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Summer Conference 2021

Description

Welcome to the homepage for the Warwick SIAM-IMA Applied Mathematics Conference 2021! We are hosting this event online using Microsoft Teams on the 8th-9th July 2021.

The conference aims at providing a relaxed environment for graduate students to showcase their research and will cover four areas of applied mathematics:

  • Climate Modelling
  • Mathematical Biology
  • Data Science/Machine Learning
  • Optimal Transport

The event will consist of four sessions, one for each of the above subject areas and will feature a senior academic as the keynote speaker followed by up to three shorter talks from current PhD students around the UK.

The conference is open to to any student or academic with a keen interest in applied mathematics and if you would like to register or give a 20 minute talk (abstract submission deadline: 28th June) in one of the above subject areas, please fill out the attached google form: https://forms.gle/eEMWQu3vQe2AEx5y5

At the end of the first day of the conference (8th July), there will be a virtual social event with a pub quiz! If you would like to attend the social, please mention this on the registration form, or email anvar.atayev@warwick.ac.uk if you forget!

Preliminary Schedule

Time Day One (Thurs 8th July) Day Two (Fri 9th July)
  Climate Modelling Data Science/Machine Learning
10:00-11:00am Keynote: Prof. Chris Budd Keynote: Dr. Francois-Xavier Briol
11:00-11:30am Nayef Shkeir George Wynne
11:30-12:00pm Thomas Gregory Takuo Matsubara
12:00-12:30pm Lunch Francesca Romana Crucinio
12:30-13:30pm Lunch Lunch
  Mathematical Biology Optimal Transport
13:30-14:00pm Anastasia Ignatieva Lunch
14:00-14:30pm Emine Atici Endes Lunch
14:30-15:00pm Connah Johnson Charlie Egan
15:00-15:30pm Coffee break Coffee break
15:30-16:30pm Keynote: Dr. Kit Yates Keynote: Dr. Susana Gomes
17:00pm Social event!  
Abstracts

Climate Modelling

Keynote: Prof. Chris Budd OBE (University of Bath) - Mathematical models for the ice ages

The ice ages are significant changes to the Earth's climate. For the last half a million years they have demonstrated a strong regularity, with large periodic changes in temperature with a 100 kilo year cycle. Before then the climate showed smaller smaller periodic changes, with a 40 kilo year cycle. Although there are many theories for this behaviour, there is as yet no fully convincing explanation for them. In this talk I will describe a number of different explanations for this phenomenon using the theory of dynamical systems. In particular I will apply recent methods from the theory of non-smooth dynamical systems to gain some insight into this complex phenomenon and other climate behaviour. In particular I will ask the question of whether the 'min-Pleistocene transition (MPT)' half a million years ago was an example of a grazing bifurcation.

Joint work with Kgomotso Susan Morupisi, Botswana University of Technology.

PhD Speaker: Nayef Shkeir - Rare events and turbulent atmospheres

Fluid in its turbulent state is one of the most complex interacting systems in physics, with an incredibly large number of strongly and nonlinearly interacting degrees of freedom. This becomes different when scale separation introduces coherent, long-lived structures into the fluid flow, which induces order and stability. I will start with an introduction into rare events, and I will present two very different scenarios of that type: (1) atmospheric jets on gas giants sustained by the turbulent background fluctuations, (2) metastability of Earth's climate on very long timescales. In these systems, the coherent structures might exist for long times in a multitude of metastable configurations, and turbulent fluctuations drive transitions between them.

PhD Speaker: Thomas Gregory - Next generation numerics for ocean modelling

As supercomputers are built with more and more cores, the limitations of traditional finite difference schemes in numerical modelling become more apparent. One issue that arises in moving to a more flexible finite element regime for ocean numerics is that Poisson's equation, which must be solved in the time stepping of such a model, is ill-conditioned on thin domains. We present a hybridized multigrid preconditioner for solving this.

Mathematical Biology

Keynote: Title - Dr. Kit Yates (University of Bath) - Adventures in Developmental Biology

In this talk I will give a number of short vignettes of work that has been undertaken in my group over the last 15 years. Mathematically, the theme that underlies our work is the importance of randomness to biological systems. I will explore a number of systems for which randomness plays a critical role. Models of these systems which ignore this important feature do a poor job of replicating the known biology, which in turn limits their predictive power. The underlying biological theme of the majority our work is development, but the tools and techniques we have built can be applied to multiple biological systems and indeed further afield. Topics will be drawn from, locust migration, zebrafish pigment pattern formation, mammalian cell migratory defects, appropriate cell cycle modelling and more. I won't delve to deeply into anyone area, but am happy to take question or to expand upon of the areas I touch on.

PhD Speaker: Anastasia Ignatieva - Recombination Dection for SARS-CoV-2

The processes of genetic mutation and recombination are fundamental drivers of viral evolution. Recombination occurs when host cells are co-infected with different strains of the same virus, and during replication the genomes are reshuffled and combined before being packaged and released as new offspring virions, now potentially possessing very different pathogenic properties. This makes the presence of recombination a crucial factor to consider when developing vaccines and treatments, but the extent of ongoing recombination within SARS-CoV-2 has remained unclear. While the effects of mutation are generally visible in sequencing data, detecting the presence of ongoing recombination is a very challenging problem. I will describe a principled and powerful statistical approach for detecting past recombination events from viral sequencing data, and present evidence for ongoing recombination in SARS-CoV-2 through the analysis of publicly available sequencing data.

PhD Speaker: Emine Atici Endes - Keratinocyte Growth Factor Based Mathematical Models of Epidermal Wound Healing

The mammalian skin is the largest organ of the body after the skeletal system and its primary function is to protect the body against the external environment. The maintenance of the skin's functionality, integrity, and strength are coordinated by specialised cells localised in three intricate layers of the skin: epidermis, dermis, and hypodermis. Understanding the fundamental cause of changes in the cells in these three skin compartments, thus provides a base for progress in interpreting injuries in the skin and its healing process.Wound healing is a normal biological and dynamic process in the human skin in which many types of cells, various cytokines, and growth factors, such as KGF, act in harmony. This process starts immediately after skin injury and is completed into three distinct but highly integrated phases: an inflammatory stage, a proliferative and migration (re-epithelialization) phase, and a long reformation and remodeling phase. Therefore, in this talk, I will give a sequence of PDE models presenting the interaction between the dermis and the epidermis, in which KGF plays an active role during the re-epithelialization.

PhD Speaker: Connah Johnson - Modelling environmental-metabolic feedback in spatially distributed bio-films

Biofilms are ubiquitous in medical settings. They can contain multiple distinct bacterial strains which complicate the task of tackling infections. Additionally, excretion of protective enzymes by bacteria within biofilms can inhibit the effects of anti-bacterials, providing regions wherein resistant strains may proliferate. It has been shown that within biofilms cross feeding between different cell types or species can support strains who would otherwise starve under substrate removal. These findings show that building a better understanding of biofilms and the dynamics within them will pay dividends in understanding bacterial infections. We seek to understand biofilm systems through mathematical modelling using our hybrid modelling platform ChemChaste. ChemChaste has been developed with the aim of modelling realistic chemical dynamics and the chemical interactions between cells via their microenvironment. Here, biofilms are modelled through coupling multiple reaction-diffusion systems to a population of individual cell agents. The cells each have their own metabolic models encoding different cell types. They can interact through the excretion and uptake of chemicals in the shared film environment. The spatial distribution of these cells and their behaviours is investigated under a range of metabolic processes and phenomena. Therein providing insights into the complex dynamics that may suggest clinical applications.

Data Science/Machine Learning

Keynote: Title - Dr. Francois-Xavier Briol (University College London) - Robust inference for expensive models with intractable likelihoods

Modern statistics and machine learning tools are being applied to increasingly complex phenomenon, and as a result make use of increasingly complex models. A large class of such models are the so-called intractable likelihood models, where the likelihood is either too computational expensive to evaluate, or impossible to write down in closed form. This creates significant issues for classical approach such as maximum likelihood estimation or Bayesian inference, which are entirely reliant on evaluations of a likelihood. In this talk, we will cover several novel inference schemes which by-pass this issue. These will be constructed from kernel-based discrepancies such as maximum mean discrepancies and kernel Stein discrepancies, and can be used either in a frequentist or Bayesian framework. An important feature of our approach is that it will be provably robust, in the sense that a small number of outliers or mild model misspecification will not have a significant impact on parameter estimation. In particular, we will show how the choice of kernel can allow us to trade statistical efficiency with robustness. The methodology will then be illustrated on a range of intractable likelihood models in signal processing and biochemistry.

PhD Speaker: George Wynne - Kernel Mean Embeddings for Functional Data Analysis

Kernel mean embeddings (KMEs) have enjoyed wide success in statistical machine learning over the past fifteen years. They offer a non-parametric method of reasoning with probability measures by mapping measures into a reproducing kernel Hilbert space (RKHS). The RKHS facilitates easy to compute, closed form expressions which makes the methodology practical and amenable to statistical analysis. Much of the existing theory and practice has revolved around Euclidean data whereas functional data has received very little investigation. Likewise, in functional data analysis (FDA) the technique of KMEs has not been explored. I will describe work which aims to bridge this gap by defining kernels which take functional inputs. In this context, KMEs offer an alternative paradigm to the common practice of projecting data to finite dimensions and employing classical finite dimensional statistical procedures. Indeed, the KME framework can handle infinite dimensional input spaces, offers an elegant theory and leverages the spectral structure of functional data. Applications include two-sample testing, goodness-of-fit testing and statistical depth.

PhD Speaker: Takuo Matsubara - Robust Generalised Bayesian Inference for Intractable Likelihoods

Generalised Bayesian inference updates prior beliefs using a loss function, rather than a likelihood, and can therefore be used to confer robustness against possible mis-specification of the likelihood. Here we consider generalised Bayesian inference with a Stein discrepancy as a loss function, motivated by applications in which the likelihood contains an intractable normalisation constant. In this context, the Stein discrepancy circumvents evaluation of the normalisation constant and produces generalised posteriors that are either closed form or accessible using standard Markov chain Monte Carlo. On a theoretical level, we show consistency, asymptotic normality, and bias-robustness of the generalised posterior, highlighting how these properties are impacted by the choice of Stein discrepancy. Then, we provide numerical experiments on a range of intractable distributions, including applications to kernel-based exponential family models and non-Gaussian graphical models.

PhD Speaker: Francesca Romana Crucinio - Product form estimators

We introduce a class of Monte Carlo estimators for product-form target distributions that aim to overcome the rapid growth of variance with dimension often observed for standard estimators. We identify them with a class of generalized U-Statistics, and thus establish their unbiasedness, consistency, and asymptotic normality. Moreover, we show that they achieve lower variances than their conventional counterparts given the same number of samples drawn from the target, investigate the gap in variance via several examples, and identify the situations in which the difference is most, and least, pronounced. We further study the estimators' computational cost and delineate the settings in which they are most efficient. We illustrate their utility beyond the setting of product-form distributions by detailing two simple extensions (one to targets that are mixtures of product-form distributions and another to targets that are absolutely continuous with respect to product-form distributions) and conclude by discussing further possible uses. Joint work with Juan Kuntz Nussio and Adam Johansen.

Optimal Transport

Keynote: Title - Dr. Susana Gomes (University of Warwick) - From linear to optimal control of falling liquid films using hierarchical models

The flow of a thin film down an inclined plane is a canonical setup in fluid mechanics, with technological applications such as manufacturing LCD screens or microchips. Mathematically, it provides a very rich framework for modelling, analysis, and control. In this talk, I will summarise a hierarchy of models for the interface of these flows, obtained from the Navier-Stokes equations using asymptotic analysis techniques. Depending on the application, we would like to robustly and efficiently manipulate these flows so that their interface has a prescribed shape (e.g. a flat interface for a smooth coating of a screen or a wavy shape to guarantee efficient cooling of microchips). I will summarise recent results using a linear feedback control methodology, showing how to stabilise any desired interfacial shape, and will present recent work examining the use of electric fields as a control mechanism, where the hierarchy of models is explored in the context of optimal control.

PhD Speaker: Charlie Egan - Optimal Transport and Non-optimal Weather

First introduced by Eady in 1949, the Eady slice equations model the formation and evolution of weather fronts. The strong discontinuities in the temperature and velocity profiles associated to weather fronts make these equations challenging to solve numerically. In this talk I will describe Cullen and Purser’s ‘geometric method’ that was developed to overcome this issue, highlighting its relation to optimal transport, and I will discuss results from our recent implementation of this numerical method, which uses state-of-the-art techniques from semi-discrete optimal transport. This is joint work with David Bourne (Heriot-Watt), Colin Cotter (Imperial), Mike Cullen (Met Office), Beatrice Pelloni (Heriot-Watt), Steve Roper (University of Glasgow) and Mark Wilkinson (Nottingham Trent).