Past Industrial and Applied Mathematics Seminar

28 January 2016
Laura Nicolaou

Respiratory illnesses, such as asthma and chronic obstructive pulmonary disease, account for one in five deaths worldwide and cost the UK over £6 billion a year. The main form of treatment is via inhaled drug delivery. Typically, however, a low fraction of the inhaled dose reaches the target areas in the lung. Predictive numerical capabilities have the potential for significant impact in the optimisation of pulmonary drug delivery. However, accurate and efficient prediction is challenging due to the complexity of the airway geometries and of the flow in the airways. In addition, geometric variation of the airways across subjects has a pronounced effect on the aerosol deposition. Therefore, an accurate model of respiratory deposition remains a challenge.

High-fidelity simulations of the flow field and prediction of the deposition patterns motivate the use of direct numerical simulations (DNS) in order to resolve the flow. Due to the high grid resolution requirements, it is desirable to adopt an efficient computational strategy. We employ a robust immersed boundary method developed for curvilinear coordinates, which allows the use of structured grids to model the complex patient-specific airways, and can accommodate the inter-subject geometric variations on the same grid. The proposed approach reduces the errors at the boundary and retains the stability guarantees of the original flow solver.

A Lagrangian particle tracking scheme is adopted to model the transport of aerosol particles. In order to characterise deposition, we propose the use of an instantaneous Stokes number based on the local properties of the flow field. The effective Stokes number is then defined as the time-average of the instantaneous value. This effective Stokes number thus encapsulates the flow history and geometric variability. Our results demonstrate that the effective Stokes number can deviate significantly from the reference value based solely on a characteristic flow velocity and length scale. In addition, the effective Stokes number shows a clear correlation with deposition efficiency.

  • Industrial and Applied Mathematics Seminar
21 January 2016
Tmoslav Plesa, John Ockendon, Hilary Ockendon

Tmoslav Plesa: Chemical Reaction Systems with a Homoclinic Bifurcation: An Inverse Problem, 25+5 min;

John Ockendon: Wave Homogenisation, 10 min + questions; 

Hilary Ockendon: Sloshing, 10 min + questions


  • Industrial and Applied Mathematics Seminar
3 December 2015
Leonid v Berlyand

We study the motion of a eukaryotic cell on a substrate and investigate the dependence of this motion on key physical parameters such as strength of protrusion by actin filaments and adhesion. This motion is modeled by a system of two PDEs consisting of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion perturbed by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters. In the subcritical regime, the well-posedness of the problem is proved (M. Mizuhara et al., 2015). Our main focus is the supercritical regime where we established surprising features of the motion of the interface such as discontinuities of velocities and hysteresis in the 1D model, and instability of the circular shape and rise of asymmetry in the 2D model. Because of properties (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why Γ-convergence techniques also can not be used. This is joint work with V. Rybalko and M. Potomkin.

  • Industrial and Applied Mathematics Seminar
26 November 2015
Adilson E Motter

Much of the recent interest in complex networks has been driven by the prospect that network optimization will help us understand the workings of evolutionary pressure in natural systems and the design of efficient engineered systems.  In this talk, I will reflect on unanticipated attributes and artifacts in three classes of network optimization problems. First, I will discuss implications of optimization for the metabolic activity of living cells and its role in giving rise to the recently discovered phenomenon of synthetic rescues. Then I will comment on the problem of controlling network dynamics and show that theoretical results on optimizing the number of driver nodes/variables often only offer a conservative lower bound to the number actually needed in practice. Finally, I will discuss the sensitive dependence of network dynamics on network structure that emerges in the optimization of network topology for dynamical processes governed by eigenvalue spectra, such as synchronization and consensus processes.  Optimization is a double-edged sword for which desired and adverse effects can be exacerbated in complex network systems due to the high dimensionality of their dynamics.

  • Industrial and Applied Mathematics Seminar
19 November 2015
Robert Style, Samuel Crew and Phil Trinh
New singularities for Stokes waves
Samuel Crew (Lincoln College) and Philippe Trinh
In 1880, Stokes famously demonstrated that the singularity that occurs at the crest of the steepest possible water wave in infinite depth must correspond to a corner of 120°. Here, the complex velocity scales like the one-third power of the complex potential. Later in 1973, Grant showed that for any wave away from the steepest configuration, the singularity moves into the complex plane, and is instead of order one-half. Grant conjectured that as the highest wave is approached, other singularities must coalesce at the crest so as to cancel the square-root behaviour. Even today, it is not well understood how this process occurs, nor is it known what other singularities may exist. 
In this talk, we shall explain how we have been able to construct the Riemann surface that represents the extension of the water wave into the complex plane. We shall also demonstrate the existence of a countably infinite number of singularities, never before noted, which coalesce as Stokes' highest wave is approached. Our results demonstrate that the singularity structure of a finite amplitude wave is much more complicated than previously anticipated, 
  • Industrial and Applied Mathematics Seminar
12 November 2015
Tiago Peixoto

Networks form the backbones of a wide variety of complex systems,
ranging from food webs, gene regulation and social networks to
transportation networks and the internet. Due to the sheer size and
complexity of many of theses systems, it remains an open challenge to
formulate general descriptions of their large-scale structures.
Although many methods have been proposed to achieve this, many of them
yield diverging descriptions of the same network, making both the
comparison and understanding of their results very
difficult. Furthermore, very few methods attempt to gauge the
statistical significance of the uncovered structures, and hence the
majority cannot reliably separate actual structure from stochastic
fluctuations.  In this talk, I will show how these issues can be tackled
in a principled fashion by formulating appropriate generative models of
network structure that can have their parameters inferred from data. I
will also consider the comparison between a variety of generative
models, including different structural features such as degree
correction, where nodes with arbitrary degrees can belong to the same
group, and community overlap, where nodes are allowed to belong to more
than one group. Because such model variants possess an increased number
of parameters, they become prone to overfitting. We demonstrate how
model selection based on the minimum description length criterion and
posterior odds ratios can fully account for the increased degrees of
freedom of the larger models, and selects the most appropriate trade-off
between model complexity and quality of fit based on the statistical
evidence present in the data.

Throughout the talk I will illustrate the application of the methods
with many empirical networks such as the internet at the autonomous
systems level, the global airport network, the network of actors and
films, social networks, citations among websites, co-occurrence of
disease-causing genes and many others.

  • Industrial and Applied Mathematics Seminar
5 November 2015
Ed Brambley

Noise limits are one of the major constraints when designing
aircraft engines.  Acoustic liners are fitted in almost all civilian
turbofan engine intakes, and are being considered for use elsewhere in a
bid to further reduce noise.  Despite this, models for acoustic liners
in flow have been rather poor until recently, with discrepancies of 10dB
or more.  This talk will show why, and what is being done to model them
better.  In the process, as well as mathematical modelling using
asymptotics, we will show that state of the art Computational
AeroAcoustics simulations leave a lot to be desired, particularly when
using optimized finite difference stencils.

  • Industrial and Applied Mathematics Seminar
29 October 2015
Michael Gomez, Jake Taylor-King, Andrew Krause, Zach Wilmott

Michael Gomez:

Title: The role of ghosts in elastic snap-through
Abstract: Elastic `snap-through' buckling is a striking instability of many elastic systems with natural curvature and bistable states. The conditions under which bistability exists have been reasonably well studied, not least because a number of engineering applications make use of the rapid transitions between states. However, the dynamics of the transition itself remains much less well understood. Several examples have been studied that show slower dynamics than would be expected based on purely elastic timescales of motion, with the natural conclusion drawn that some other effect, such as viscoelasticity, must play a role. I will present analysis (and hopefully experiments) of a purely elastic system that shows similar `anomalous dynamics'; however, we show that here this dynamics is a consequence of the ‘ghost’ of the snap-through bifurcation.

Andrew Krause:

Title: Fluid-Growth Interactions in Bioactive Porous Media   
Abstract: Recent models in Tissue Engineering have considered pore blocking by cells in a porous tissue scaffold, as well as fluid shear effects on cell growth. We implement a suite of models to better understand these interactions between cell growth and fluid flow in an active porous medium. We modify some existing models in the literature that are spatially continuous (e.g. Darcy's law with a cell density dependent porosity). However, this type of model is based on assumptions that we argue are not good at describing geometric and topological properties of a heterogeneous pore network, and show how such a network can emerge in this system. Therefore we propose a different modelling paradigm to directly describe the mesoscopic pore networks of a tissue scaffold. We investigate a deterministic network model that can reproduce behaviour of the continuum models found in the literature, but can also exhibit finite-scale effects of the pore network. We also consider simpler stochastic models which compare well with near-critical Percolation behaviour, and show how this kind of behaviour can arise from our deterministic network model.

Jake Taylor-King
Title:A Kinetic Approach to Evolving Spatial Networks, with an Application to Osteocyte Network Formation 
Abstract:We study an evolving network where the nodes are considered as represent particles with a corresponding state vector. Edges between nodes are created and destroyed as a Poisson process, and new nodes enter the system. We define the concept of a “local state degree distribution” (LSDD) as a degree distribution that is local to a particular point in phase space. We then derive a differential equation that is satisfied approximately by the LSDD under a mean field assumption; this allows us to calculate the degree distribution. We examine the validity of our derived differential equation using numerical simulations, and we find a close match in LSDD when comparing theory and simulation. Using the differential equation derived, we also propose a continuum model for osteocyte network formation within bone. The structure of this network has implications regarding bone quality. Furthermore, osteocyte network structure can be disrupted within cancerous microenvironments. Evidence suggests that cancerous osteocyte networks either have dendritic overgrowth or underdeveloped dendrites. This model allows us to probe the density and degree distribution of the dendritic network. We consider a traveling wave solution of the osteocyte LSDD profile which is of relevance to osteoblastic bone cancer (which induces net bone formation). We then hypothesise that increased rates of differentiation would lead to higher densities of osteocytes but with a lower quantity of dendrites. 




  • Industrial and Applied Mathematics Seminar
22 October 2015
Alex Cayco Gajic

Feedforward layers are integral step in processing and transmitting sensory information across different regions the brain. Yet experiments reveal the difficulty of stable propagation through layers without causing neurons to synchronize their activity. We study the limits of stable propagation in a discrete feedforward model of binary neurons. By analyzing the spectral properties of a mean-field Markov chain model, we show when such information transmission persists. Addition of inhibitory neurons and synaptic noise increases the robustness of asynchronous rate transmission. We close with an example of feedforward processing in the input layer to cerebellum. 

  • Industrial and Applied Mathematics Seminar
15 October 2015
Arjen Doelman

We consider the impact of spatial heterogeneities on the dynamics of 
localized patterns in systems of partial differential equations (in one 
spatial dimension). We will mostly focus on the most simple possible 
heterogeneity: a small jump-like defect that appears in models in which 
some parameters change in value as the spatial variable x crosses 
through a critical value -- which can be due to natural inhomogeneities, 
as is typically the case in ecological models, or can be imposed on the 
model for engineering purposes, as in Josephson junctions. Even such a 
small, simplified heterogeneity may have a crucial impact on the 
dynamics of the PDE. We will especially consider the effect of the 
heterogeneity on the existence of defect solutions, which boils down to 
finding heteroclinic (or homoclinic) orbits in an n-dimensional 
dynamical system in `time' x, for which the vector field for x > 0 
differs slightly from that for x < 0 (under the assumption that there is 
such an orbit in the homogeneous problem). Both the dimension of the 
problem and the nature of the linearized system near the limit points 
have a remarkably rich impact on the defect solutions. We complement the 
general approach by considering two explicit examples: a heterogeneous 
extended Fisher–Kolmogorov equation (n = 4) and a heterogeneous 
generalized FitzHugh–Nagumo system (n = 6).

  • Industrial and Applied Mathematics Seminar