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Mathematics Institute

Applied Maths Seminar 2024-2025

Organisers: Clarice Poon and Ellen Luckins

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 Clarice Poon or Ellen Luckins 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 2

Term 1

Abstracts

Term 1

Week 1.
(a) Daniel Booth (Warwick) -- A Hele-Shaw Newton's Cradle

In this talk, I study the motion of bubbles in a Hele-Shaw cell under a uniform background flow. I focus on a distinguished limit in which the bubble is approximately circular in plan view. Each bubble’s velocity is determined by a net force balance incorporating the Hele-Shaw viscous pressure and drag due to the thin films separating the bubble from the cell walls. The qualitative behaviour of the system is found to depend on a dimensionless parameter $\delta \propto \mathrm{Ca}^{1/3}R/h$, where $\mathrm{Ca}$ is the capillary number, $R$ is the bubble radius and $h$ is the cell height. First, it is found that an isolated bubble travels faster than the external fluid if $\delta>1$ or slower if $\delta<1$, and the theoretical dependence of the bubble velocity on $\delta$ is found to agree well with experimental observations. Then, in a system of two bubbles of different radii on different streamlines of the background flow, it is found that the larger bubble can overtake the smaller one, and they avoid contact by rotating around each other while passing. Finally, in a train of three identical bubbles travelling along the centre line, the middle bubble either catches up with the one in front (if $\delta >1$) or is caught by the one behind (if $\delta <1$), forming what we term a Hele-Shaw Newton's cradle.

(b) Ellen Jolley (Warwick) -- Modelling fluid-particle interactions for aircraft icing applications

As an aircraft flies through cloud at temperatures below freezing, it encounters ice particles and supercooled droplets, which results in the accretion of ice onto its surfaces and hence deformation of its aerodynamic shape. This can, in worst cases, cause series accidents. Mathematical modelling can inform new predictive codes and improved safety tests. In this talk I will introduce a model for the motion of individual ice particles near a surface in a high Reynolds number flow, revealing an array of different possible motions including collision with the wall, departing far from the wall and some others in between. I will also discuss a model for a particle interacting with a surface coated in water, as is common in icing conditions, enabling the particle to ‘skim’ along the water surface.

Week 2. Tim LaRock (Oxford) -- Encapsulation Structure and Dynamics in Empirical Hypergraphs

Hypergraphs are a powerful modelling framework to represent complex systems where interactions may involve an arbitrary number of nodes, rather than pairs of nodes as in traditional network representations. In this talk we will explore the extent to which smaller hyperedges are subsets of larger hyperedges in real-world and synthetic hypergraphs, a property that we call encapsulation. Building on the concept of line graphs, we develop measures to quantify the relations existing between hyperedges of different sizes and, as a byproduct, the compatibility of the data with a simplicial complex representation–whose encapsulation would be maximum. We then turn to the impact of the observed structural patterns on diffusive dynamics, focusing on a variant of threshold models that we call encapsulation dynamics, and demonstrate that non-random patterns can accelerate the spreading in the system.

Week 3. Karen Meyer (Dundee) -- Persistence in Solar Physics

Persistence, or long memory, is of longstanding interest in solar physics, having first been identified in time series of sunspot numbers in the seminal paper by Mandelbrot and Wallis (1969): “Some long‐run properties of geophysical records”. They used a method called Rescaled Range Analysis (R/S) to determine a Hurst exponent, H=0.93, which is indicative of strong persistence. It has since been suggested that for sunspot numbers, and indeed most times series of solar quantities, R/S is not an appropriate method for estimating persistence due to the non-stationary nature of the time series. Detrended fluctuation analysis (DFA) has been proposed as a more suitable method for estimating persistence, and has since been widely used in the analysis of solar and geo-physical time series. However, DFA is known to introduce uncontrolled bias and is in fact inappropriate for non-stationary processes (Bryce & Sprague, 2012).

Here, we assume an alternative class of long-memory models, more commonly found in statistics and econometrics: fractionally integrated processes. We revisit solar physics time series such as sunspot number and total solar irradiance with more robust estimators, and identify higher persistence than previous studies, as well as persistence over timescales significantly shorter than previously identified.

We also consider persistence in time series of quantities derived from solar physics simulations, demonstrating that these simulations capture the memory structure that is present in the observational input data. Further, we provide an algorithm for the quantitative assessment of simulation burn-in: the time after which a quantity has evolved away from its arbitrary initial condition to a physically more realistic state.

Week 4. David Bourne (Heriot Watt) -- Optimal transport theory and the compressible semi-geostrophic equations

The semi-geostrophic equations are a simplified model of large-scale atmospheric flows and frontogenesis. In this talk I will discuss existence and numerical approximation of weak solutions of the semi-geostrophic equations for a compressible fluid. This is joint work with Charlie Egan (Göttingen), Théo Lavier and Beatrice Pelloni (Heriot-Watt), and Quentin Mérigot (Université Paris-Saclay).

Week 5. Ran Holtzman (Coventry) -- Nonequilibrium flow in disordered media: Memory, hysteresis, and energy dissipation

Fluid-fluid displacements in disordered porous media is ubiquitous in a wide range of applications across scales, from fuel cells to subsurface water and energy resources. Common to many of these systems is their out-of-equilibrium macroscopic behaviour, including the emergence of instabilities, preferential pathways, and path- and rate-dependency, as a result of coupled mechanisms at much finer scales. After introducing my approach to study such systems, I will expand on a fundamental interdisciplinary problem: hysteresis and energy dissipation in disordered media.

I will present an ab-initio model of quasistatic fluid-fluid displacement in an imperfect Hele-Shaw cell, with random gap spacing caused by "defects". In contrast with existing (phenomenological) approaches, all our model parameters have a clear, identifiable physical meaning. We establish a quantitative link between the microscopic capillary physics, spatially-extended collective events (avalanches), and large-scale hysteresis in terms of capillary pressure-saturation (PS) in drainage and imbibition. We show that this dissipation is due to abrupt changes in the interface configuration (Haines jumps), and deduce the relative importance of viscous dissipation from comparison with experiments. We distinguish between “weak” (reversible interface displacement, exhibiting no hysteresis and dissipation) and “strong” (irreversible) defects. Remarkably, we show that cooperative effects mediated by interfacial tension lead to the emergence of irreversible, dissipative jumps among entities (defects), which are by themselves non-dissipative (“weak”). We establish a critical separation distance, analytically and numerically, verified by a proof-of-concept experiment. This nonintuitive finding questions the validity of the widely-used compartment models which rely on the existence of noninteracting hysteretic units.

Week 6. Dante Kalise (Imperial) -- Beyond Mean Field: Advanced Control and Optimization for Large-Scale Interacting Particle Systems
This talk is about novel directions on optimization and control of large-scale agent-based models beyond the paradigm of mean field control and games. We will discuss some novel directions such as an optimal control formulation of global optimization problems and the use of consensus-based optimization methods, and the optimal stabilization of McKean-Vlasov PDEs. In each case, we will see that a fundamental building block is the solution of nonlinear transport or Hamilton-Jacobi-Bellman type PDE. We will discuss the construction of numerical schemes based on polynomial approximation, spectral methods and deflation operators.
Week 7. Tatiana Bubba (Ferrara) -- Regularisation of tomographic inverse problems through multiresolution systems
Tomographic imaging allows to reconstruct images of hidden structure in an object by taking thereof projections: it finds applications in healthcare (medical imaging) and industry (production quality control), just to name a few. Like every inverse problem, tomography is ill-posed and very challenging to solve. In general, measurements are scarce and noisy, yielding an unstable problem which calls for accurate modelling and for complementing the insufficient data with some prior information which may be available on the solution. Traditionally this has been answered through regularisation theory, with sparsity promoting regularisation becoming dominant in the last decades.
In this talk, I will focus on some applications of limited data tomography where classical regularisation strategies can be coupled with ideas coming from multi resolution systems (such as wavelet and shearlets) and data-driven techniques. The common denominator will be the interplay between sparse regularization theory, harmonic analysis, microlocal analysis and machine learning: this allows to derive theoretical guarantees for the different case studies. The approaches proposed are tested on both simulated and measured data, showcasing the advantages of this strategy in practice.
Week 8. Kirsty Wan (Exeter) -- Multiciliary coordination across scales
Cilia are hair-like protrusions found on cells that facilitate various physiological flows, whether external (outside the organism) or internal (such as feeding or mucociliary clearance). When multiple cilia are in close proximity, they interact, leading many types of local and global coordination patterns. These interactions, which can occur through fluid or via elastic/cytoskeletal linkages, are often complex and system-dependent. This talk will explore different strategies of ciliary coordination and propulsion across diverse organisms, from single-celled protists to marine invertebrate larvae. We'll discuss how cilia can move in synchrony, maintain specific synchronization patterns, or beat metachronously on topologically interesting structures.
Week 9. Giovanni S. Alberti (Genoa) -- Learning the optimal regularizer for inverse problems
In this talk, we consider the problem of learning the optimal regularizer for linear inverse problems modeled in separable Hilbert spaces. In the context of generalized Tikhonov regularization, we characterize the optimal regularizer and derive generalization estimates, in both supervised and unsupervised settings. In the context of sparsity promoting regularization, we derive generalization estimates for learning the optimal “change of basis” in the $\ell^1$ penalty term. We also consider the Bayes estimator associated to a suitable prior modeling (group) sparsity, and show that it can be written as a shallow neural network with a specific attention mechanism. The weights of the network can then be learned by leveraging the well-established training algorithms for NN, yielding state-of-the-art performance for dictionary learning tasks.
Week 10. Kostas Zygalakis (Edinburgh) -- Optimization algorithms and differential equations: theory and insights

The ability of calculating the minimum (maximum) of a function lies in the heart of many applied mathematics applications. In this talk, we will connect such optimization problems to the large time behaviour of solutions to differential equations. In addition, using a control theoretical formulation of these equation, we will utilise a set of linear matrix inequalities (applicable in the case of strongly convex potentials) to establish a framework that allow us to deduce their long-time properties as well as deducing the long time properties of their numerical discretisations. using this framework, we give an alternative explanation for the good properties of Nesterov method for strongly convex functions, as well as highlight the reasons behind the failure of the heavy ball method. If there is time I will also discuss recent work that highlights how to extend these ideas in a non-Euclidean setting

References:

[1] P. Dobson, J. M. Sanz-Serna, K. C. Zygalakis Accelerated optimization algorithms and ordinary differential equations: the convex non Euclidean case, arXiv:2410.19380, (2024)

[2] P. Dobson, J. M. Sanz-Serna, K. C. Zygalakis On the connections between optimization algorithms, Lyapunov functions, and differential equations: theory and insights. SIAM J. Optim. (to appear), (2024).

[3] J. M. Sanz Serna, K.C. Zygalakis, The connections between Lyapunov functions for some optimization algorithms and differential equations. SIAM J. Numer. Anal., 59(3), 1542–1565, (2021).

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