Skip to main content Skip to navigation

2021-22

Organiser: Susana Gomes

The Applied Maths Seminars are held on Fridays 12:00-13.00. This year the seminar will be hybrid: you can choose to attend in person in room B3.02 or on MS Teams. The team for the seminar is the same as last year, but if you are not a member, you can send a membership request via MS Teams or email the organiser.

Please contact Susana Gomes if you have any speaker suggestions for term 2.

Seminar Etiquette: Here is a set of basic rules for the seminar.

  • Please keep your microphone muted throughout the talk. If you want to ask a question, please raise your hand and the seminar organiser will (a) ask you to unmute if you are attending remotely or (b) get the speaker's attention and invite you to ask your question if you are in the room.
  • If you are in the room with us, the room microphones capture anything you say very easily, and this is worth keeping in mind ☺️
  • You can choose to keep your camera on or not. Colleagues in the room will be able to see the online audience.
  • Please let me know if you would like to attend any specific talk in person and/or have lunch with any of the speakers who are coming to visit us so that I can make sure you have a place in the room.

Term 1


Abstracts


Term 1

Week 1. Cicely Macnamara (Glasgow) - An agent-based model of the tumour microenvironment

The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Cancer cells can arise from any type of cell in the body; cancers can grow in or around any tissue or organ making the disease highly complex. My research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modelling. In this talk I shall present a 3D individual-based force-based model for tumour growth and development in which we simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent is fully realised, for example, cells are described as viscoelastic sphere with radius and centre given within the off-lattice model. Interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells, as well as intercellular aspects such as cell phenotypes.

Week 2. Nabil Fadai (Nottingham) - Semi-infinite travelling waves arising in moving-boundary reaction-diffusion equations

Travelling waves arise in a wide variety of biological applications, from the healing of wounds to the migration of populations. Such biological phenomena are often modelled mathematically via reaction-diffusion equations; however, the resulting travelling wave fronts often lack the key feature of a sharp ‘edge’. In this talk, we will examine how the incorporation of a moving boundary condition in reaction-diffusion models gives rise to a variety of sharp-fronted travelling waves for a range of wavespeeds. In particular, we will consider common reaction-diffusion models arising in biology and explore the key qualitative features of the resulting travelling wave fronts.

Week 3. Jemima Tabeart (Edinburgh) - Parallelisable preconditioners for saddle point weak-constraint 4D-Var

The saddle point formulation of weak-constraint 4D-Var offers the possibility of exploiting modern computer architectures. Developing good preconditioners which retain the highly-parallelisable nature of the saddle point system has been an area of recent research interest, especially for applications to numerical weather prediction. In this presentation I will present new proposals for preconditioners for the model and observation error covariance terms which explicitly incorporate model information and correlated observation error information respectively for the first time. I will present theoretical results comparing our new preconditioners to existing standard choices of preconditioners. Finally I will present two numerical experiments for the heat equation and Lorenz 96 and show that even when our theoretical assumptions are not completely satisfied, our new preconditioners lead to improvements in a number of settings.

Week 4. Jamie Foster (Portsmouth) - Mathematical modeling of brewing espresso

We give a brief introduction to current practices in cafe-style espresso extraction. Making the tastiest cup is considered an art rather than a science, however, processes can be made more systematic by better understanding the physical processes underlying brewing. We develop a physics-based mathematical model for the extraction process. Owing to the disparity in lengthscales between that of a grain within the "puck" and the depth of the whole "puck", the model can be systematically reduced via asymptotic homogenisation. The reduced model can then be solved by numerical methods. We will show that the model is able to reproduce some of the trends that are observed in experimental data and in practice. We also discuss possible strategies to make espresso more efficiently and more reproducibly.

Week 5. Louise Dyson (Warwick) - From ants to COVID-19: applications of mathematical modelling in biology and epidemiology

Like many applied mathematicians, I am interested in a variety of ways mathematical modelling may be used to understand the world. In this talk I will discuss two applications. Firstly investigating the phenomenon of noise-induced bistable states, inspired by the organisation of ant colonies, and extending this to consider how coupling together multiple bistable systems may induce synchronisation. Secondly describing my recent work during the COVID-19 pandemic analysing and modelling variants of concern.

Week 6. Upanshu Sharma (Berlin) - Variational structures beyond gradient flows

Gradient flows is an important subclass of evolution equations, whose solution dissipates an energy ‘as fast as possible’. This distinguishing feature endows these equations with a natural variational structure, which has received enormous attention over the last two decades. In recent years, it has become clear that if the gradient-flow equation originates from an underlying (reversible) stochastic particle system, then often the aforementioned variational structure is an exact decomposition of the large-deviation rate functional for the particle system. However, this decomposition and the corresponding gradient-flow structure breaks down if the underlying particles have additional non-dissipative effects (for instance in the case of non-reversible independent particles), even though the large-deviation rate functional is still available. Using the guiding example of independent non-reversible Markov jump particles, in this talk, I will discuss the various features of the rate functional and how it connects to relative entropy and Fisher information. Furthermore, I will show that if the underlying particle system is augmented with fluxes, then it is possible to derive gradient-flow-type structures in the non-reversible setting.

Week 7. Olga Mula (Paris Dauphine) - Optimal State and Parameter Estimation Algorithms and Applications to Biomedical Problems

In this talk, I will present an overview of recent works aiming at solving inverse problems (state and parameter estimation) by combining optimally measurement observations and parametrized PDE models. After defining a notion of optimal performance in terms of the smallest possible reconstruction error that any reconstruction algorithm can achieve, I will present practical numerical algorithms based on nonlinear reduced models for which we can prove that they can deliver a performance close to optimal. The proposed concepts may be viewed as exploring alternatives to Bayesian inversion in favor of more deterministic notions of accuracy quantification. I will illustrate the performance of the approach on simple benchmark examples and we will also discuss applications of the methodology to biomedical problems which are challenging due to shape variability.

Week 8. Clarice Poon (Bath) - Smooth bilevel programming for sparse regularization

Nonsmooth regularisers are widely used in machine learning for enforcing solution structures (such as the l1 norm for sparsity or the nuclear norm for low rank). State of the art solvers are typically first order methods or coordinate descent methods which handle nonsmoothness by careful smooth approximations and support pruning. In this work, we revisit the approach of iteratively reweighted least squares (IRLS) and show how a simple reparameterization coupled with a bilevel resolution leads to a smooth unconstrained problem. We are therefore able to exploit the machinery of smooth optimisation, such as BFGS, to obtain local superlinear convergence. The result is a highly versatile approach, handles both convex and non-convex regularisers, and different optimisation formulations (e.g. basis pursuit and lasso). We show that this is able to significantly outperform state of the art methods for a wide range of problems.

Week 9. Tom Montenegro-Johnson (Birmingham) - Soft, long, and thoroughly absorbent: unpublished works on looped active filaments and responsive hydrogels

Modern manufacturing techniques are now sufficiently advanced to create swimming microbots made of soft, programmable materials and catalytic fluid-driving components. This combination can endow microbots with a wealth of complex dynamical behaviours, arising from the multiphysical interactions of heterogenous materials, fluid domain, and swimmer geometry.

I will discuss 2 recent works in this field. In the first, we extend our recent slender phoretic theory to model and anbalyse chemically-active autophoretic slender loops, including a theme with variations on the torus, and a trefoil knot. In the second, we consider the swelling and shrinking dynamics of a thermoresponsive sphere of hydrogel. An effort will be made to relate the two in a wider context.

Week 10. Peter Baddoo (MIT) - Integrating physical laws into data-driven system identification

Incorporating partial knowledge of physical laws into data-driven architectures can improve the accuracy, generalisability, and robustness of the resulting models. In this work, we demonstrate how physical laws – such as symmetries, invariances, and conservation laws – can be integrated into the dynamic mode decomposition (DMD). DMD is a widely-used data analysis technique that extracts low-rank modal structures and dynamics from high-dimensional measurements. However, DMD frequently produces models that are sensitive to noise, fail to generalise outside the training data, and violate basic physical laws. Our physics-informed DMD (piDMD) optimisation restricts the family of admissible models to a matrix manifold that respects the physical structure of the system. We focus on five fundamental physical properties – conservative, self-adjoint, local, causal, and shift-invariant – and derive closed-form solutions and efficient algorithms for the corresponding piDMD optimisations. With fewer degrees of freedom, piDMD models are less prone to overfitting, require less training data, and are often less computationally expensive to build than standard DMD models. We demonstrate piDMD on a range of challenging problems in the physical sciences, including travelling-wave systems, the Schrödinger equation, solute advection-diffusion, and three-dimensional transitional channel flow. In each case, piDMD significantly outperforms standard DMD in metrics such as spectral identification, state prediction, and estimation of optimal forcings and responses. We also demonstrate how to model high-dimensional nonlinear structures via kernel regression.