probabilistic methods to time-parallelisation - Kamran Pentland (MathSys)

21th January 2020

Hierarchical structure in a condensed zero-range process - Simon Gabriel (Maths CDT) Infinite Horizon Stochastic Differential Utility - Joe Jerome (Statistics)
28th January 2020 Mathematically Modelling Metal Sheet Spinning - Hanson Bharth (MASDOC) Minimax rates in change point detection problems - Mengchu Li (Statistics)
4th February 2020 Mnerh Alqahtani (Maths) Bhavan Chahal (MathSys)
11th February 2020 Laura Guzmn Rincon (MathSys) Francesca Crucinio (Statistics)
18th February 2020 Dom Brockington (MASDOC) Matt Coates (MASDOC)
25th February 2020 Diogo Caetano (MASDOC)  
4th March 2020    
11th March 2020    
18th March 2020 Dimitris Lygkonis (MASDOC) Emanuele Zuccoli (Maths)
Abstracts

Week 1 (Talk 1) - Helicity for volume-preserving Anosov flows - Solly Coles (Maths)

The helicity of a volume-preserving flow measures the extent to which flow lines tangle around each other. Helicity is a useful invariant which helps to answer certain variational problems in magnetohydrodynamics. Due to Arnold, one can characterise helicity in terms of the linking numbers of knots constructed by closing up trajectories of the flow with geodesic arcs. In this talk we will describe Arnold's characterisation, followed by a new characterisation for the case of Anosov flows, in terms of the linking of periodic trajectories.

Week 1 (Talk 2) - Stochastic parareal: a novel application of probabilistic methods to time-parallelisation - Kamran Pentland (MathSys)

Most modern techniques used to numerically integrate partial differential equations rely on parallelising aspects of the spatial dimension and, whilst highly efficient, are reaching scale up limits. Further numerical speed-up is often limited by the sequential time stepping method used. Introducing parallelism into the temporal dimension is not an obvious route one chooses due to the inherent sequential nature of time (i.e. future solutions states depend upon previous states). Having only been developed over the last few decades, time-parallel methods enable us to do just this and are gaining popularity in a number of applications. In this talk we will introduce the parareal algorithm (an existing robust parallel-in-time numerical integrator) that solves initial value problems in parallel. It converges deterministically at a given rate (yielding parallel speed-up) which we aim to improve upon by incorporating probabilistic methods - thus we develop a stochastic parareal algorithm.

Week 2 (Talk 1) - Hierarchical structure in a condensed zero-range process - Simon Gabriel (Maths CDT)

Motivated by the physics of condensed matter, we study the limiting dynamics of stochastic particle systems on a microscopic and macroscopic scale. A particular simple but interesting toy model is the zero-range process. Most classical results show that the condensate is a single macroscopic cluster, however, more sophisticated formations are possible.

In this talk we will introduce the zero-range process, define condensation and discuss conditions to see a Poisson-Dirichlet distributed condensate in the thermodynamic limit.

Week 2 (Talk 2) - Infinite Horizon Stochastic Differential Utility - Joe Jerome (Statistics)

Stochastic differential utility has been widely studied since its formulation by Duffie and Epstein in 1992. It allows modelling of a much wider range of risk and intertemporal preferences and therefore provides a natural extension to the Merton problem for time-additive utility. However, whilst the finite time horizon problem is now fairly well understood, few have investigated the infinite horizon `lifetime' problem. In our paper we provide a novel formulation of the lifetime problem, highlighting and explaining the role of the transversality condition. We then discuss the parameters governing the agent's preferences, and show that certain parameter combinations considered in the literature are ill-posed over the infinite horizon.

We prove existence of a finite valued utility process for a large class of consumption streams and then show that, by considering a natural generalisation, we may assign a meaningful utility to any non-negative progressively measurable process. This means that, regardless of the choice of financial market, self-financing consumption streams are always evaluable.

Finally, we show existence and uniqueness of an optimal strategy in a Black-Merton-Scholes market.

Week 3 (Talk 1) - Mathematically Modelling Metal Sheet Spinning - Hanson Bharth (MASDOC)

Metal is a material which is used in abundance and the efficient use of it has been imperative to the advancement of civilization since the Bronze Age. New technology and engineering processes have been developed for metal spinning, but progress has been foiled by our understanding of the physics. Metal forming is an extremely energy-intensive process so it is vital that we iron this out. In this talk, I shall present an approximate model for metal sheet spinning which is then formulated into a matrix Wiener-Hopf problem. To alloy us to find a solution we borrow some tools from orthogonal polynomials to compute the Wiener-Hopf decompositions. Finally, I shall discuss a hidden free-boundary problem and some simulations to demonstrate.

Week 3 (Talk 2) - Minimax rates in change point detection problems - Mengchu Li (Statistics)

In this talk, I will introduce the change point detection problem in general and focus on the simple univariate mean change point problem. Some known minimax results in change point detection and localisation will be presented as well as the celebrated wild binary segmentation framework, which has been shown to be minimax optimal (up to log factor). The focus will then shift to the robust mean change point detection problem. Some preliminary results on the minimax rates of the problem will be presented, complemented by a simple heuristic algorithm.

Week 4 (Talk 1) - Mnerh Alqahtani (Maths)

TBC.

Week 4 (Talk 2) - Bhavan Chahal (MathSys)

TBC.

Week 5 (Talk 1) - Laura Guzmn Rincon (MathSys)

TBC.

Week 5 (Talk 2) - Francesca Crucinio (Statistics)

TBC.

Week 6 (Talk 1) - Dom Brockington (MASDOC)

TBC.

Week 6 (Talk 2) - Matt Coates (MASDOC)

TBC.

Week 7 (Talk 1) - Diogo Caetano (MASDOC)

TBC.

Week 7 (Talk 2) - TBC

TBC.

Week 8 (Talk 1) - TBC

TBC.

Week 8 (Talk 2) - TBC

TBC.

Week 9 (Talk 1) - TBC

TBC.

Week 9 (Talk 2) - TBC

TBC.

Week 10 (Talk 1) - Dimitris Lygkonis (MASDOC)

TBC.

Week 10 (Talk 2) - Emanuele Zuccoli (Maths)

TBC.

Term 1
Date Talk 1 Talk 2
15th October 2020

Dislocations and Grain Boundaries: A Short Account - Anvar Atayev (MASDOC)

Predicting Asymptotic Behaviour of Matched Solutions - Matthew King (MASDOC)
22nd October 2020 A phase field model for raft formation on biological membranes - Luke Hatcher (MASDOC) N/A
29th October 2020 Social event  
5th November 2020 Meta stability in atmospheric jets - Nayef Shkeir (MathSys) Point particle interactions on surface biomembranes - Philip Herbert (MASDOC)
12th November 2020 Recombination detection for viral genetic data - Ana Ignatieva (Statistics) Diffusion Limits at Small Times for Coalescent Processes with Mutation and Selection - Phil Hanson (MASDOC)
19th November 2020 A family of continuous-time dynamical systems with a trivial CLT - Nicolò Paviato (MASDOC) Kesten Processes and Wealth Generation - Samuel Forbes (MathSys)
26th November 2020 The fully nonconforming virtual element method for fourth order perturbation problems - Alice Hodson (MASDOC) Why are living systems cellular? Modelling reaction-diffusion systems in biochemical systems - Connah Johnson (MathSys)
3rd December 2020 Transaction tax in a general equilibrium model - Osian Shelley (MASDOC) Fixing Bias in Zipf Estimators using approximate Bayesian computation - Charlie Pilgrim (MathSys)
10th December 2020 Differentiating random and deterministic uncertainty in Reinforcement Learning - Jake Thomas (MathSys) Exact MCMC inference for Wright-Fisher diffusions - Jaro Sant (MASDOC)
Abstracts

Week 2 (Talk 1) - Dislocations and Grain Boundaries: A Short Account - Anvar Atayev (MASDOC)

First described by Vito Volterra in 1907 and formally discovered independently by Egon Orowan, Michael Polanyi and Geoffrey Taylor in 1934, dislocations, a type of crystal defect and their derivative structures, such as grain boundaries, have been studied extensively by the mathematical and engineering communities since their inception. In this seminar, we provide a short and non-technical overview of dislocations and grain boundaries, describe how they're formed, how they're modelled and why they are interesting to study. Furthermore, a short discussion on the problem of equilibrium configurations of small angle grain boundaries from a view of discrete dislocation dynamics will be presented.

Week 2 (Talk 2) - Predicting Asymptotic Behaviour of Matched Solutions - Matthew King (MASDOC)

When solving complex ODE's, a solution may be limited by a radius of convergence. In these instances, a second solution may be proposed that is valid across the remainder of the domain. A Full solution can then be offered by performing matching between these solutions. Under certain asymptotic limits it may not be the case that these solutions follow the same behaviour, and to be able to make predictions for the full solution in these limits we must understand the matching process. Motivated by an example from aeroacoustics this talk will look at when these complications may occur, why the matching may not behave as might be initially expected, and how correct predictions may still be made.

Week 3 - A phase field model for raft formation on biological membranes - Luke Hatcher (MASDOC)

In this talk we introduce and develop a model for phase separation on biological membranes. Motivated by observations of subdomains (rafts) on biological membranes which compartmentalise cellular processes we propose a model which couples the Helfrich energy to a Cahn-Hilliard energy. Using a perturbation method, we describe the geometry of the membrane as a graph over a sphere. The resulting energy is a small deformation functional that is coupled to the interface. 

We first explore the energy for a diffuse interface. We begin by focusing on the equilibria of the energy functional and use a gradient flow to numerically verify that the model predicts the formation of stable raft-like structures. We will discuss the parameter dependence of these subdomains. Subsequently, by calculating a Γ-limit, we relate the diffuse interface approach to a sharp interface approach.  

Finally, we use Onsager's variational principle to address the non-equilibrium dynamics of a membrane. We obtain a Cahn-Hilliard equation with degenerate mobility coupled to a small deformation equation. We will discuss the challenges in proving an existence result for this problem and how they can be overcome. Again, we will consider the free boundary problem for the corresponding sharp interface approach.

Week 4 - Social event

This week's talks have been postponed due to speaker availability. Instead we will be meeting for a coffee and a catch-up to discuss our plans for events this academic year and to ask you what sort of things you would be interested taking part or joining in with! Please do come along, all are welcome!

Week 5 (Talk 1) - Meta stability in atmospheric jets - Nayef Shkeir (MathSys)

Turbulence in atmospheres, oceans and plasma flows leads to coherent large-scale jets that persist for long-times. These jets may be steady or transition between several meta-stable jet configurations. The main question is: Under what conditions, and from what mechanisms, can the system switch, and with what probability, i.e. how likely is it that Jupiter looses one of its jets. In this talk, we present a study of the dynamics of these atmospheric jets on large rotating Jovian planets where we can apply the stochastically forced two-dimensional barotropic equation and its various approximations.

Week 5 (Talk 2) - Point particle interactions on surface biomembranes - Philip Herbert (MASDOC)

In this talk, we discuss a model for protein interactions on a near spherical biomembrane. This is motivated by the fact that proteins are responsible for many vital task and, at large distances, any interactions are believed to be predominantly membrane mediated. We begin by discussing a simplified model for the membrane and the proteins. This is followed by outlining a method for numerically approximating the membrane-protein system. We conclude with the differentiability of the energy of the membrane-protein system, which may be utilised for a gradient descent algorithm.

Week 6 (Talk 1) - Recombination detection for viral genetic data - Ana Ignatieva (Statistics)

The processes of genetic mutation and recombination are fundamental drivers of viral evolution. Mutation events produce small changes within the genome during replication, and are generally visible in sequencing data. Recombination, on the other hand, occurs when genetic material from two parent particles is mixed together before being passed on to the offspring -- with the potential to rapidly and drastically change its pathogenic properties. The detection of recombination events from a sample of genetic data is a very challenging problem. I will give a brief overview of the combinatorial, algorithmic and statistical aspects of recombination detection, and discuss recent work combining these approaches with an application to viral sequencing data.

Week 6 (Talk 2) - Diffusion Limits at Small Times for Coalescent Processes with Mutation and Selection - Phil Hanson (MASDOC)

We introduce several backwards in time models present in population genetics that model lines of ancestry when a population is subject to random mutation and natural selection. When considering the ancestry of an infinite population we find that the number of ancestors instantaneously becomes finite as soon as we look into the past. We consider how quickly this happens (and what "quickly" means in this context) and consider these processes close to zero, characterising their asymptotic mean and second order fluctuations.

Week 7 (Talk 1) - A family of continuous-time dynamical systems with a trivial CLT - Nicolò Paviato (MASDOC)

Great interest has been shown in proving limit laws, such as the central limit theorem and Donsker's invariance principle, for a large class of discrete and continuous-time systems. To study their rates of convergence we consider a martingale method introduced by Gordin, which in some cases leads to an unforeseen result. In this talk we will show some of the difficulties that arise in dealing with semiflows that have a contracting family of transfer operators. The work presented here was done in collaboration with Prof. Ian Melbourne.

Week 7 (Talk 2) - Kesten Processes and Wealth Generation - Samuel Forbes (MathSys)

Kesten processes are discrete stochastic multiplicative processes which have wide applicability in fields such as economics, social science and neuroscience. Results on the convergent case were studied rigorously in the 1970s by Harry Kesten (1931-2019). However it has only been fairly recently that the non-convergent case has been studied. I will introduce the theory of Kesten processes as well as show simulations and discuss possible applications to wealth generation.

Week 8 (Talk 1) - The fully nonconforming virtual element method for fourth order perturbation problems - Alice Hodson (MASDOC)

In recent years, the discretisation of partial differential equations via the virtual element method (VEM) has seen a rapid increase. The virtual element method was first introduced to solve second order elliptic problems and is a generalisation of both finite element and mimetic finite difference methods. VEM spaces can easily be constructed to enforce desirable properties of the discrete functions even on general polygonal meshes which makes the approach very interesting for a wide range of problems. In this talk, we present a class of nonconforming virtual element methods for a general fourth order PDE in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. We analyse these nonconforming VEMs and look at their application to fourth order perturbation problems. Finally, we showcase the flexibility of our method by demonstrating the ease with which our approach can handle nonlinear fourth order problems via some additional numerical experiments..

Week 8 (Talk 2) - Why are living systems cellular? Modelling reaction-diffusion systems in biochemical systems - Connah Johnson (MathSys)

From the most complex higher organisms to the simplest bacterium, cells are a basic structure of living systems. Cells act as a container for cell bound chemicals, utilise their own internal chemical reaction systems, and dynamically evolve through growth and division events. But what impact does the introduction of cell entities have on the local and global chemical concentration fields? To model the role cells play in shaping their environment we develop a hybrid discrete-continuous software suite. In the simulations cells are modelled as discrete agents coupled to a continuous parabolic PDE domain with a focus on implementing realistic chemical reactions. We show the local perturbation of chemical fields due to cell agents. These perturbations may lead to altered local chemistry and reaction specification in living systems, potential drivers of biological complexity.

Week 9 (Talk 1) - Transaction tax in a general equilibrium model - Osian Shelley (MASDOC)

In this talk, we consider the effects of a quadratic tax rate levied against two agents with heterogeneous risk aversions in a continuous-time, risk-sharing equilibrium model. The goal of each agent is to choose a trading strategy which maximises the expected changes in her wealth, for which an optimal strategy exists in closed form, as the solution to an FBSDE.

This tractable set-up allows us to analyse the utility loss incurred from taxation. In particular, we show why in some cases an agent can benefit from the taxation before redistribution. Moreover, when agents have heterogeneous beliefs about the traded asset, we discuss if taxation and redistribution can dampen speculative trading and benefit the agents, respectively.

No knowledge of stochastic calculus required.

Week 9 (Talk 2) - Fixing Bias in Zipf Estimators using approximate Bayesian computation - Charlie Pilgrim (MathSys)

Zipf’s law describes a relationship between the number of occurrences of a word in a book and that word's occurrence ranking. I will show how the most popular estimator to fit Zipf models is biased. I will then go on to describe a population Monte Carlo algorithm combined with approximate Bayesian computation that can fit Zipf models with much less bias.

Week 10 (Talk 1) - Differentiating random and deterministic uncertainty in Reinforcement Learning - Jake Thomas (MathSys)

It is common practice in statistics to model all uncertainty using the structure of probability theory and random phenomena. However, in many circumstances uncertainty is not inherently random, instead it is due to a lack of information about a deterministic quantity. In this talk I will describe how possibility theory can be used to give a more accurate picture of deterministic uncertainty and outline how this can be applied in the context of reinforcement learning.

Week 10 (Talk 2) - Exact MCMC inference for Wright-Fisher diffusions - Jaro Sant (MASDOC)

Inferring genetically relevant features such as selection, mutation and effective population size from population-wide data has been a perennial problem for geneticists. Most of the traditional methods used are based solely on present day genetic information which greatly impairs the inference as the data used is essentially a static snapshot of the population being considered. Recent advances in gene sequencing as well as improvements in the technologies required to retrieve DNA from old remains such as fossils (called ancient DNA (aDNA)), have allowed for the creation of genetic time series datasets spanning several centuries. Such datasets potentially hold a wealth of information with regards to how several genetic factors and phenomena have influenced and helped shape the population upon which they act. However, eliciting such information from the data requires the development of more intricate statistical procedures. To make matters worse, the transition density of the Wright-Fisher diffusion (which is somewhat the "standard" diffusion of choice to model changes in allele frequencies) is analytically unavailable, leading to an intractable likelihood. In this talk, we propose a Markov Chain Monte Carlo (MCMC) setup which takes into account both the temporal nature of the observations as well as the intractability of the likelihood, whilst allowing for an exact inferential treatment of the parameters of interest (i.e. without the need to resort to any approximations).