Applied Mathematics Seminars in 2019

Abstract: In this talk I'll give an overview of work in the uncertainty quantification of PDEs with random input data, where the main objective is to compute expected values of quantities of interest derived from the solutions of the PDEs. I'll give some practical examples and then I'll explain how, via parametrization, the random PDE can be written as a parametrized family of deterministic PDEs with parameter lying in a possibly (very) high dimensional space. Such problems can then be solved by sampling the PDE (often many times over) and then averaging, to obtain expected values.

A successful algorithm then consists of (a) making good choices of points in high-dimensional parameter space at which to sample the data, (b) computing the samples of the data, and (c) fast computation of samples of the PDE, very many of which may be needed.

 In recent years there are many successful algorithms combining (a), (b) and (c) for some classes of PDEs, particularly the diffusion equation, and I'll describe a method which uses quasi-Monte Carlo for (a), circulant embedding for (b) and algebraic multigrid for (c). Recently I've been working on the frequency domain wave equation, which arises in the study of waves in random media. There the problems which arise are much more difficult particularly because there is no nice method to achieve (c), and so there are many open problems. I'll present some recent progress in this area.

Abstract: We revisit the tears of wine problem for thin films in water-ethanol mixtures and present a new model for the climbing dynamics. The new formulation includes a Marangoni stress balanced by both the normal and tangential components of gravity as well as surface tension which lead to distinctly different behavior. The combined physics can be modeled mathematically by a scalar conservation law with a nonconvex flux and a fourth order regularization due to the bulk surface tension. Without the fourth order term, shock solutions must sastify an entropy condition - in which characteristics impinge on the shock from both sides. However, in the case of a nonconvex flux, the fourth order term is a singular perturbation that allows for the possibility of undercompressive shocks in which characteristics travel through the shock. We present computational and experimental evidence that such shocks can happen in the tears of wine problem, with a protocol for how to observe this in a real life setting.

Abstract: Optimal transportation is a very rich and well-established field in mathematics. I consider here its variant where the transport has a direction and an additional martingale, or barycentre preservation, constraint. I will explain how this problem, called the Martingale Optimal Transport (MOT), arises naturally in (robust) financial mathematics and how it links with the classical Skorokhod embedding problem in probability. I will then discuss some recent results on structure of martingale transports. Finally, I will present recent advances on numerical methods for such problems via Linear Programming and/or deep Neural Networks methods.
Based on joint works with Pietro Siorpaes and with Gaoyue Guo.

Abstract: There are a number of example problems arising in triple-deck theory which we have attempted to solve recently where the solution technique seems to generate unexplained behaviour. For example in the classic supersonic compression ramp problem discussed in Logue, Gajjar & Ruban (2014) wavepackets appear in the results which do not appear to be related to anything physical. A similar situation arises in trying to solve the classic initial value vibrator problem of Terentev (1981). In this talk we discuss a closely related problem of boundary layer flow past localised heating elements. The linearised initial-value problem is solved analytically as well as numerically, and we will discuss the strange wavepacket behaviour which is also present.

Abstract: Illegal harvest of wildlife (poaching) is one of the greatest threats to biodiversity. Most countries try to reduce poaching by increasing law enforcement to catch and punish poachers. But despite best efforts from police, poaching is more frequent now than ever. In this talk, we present simple ordinary differential equation models of poachers and wildlife, to explore why law-enforcement has failed to stem the poaching problem. We then use these models to project the performance of controversial, alternative, management actions, such as, campaigns to reduce consumer demand for illegal wildlife products, and legalising trade of these products.

Abstract: We explore the work of my recent PhD thesis, namely, theory surrounding the singularity at a simultaneous binary collision in the 4-body problem. It is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than  differentiable with respect to initial conditions. Through an example based analysis of planar systems, this concept of block regularisation is defined. Then, this curious loss of differentiability is investigated through a blow-up procedure and a new proof of the  regularity in the collinear problem is provided. In the process, it is revealed that the critical manifold from the blow-up consists of two manifolds of normally hyperbolic saddle singularities which are connected by a manifold of heteroclinics. By utilising recent work on transitions near such objects and their normal forms, an asymptotic series of the transition past the singularity is explicitly computed. It becomes remarkably apparent that the finite differentiability at  is due to the inability to construct a set of integrals local to the simultaneous binary collision.

Abstract: Systems with small parameters are often studied using asymptotic techniques. Despite the ubiquity of these techniques, many classical asymptotic methods are unable to capture behaviour that occurs on an exponentially small scale, which lies "beyond all orders" of power series in the small parameter. Typically this does not cause any issues; this behavior is too small to have a measurable impact on the overall behaviour of the system. I will showcase two systems in which exponentially small contributions have a significant effect on the overall system behaviour.

 The first system, which I will discuss in detail, will be nonlinear waves propagating through particle chains with periodic masses. I will show that it is typically possible for Toda and FPUT lattices for certain combinations of parameters - determined by the exponentially small system behaviour - to produce solitary waves that propagate indefinitely. The second system, which I will discuss more briefly, will be the shape of bubbles in a steadily translating Hele-Shaw cell. By studying exponentially small effects, it is possible to construct exotic bubble shapes which correspond to recent laboratory experiments.

Abstract: Rhythms in the brain are vital for all aspects of physiological development and function. At the cellular level, neural rhythms typically manifest as electrical signals known as bursts, consisting of long periods of inactivity interspersed with rapid trains of closely spaced action potentials. Bursts are the basic units of neural information and have been proposed to support numerous functional roles, such as synchronization between neuronal populations, attention, synaptic plasticity, and memory and awareness. In this seminar, we examine the dynamics of bursting from the viewpoint of canard (French: duck) theory. We explain the origins and properties of bursting in various contexts, such as in hormone and neurotransmitter secretion in the pituitary gland. We also discuss how the predictions from canard theory can be tested in vitro.

Abstract: Let's first prove that negative probabilities are OK! The question is when? and how? Answer: when modelling the dynamics of a high-D system by a low-D system---such model reduction is the theme of this talk. 

Then let's discuss how averaging is unsound despite many claiming it is exact! Of course, such averaging underpins many conservation PDEs in space, and hence these PDEs may mislead!

Lastly, let's see how non-autonomous/stochastic systems have to be modelled by uncertain variables! Examples discussed include quasi-stationary probability, thin fluid films, shear dispersion, spatial birth and death, lattice systems, Brownian motion, and population models. Remember these paradoxes and their resolution whenever you consider dynamics at multiple levels of model resolution.

Abstract: Reaction-nonlinear diffusion models arising in the context of cell migration and population dynamics can exhibit the property of aggregation – or backward diffusion. While this is physically relevant, mathematically it causes such models to break down. The aggregation causes shocks to form, and the solutions are no longer computable.

To account for shocks, modellers have employed the technique of regularisation – adding additional small higher order terms to these models to smooth out the shocks. These regularisation techniques have been widely employed in models of chemical phase-separation, though they have gone relatively unnoticed in biological models until very recently.

We have developed techniques from the field of geometric singular perturbation theory to resolve similar issues of shock formation in a different class of models, so-called advection-reaction models (hyperbolic balance laws). In this presentation, we will tackle the question of existence and formation of shocks in regularised reaction-nonlinear diffusion models using geometric singular perturbation theory.

Abstract: We study the discrete nonlinear wave equation in arbitrary finite networks. This is a general model, where the usual continuum Laplacian is replaced by the graph Laplacian. We consider such a wave equation with a cubic on-site nonlinearity which is the discrete \Phi^4 model, describing a mechanical network of coupled nonlinear oscillators or an electrical network where the components are diodes or Josephson junctions.

In the first part, we investigate the extension of the linear normal modes of the graph Laplacian into nonlinear periodic orbits. Normal modes -whose Laplacian eigenvectors are composed uniquely of {1}, {-1,1} or {-1,0,1}- give rise to nonlinear periodic orbits for the discrete \Phi^4 model. We perform a systematic linear stability (Floquet) analysis of these orbits and show the modes coupling when the orbit is unstable. Then, we characterize graphs having Laplacian eigenvectors in {-1,1} and {-1,0,1} using graph spectral theory.

In the second part, we investigate periodic solutions that are exponentially (spatially) localized. Assuming a large amplitude localized initial condition on one node of the graph, we approximate its evolution by the Duffing equation. The rest of the network satisfies a linear system forced by the excited node. This approximation is validated by reducing the discrete \Phi^4 equation to the discrete nonlinear Schrodinger equation and by Fourier analysis. These results relate nonlinear dynamics to graph spectral theory.

Abstract: The facial structure of convex sets can be surprisingly complex, and unexpected irregularities of the arrangements of faces give rise to badly behaved sets and various counterexamples. In this talk I will focus on specific properties of facial structure that capture irregularities in the facial structure of the set (dimensions of faces, singularity degree, facial exposure and facial dual completeness). I will also talk about some classic results related to faces of convex sets, mention some new results and counterexamples and will relate this to several open problems in convex algebraic geometry and the geometry of polytopes.

Abstract: Many problems in biology, physics and engineering involve predicting and controlling complex systems, loosely defined as interconnected system-of-systems. Such systems can exhibit a variety of interesting non-equilibrium features such as emergence and phase transitions, which result from mutual interactions between nonlinear subsystems. Modelling these systems is a task in-and-of itself, as systems can span many physical domains and evolve of multiple time scales. Nonetheless, one wishes to analyse the geometry of these models and relate both qualitative and quantitative insights back to the physical system. Beginning with the modelling and analysis of a coupled optomechanical systems, this talk presents some recent results concerning the existence and stability of emergent oscillations. This forms the basis for a discussion of new directions in symbolic computational techniques for complex physical systems as a means to discuss emergence more generally.

Abstract: A technique has been developed to fit all types of geomagnetic storms identified in the Dst. A lognormal fitting procedure may be used to describe any storm by setting the lognormal standard deviation to greater than 0.9. The fit needs to be constrained around the peak time and the scaling factor determined. This will enable an autonomous method for fitting any type of storm within the Dst. The unique factor identifying the relationship between the main and recovery phase of a storm is the lognormal mean.

Abstract: In this talk we will consider two and three dimensional boundary layer flows. The stability of both the flat plate and rotating disk boundary layers will be discussed in the context of fluids that exhibit a variational viscosity. In particular, we will present linear stability results for fluids with viscosity that varies as a function of temperature. Numerical results (neutral curves, growth rates and energy analyses) will be supported by asymptotic predictions at large Reynolds numbers. The influence of an enforced axial flow will also be discussed in the context of Chemical Vapour Deposition (CVD).

Abstract: We study the Schlesinger system in the case when the unknown matrices of arbitrary size (p×p) are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference q, the same for all matrices. We show that such a system possesses a family of solutions expressed via periods of meromorphic differentials on the Riemann surfaces of superelliptic curves. We determine the values of the difference q, for which our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the (2 × 2)-case, we obtain explicit sequences of rational solutions and one-parameter families of rational solutions of Painleve VI equations. This is a joint work with Renat Gontsov and Vasilisa Shramchenko.

Abstract: Calcium plays a central role in how our hearts beat. Each heartbeat is governed by the cyclic rise and fall of calcium in the cell cytoplasm through various co-ordinated and tightly regulated electrical and chemical processes. Calcium also plays a crucial role in determining when our heart muscle will grow in order to increase the force with which the heart beats as long-term demand for blood supply is increased. This process of cell and heart muscle growth is termed hypertrophy and is analogous to how our skeletal muscles grow through weight training. Exactly how calcium can regulate beat-to-beat muscle contraction and also send a signal to the cell nucleus for long-term growth is unclear. In this talk I will present our research into how the spatial organisation of ion-channels that govern calcium concentration in the cytoplasm affect calcium dynamics in the cell for beat-to-beat contraction and hypertrophic growth.