Organiser: Susana Gomes and Shreyas Mandre (from term 2)
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 organisers.
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.
Please note that we are still waiting for confirmation of restrictions on the new year to confirm whether our speakers will be able to come in person in term 2.
Problems in viscous fingering (abstract)
|2||21 Jan||Balasz Kovacs (Regensburg)||online||
A convergent algorithm for the interaction of mean curvature flow and diffusion (abstract)
|3||28 Jan||Mark Opmeer (Bath)||online||Model Reduction and the Numerical Solution of Lyapunov Equations (abstract)|
|4||04 Feb||Nastassia Pouratier Duteil (Inria Paris)||TBC||
Continuum limits of collective dynamics with time-varying weights (abstract)
|5||11 Feb||Marta Catalano (Warwick Statistics)||F2F (if possible)||
|6||18 Feb||Adam Townsend (Durham)||TBC||
|Ben Goddard (Edinburgh)||TBC||
|8||04 Mar||Kerstin Lux (TU Munich)||F2F (if possible)||TBC (abstract)|
|9||11 Mar||Thomasina Ball (Warwick)||F2F (if possible)||TBC (abstract)|
|10||18 Mar||Matthew Colbrook (Cambridge/École Normale Supérieure)||F2F (if possible)||
|1||08 Oct||CIcely Macnamara (Glasgow)||online||An agent-based model of the tumour microenvironment (abstract)|
|2||15 Oct||Nabil Fadai (Nottingham)||F2F||
Semi-infinite travelling waves arising in moving-boundary reaction-diffusion equations (abstract)
|3||22 Oct||Jemima Tabeart (Edinburgh)||F2F||Parallelisable preconditioners for saddle point weak-constraint 4D-Var (abstract)|
|4||29 Oct||Jamie Foster (Portsmouth)||F2F||
Mathematical modeling of brewing espresso (abstract)
|5||05 Nov||Louise Dyson (Warwick)||online||
From ants to COVID-19: applications of mathematical modelling in biology and epidemiology (abstract)
|6||12 Nov||Upanshu Sharma (Berlin)||F2F (will visit the whole week)||
Variational structures beyond gradient flows (abstract)
|Olga Mula (Paris Dauphine)||online||
Optimal State and Parameter Estimation Algorithms and Applications to Biomedical Problems (abstract)
|8||26 Nov||Clarice Poon (Bath)||online||Smooth bilevel programming for sparse regularization (abstract)|
|9||03 Dec||Tom Montenegro-Johnson (Birmingham)||F2F (will also give the colloquium)||Soft, long, and thoroughly absorbent: unpublished works on looped active filaments and responsive hydrogels (abstract)|
|10||10 Dec||Peter Baddoo (MIT)||F2F||Integrating physical laws into data-driven system identification (abstract)|
Growth of complex fingers at the interface of air and a viscous fluid in the narrow gap between two parallel plates is an archetypical problem of pattern formation. This problem is quasi-two-dimensional and its key information is encapsulated in the interface, so it is relatively simple to study both experimentally and theoretically compared to many other nonlinear phenomena. In this talk we consider two variants of this system that allows us to ask different questions, one of the fundamental nature and one of practical importance.
Firstly, we consider the fingering between plates which are subject to a time-dependent (power-law) plate separation. We demonstrate that in this system the interface can evolve in a self-similar fashion such that the interface shape at a given time is simply a rescaled version of the shape at an earlier time. These novel, self-similar solutions are linearly stable but they only develop if the initially circular interface is subjected to unimodal perturbations. Conversely, the application of nonunimodal perturbations (e.g., via the superposition of multiple linearly unstable modes) leads to the development of complex, constantly evolving finger patterns similar to those that are typically observed between rigid stationery plates. We discuss these finding in the context of understanding disordered front propagation more generally.
Secondly, we demonstrate that the lubrication flow in the narrow gap between boundaries, in which one of the rigid plates has been replaced by a confined elastic solid, can `choke' at high flow rates, due to the deforming solid making contact with the plate and sealing the gap. This phenomenon is important in the context of designing soft microfluidics and predicting behaviour of deformable porous media. We show that, depending on the finger morphology, the fingering instability can either promote or suppress the choking in experiments compared with analogous axisymmetric simulations at the same control parameters. We discuss physical mechanisms that lead to this behaviour.
Week 2. Balasz Kovacs (Regensburg) - A convergent algorithm for the interaction of mean curvature flow and diffusion
In this talk we will present an evolving surface finite element algorithm for the interaction of forced mean curvature flow and a diffusion process on the surface.
The evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by the above coupled geometric PDE system. The coupled system is inspired by the gradient flow of a coupled energy, we will use this model for introductory purposes.
We will present two algorithms, based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface.
For one of the numerical methods we will give some insights into the stability estimates which are used to prove optimal-order $H^1$-norm error estimates for finite elements of polynomial degree at least two.
We will present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.
The talk is based on joint work with C.M. Elliott (Warwick) and H. Garcke (Regensburg).
We will consider Model Reduction in the context of partial differential equations where the aim is to approximate the PDE by a (low-dimensional) system of ODEs with little loss of accuracy.
Based on singular value analysis of an associated Hankel operator, we will show when this is possible and when it is not (essentially: it is possible for parabolic PDEs and isn't possible for hyperbolic PDEs).
The actual computation of the reduced order model requires the numerical solution of an (operator) Lyapunov equation. We will analyse a numerical algorithm which efficiently does this.
Week 4. Nastassia Pouradier Duteil (Inria Paris) - Continuum limits of collective dynamics with time-varying weights
In this talk, we will derive the mean-field limit of a collective dynamics model with time-varying weights. The limit equation is a transport equation with source, where the (non-local) transport term corresponds to the position dynamics, and the (non-local) source term comes from the weight redistribution among the agents. We show existence and uniqueness of the solution and introduce a new empirical measure (in the position space) taking into account the weights. Continuity with respect to the initial data allows us to prove the convergence of the microscopic system to the macroscopic equation. This mean-field limit can be derived if the particles' dynamics preserve indistinguishability. If they do not, another point of view consists of deriving the so-called graph limit. We will introduce the graph limit and show the subordination of the mean-field limit to the graph limit equation.
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.
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.
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.
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.
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.
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Mathematics Research Centre
University of Warwick
Coventry CV4 7AL - UK