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Applied Maths Seminar 2023-2024

Organisers: Ferran Brosa Planella and Thomasina Ball

The Applied Maths Seminars are held on Fridays 12:00-13.00. This year the seminar will be hybrid (at least for Term 1): 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 organisers.

Please contact Ferran Brosa Planella or Thomasina Ball if you have any speaker suggestions for future terms.

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 us know if you would like to meet 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


Term 1

Week 1. Francesca Arrigo (Strathclyde) - f(A)bulous networks: an overview of matrix functions in network science

Though seemingly they belong to two different worlds, matrix functions and network science have some degree of overlap thanks to a very simple fact; powers of the adjacency matrix count traversals in the underlying network. This concept in turn allows for the definition of centrality measures in terms of entries (or sums thereof) of functions of the adjacency matrix.

In this talk, after reviewing basic definitions, we will give an overview of popular walk-based centrality measures in networks, emphasizing the role of matrix functions and of expressions of the form f(A)b and c^T f (A)b. We will further discuss nonbacktracking walk-based centralities and describe challenges and open problems.

Week 2. Lyndon Koens (Hull) - Modelling Microscopic Swimmers

Microscopic swimmers exist in diverse and complex environments and use many different forms of propulsion and feeding. Biological microswimmers play key roles in many ecosystems, while artificial microswimmers could be used for targeted drug delivery, or the development of new keyhole surgeries. Modelling and understanding the behaviour of such swimmers is however a non-trivial task. In this talk I will provide a brief overview of the swimming problem before talking about some of my recent work on modelling the behaviour of microscopic swimmers.

Week 3. Oana Lang (Imperial) - Eulerian calibration for stochastic models

Stochastic partial differential equations are widely used to model the evolution of uncertain dynamical systems in geophysical fluid dynamics. For a judicious modelling of the evolution of a fluid flow, the noise term needs to be properly calibrated. Lagrangian methods have been developed in this sense, where particle trajectories are simulated starting from each point on both the physical grid and its refined version, then the differences between the particle positions are used to calibrate the noise. This is computationally expensive and not fully justified from a theoretical perspective. We propose an Eulerian alternative which is tested on a stochastic rotating shallow water system and it can be applied to a general class of stochastic models. This is joint work with Dan Crisan and Alex Lobbe and the results are being published in “Noise calibration for SPDEs: a case study for the rotating shallow water model” (Foundations of Data Science), arXiv:2305.03548.

Week 4. Thomas Hudson (Warwick) - Crystal evolution at the nanoscale: Elasticity, plasticity and dislocations

Dislocation motion is a key feature of crystal plasticity at the smallest scales, and many mathematical challenges must be overcome to establish accurate, well-posed theories which connect our knowledge about dislocation motion at the nanoscale with the predictive power of macroscopic models.

I will present a range of recent work with my Warwick-based collaborators on the formulation and fitting of a range of models at the interface between atomistic and continuum theories. The first of these works focuses on the formulation of a new space-time geometric model for dislocation motion with Filip Rindler. If time permits, I also hope to discuss ongoing work on the data-driven fitting and simulation of reduced models for dislocation motion, starting from both atomistic and continuum theories; the latter projects are work with Geraldine Anis, Peter Brommer, Joseph Duque Lopez and James Kermode.

Week 5. Edwina Yeo (UCL) - Crystal evolution at the nanoscale: Elasticity, plasticity and dislocations

A wide range of biological and therapeutic species change their behaviour, geometry or propensity to adhere in response to external stimuli. For instance, bacteria modify their swimming behaviour close to fluid boundaries and proteins unfold differently depending on local flow structure. Predicting the behaviour of these species in macroscale fluid flows is vital to mitigate and understand effects such as aggregation or surface adhesion.

In this talk, I will present two examples of incorporating discrete microscale behaviour into continuum models via mean-field modelling. Firstly, I include microscale protein unfolding to predict blood clot initiation and secondly, I include microscale magnetic interactions to predict the macroscale aggregation and transport of magnetic nanoparticles. I will discuss the benefits of upscaling for model parameterisation and prediction as well as the limitations of the resulting macroscale models.

Week 6. Luke Davis (UCL) - Active matter under control: Insights from response theory


Active constituents burn fuel to sustain individual motion, giving rise to collective effects that are not seen in systems at thermal equilibrium, such as phase separation with purely repulsive interactions. There is a great potential in harnessing the striking phenomenology of active matter to build novel controllable and responsive materials that surpass passive ones. Yet, we currently lack a systematic roadmap to predict the protocols driving active systems between different states in a way that is thermodynamically optimal. Equilibrium thermodynamics is an inadequate foundation to this end, due to the dissipation rate arising from the constant fuel consumption in active matter. In this talk I will walk you through the key derivation steps and implementation of our versatile framework for the thermodynamic control of active matter. Combining recent developments in stochastic thermodynamics and nonequilibrium response theory, our approach shows how to find the optimal control for either continuous- or discrete-state active systems operating arbitrarily far from equilibrium. Our results open the door to designing novel active materials which are not only built to stabilize specific nonequilibrium collective states, but are also optimized to switch between different states at minimum dissipation.

Week 7. Boris Shraiman (UCSB) - Physics of Morphogenesis

Morphogenesis is a developmental process through which plants and animals acquire their shape and form. Although Biology has identified many of the key genes and cellular mechanisms of morphogenesis, the question of how Living Matter encodes the geometry of the shapes that it generates remains an open problem. This talk will focus on the interplay of physical forces and genetically encoded regulation that underly morphogenic processes. Specifically, the talk will describe how mechanical self-organization on cellular scale acts to convert spatial patterns of developmental gene expression into controlled transformation of tissue shape. We will see that i) simple ideas from Physics go far in explaining non-trivial behavior of tissues, and that ii) “active mechanics” encountered in tissue morphogenesis pushes the envelope of continuum mechanics beyond what we have learned from textbooks.

Week 8. Michael Faulkner (Warwick) - Fast sampling at phase transitions in statistical physics
Sampling algorithms are commonplace in statistics and machine learning – in particular, in Bayesian computation – and have been used for decades to enable inference, prediction and model comparison in many different settings. They are also widely used in statistical physics, where many popular sampling algorithms first originated [1, 2]. At a high level, the goals within each discipline are the same – to sample from and approximate statistical expectations with respect to some probability distribution – but the motivations, nomenclature and methods of explanation differ significantly. This has led to challenges in communicating between the fields, and indeed the fundamental goals of one field are often misunderstood in the other. In this talk, we elucidate statistical physics for the statistician, emphasising that probability models are studied as functions of thermodynamic hyperparameters such as the temperature. This is particularly useful for characterising phase transitions, ie, boundaries in thermodynamic hyperparameter space between distinct thermodynamic phases.
We then move on to sampling algorithms, with a particular focus on the behaviour of the Metropolis algorithm [1] when simulating the 2D Ising and 2DXY models of magnetism. Metropolis dynamics are metastable in the low-temperature phase of each model, mixing between states of equal probability density on a timescale the diverges with system size (proportional to the dimensionality of parameter space). Moreover, the Metropolis algorithm also suffers from the closely related phenomenon of critical slowing down at phase transitions. These strongly correlated dynamics are characterised by asymptotically long integrated autocorrelation times, due to a flattening of the target density that essentially results from the system trying to exist simultaneously in both thermodynamic phases. Indeed, these key aspects of statistical physics have led to innovations in sampling algorithms that inform the Bayesian world. In particular, we present the Swendsen—Wang [3], Wolff [4] and event-chain Monte Carlo [5-7] algorithms. The first two simulate the 2D Ising model and were developed in response to the metastability and critical slowing down of the Metropolis algorithm. They circumvent both phenomena to mix with low autocorrelation and independent of system size. We then show that event-chain Monte Carlo similarly circumvents the low-temperature Metropolis metastability of the 2DXY model [7] and discuss its potential utility in bypassing an hypothesised critical slowing down at the phase transition. This talk is based on a recent review paper on the subject [8].
[1] Metropolis et al., J. Chem. Phys. 21 1087 (1953)
[2] Alder & Wainwright, J. Chem. Phys. 27 1208 (1957)
[3] Swendsen & Wang, Phys. Rev. Lett. 58 86 (1987)
[4] Wolff, Phys. Rev. Lett. 62 361 (1989)
[5] Bernard, Krauth & Wilson, Phys. Rev. E 80 056704 (2009)
[6] Michel, Mayer & Krauth, EPL (Europhys. Lett.) 112 20003 (2015)
[7] Faulkner, arXiv:2209.03699 (2022)
[8] Faulkner & Livingstone, arXiv:2209.03699 (2022)
Week 9. Matteo Icardi (Nottingham) - Homogenisation and model reduction for transport in porous media

We propose a new model reduction method to reduce the complexity of multiscale scalar transport problems with a dominant axial dynamic. Our approach combines the Hierarchical Model (HiMod) reduction and a two-scale asymptotic homogenisation technique. We extend the two-scale asymptotic expansion to any desired order and obtain a differential recursive formula for the high-order correctors. These are then used as a modal basis for the fast variable, which approximates the transverse dynamics of the flow. We use finite element discretisation to model the leading stream. We name this method HiPhomε (High-order Projection-based Homogenisation). We will present examples on both steady and unsteady advection-diffusion-reaction scenarios. The numerical results demonstrate that HiPhomε outperforms standard homogenised models and the classical HiMod with spectral modes, in terms of accuracy and convergence rate. The method proposed extends the reliability of standard homogenised solutions to transient and pre-asymptotic regimes and has applications in various fields, such as hydraulics and haemodynamics. A reformulation of the method to derive closed high-order homogenised models for general porous media applications will also be discussed.