Applied Mathematics MSc Projects 2017–2018

Imperial College London

Last revised: December 4, 2017

Professor Mauricio Barahona

General topics: Dynamics and graph theory. Network analysis. Stochastic processes on graphs.

Optimization. Dimensionality reduction, geometric projections for high-dimensional data. Com-

munity detection on graphs. Deep learning.

Areas of application: Social networks, financial and economic data, coarse-graining and segmenta-

tion of images, bioinformatics

Some examples of possible projects:

1. Theory of graph-theoretical data analysis: a series of possible projects on the conceptual and

mathematical extensions of techniques for the representation of data as graphs, and the coarse-

graining of such representations. These include topics such as: * time-varying networks and their

partitions: detecting break points using multidimensional algorithms * development of notions of

robustness for multiscale graph partitions: node and edge deletion, statistical predictability and

bootstraps * optimal sparsification of graphs that preserve structural and spectral properties *

anomaly detection using graph-theoretical notions of manifold reconstruction

2. Finding roles and communities in directed networks based on flow profiles: Application to

Twitter networks (to find roles in information propagation. This project consists of the analysis of

networks constructed from a large database of ‘tweets’, collected over a period of more than a year.

We will construct sequences of networks (e.g., ‘retweet’, word adjacencies, followers) encompassing

small periods of time and analyse their features as they change in time. The methods we use rely on

concepts from graph theory, dynamical systems, and stochastic processes. Partly in collaboration

with: colleagues at Oxford, and at the Big Data Analysis Unit at Imperial.

3. Nonlinear dimensionality reduction for high-dimensional data: Application of recently developed

techniques in our group for the analysis of high dimensional data using graph theoretical techniques

linked to geometric constructions, as well as the use of diffusion dynamics on graphs for community

detection. Application to: (a) transcriptomics profiles of cellular responses to chemical compounds

that can originate cancer, (b) single-cell RNA profiles of cell types related to development and

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stem cells, (c) analysis of behavioral time-series of motion of C elegans: Over 10,000 videos of

freely moving C. elegans, a nematode worm, with sufficient resolution will be analysed to obtain

reduced representations of complex postural times series. What is the dimensionality of motional

behavior? Partly in collaboration with: Syngenta (a), the Sanger Institute at Cambridge (b), Dr

Andre Brown Imperial MRC Clinical Sciences Centre (c).

4. Dimensionality reduction and machine learning: a graph-theoretical perspective. The project will

explore the mathematical underpinnings of various dimensionality reduction and machine learning

methods in discrete settings, such as graphs and networks, focusing in particular on connections

between underlying geometry, dynamics on the graph, spectral properties, and graph topology at

multiple scales. Theoretical tools and ideas from statistical physics, spectral graph theory, and

wavelet analysis, will be brought to bear on underlying mathematical questions, as well as on

specific methods of interest, for example the neural network architectures used in deep learning

algorithms. The project will also involve developing better understanding and improvement of

current techniques for clustering/community detection and graph sparsification, and applications

to multiplayer graphs and other hierarchical data. There will also be ample opportunities to work

with real data and implement novel techniques and algorithms.

With Dr Asher Mullokandov.

6.Temporal Graph Signal Analysis - maximal signal compressibility/decomposition:

The dynamics taking place on a network and its structure are intimately connected. It is possible to

use a Fourier-like theory to decompose the signal on the nodes of a network using e.g. the eigenbasis

of the Laplacian of said network and there exists a natural analogy between the eigenvalues of the

Laplacian and the notion of spatial frequencies.

In this project, we are interested in in exploring the effect of the basis used on the characterisation

of the signal. Two possible directions are: i) is it possible to increase signal spectral representation

compressibility, in terms of the frequencies significantly present in the spatial power spectrum,

by altering the underlying network for example by ”increasing its symmetries”? This would be

particularly helpful in the context of understanding network (quasi)-symmetries and their role in

signal characterisation. ii) is there a universal basis , i.e. a (class of) network(s) that possesses

key topological features that allows for an informative decomposition of the signal on any network?

Such a basis would be particularly useful when the network supporting the signal is partially/not

known?

With Dr Paul Expert.

7. Effect of multi-failure in graphs

Graphs are often used to describe complex physical interactions and energy exchanges in a variety

of domains, from power grids to molecules. An important question to tackle in this context is to

precisely understand the effect of affecting, or even destroying a particular connection to the rest

of the graph. Recently, a powerful method has been developed using diffusion on graphs to obtain

this information. The aim of this project is to extend this method to the case where two or more

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connections are affected of destroyed, and apply it to real world example to assess the importance

of such events. For power grid networks, such events can lead to cascading of failures resulting in

a blackout of part of the network.

Requirements: basic notions of graph theory and knowledge of Python programming

With Dr Alexis Arnaudon.

8. Helicity of simplicial complexes

Simplicial complexes are discrete structures composed of a point, lines, faces, etc... with a specific

relationship between them such that they can be seen as discrete manifolds. This implies that

a discrete notion of differential calculus exists on these structures. In fact, simplicial complexes

are often used as discretization of smooth manifolds such that the original differential structure

is approximately preserved. One of the main application of such approach is for simulations of

fluid dynamics. This project will not directly be about fluid dynamics but will be to implement

the continuous notion of helicity of flow lines to simplicial complexes. This notion relates to

how entangled flow lines of a fluid are and can surprisingly be computed only using differential

geometric tools. Helicity provides important topological information about the simplicial complex,

when assumed to be be a discretization of a continuous space but more importantly when it is

obtained by some real world three dimensional datasets, in which case we can learn about the

topology of the data itself.

Requirements: basic notions of differential geometry and graph theory, knowledge of Python pro-

gramming

With Dr Alexis Arnaudon.

9. Metabolic Flux analysis and directed graphs — see Dr Diego Oyarzun

10. Stochastic dynamics of structured populations see Dr Philipp Thomas

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Quantum random walks and topological phases – Dr Ryan Barnett

In this project, we will investigate a quantum version of the classical random walk – the quantum

random walk. The quantum walker explores simultaneously all of the possible paths of her classical

counterpart.

A particularly interesting situation, receiving recent attention is as follows. The walker carries with

her a spin (you can think of this as a vector). The walker initially stands at the origin and her spin

points up. The spin is rotated by some specified angle (step 1). Next the walker takes a quantum

step both to the left and right – the probability to go left (right) is proportional to the up (down)

spin component. Step 1 and step 2 are then repeated N times. In [1] it was realised that some of

these random walks (for different initial conditions) can have peculiar features – the walker can be

confined to only a small region of the lattice. This is due to the underlying topological structure

of the equations governing the dynamics.

Initial

Step 1

Step 2

After becoming familiar with a few recent results/papers on this topic, the aim of the project is

to find variations of these models that map onto established topological problems in condensed

matter physics. In particular, by adding some additional slow time dependence, the realisation of

the Thouless charge pump [2] will be sought.

[1] Takuya Kitagawa, Mark S. Rudner, Erez Berg, and Eugene Demler, “Exploring topological

phases with quantum walks” Phys. Rev. A 82, 033429 (2010)

[2] F. Duncan M. Haldane, “Nobel Lecture: Topological quantum matter” Rev. Mod. Phys. 89,

040502 (2017)

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Can superfluids rotate rigidly? – Dr Ryan Barnett

A superfluid is a substance that flows with zero viscosity. When mechanically rotated (e.g. when

in a spinning bucket) superfluids typically will form a vortex lattice, in stark contrast to classical

fluids. This is because the velocity of a superfluid is the gradient of a phase having quantum

mechanical origins: v = ~m∇θ. As a result, the voticity, ∇ × v, is zero everywhere except the

positions marking the vortex centres (where the phase is ill-defined). Rigid rotation – where ∇×v

is a non-zero constant everywhere – clearly cannot be obtained for this case.

The above does not apply when the atoms forming the superfluid have internal spin degrees of

freedom. For this case, the vorticity is related to the spin direction, denoted by unit vector n, as

∂xvy − ∂yvx =~

2mn · (∂xn× ∂yn)

(restricting to two spatial dimensions for simplicity). This elegant and geometric equation is known

as the Mermin-Ho relation.

The aim of this project is to investigate if spinor fluids can rotate in ways similar to their classical

counterpart. That is, can such steady state solutions be found in the rotating frame of reference?

A recent affirmative result along these lines was obtained in [1]. Spinor superfluids with spin-orbit

coupling – a topic of considerable recent experimental progress – is likely a crucial ingredient and

will be investigated in this context.

[1] Sandro Stringari, “ Diffused Vorticity and Moment of Inertia of a Spin-Orbit Coupled Bose-

Einstein Condensate” Phys. Rev. Lett. 118, 145302 (2017)

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John W. Barrett

Both projects concern macroscopic fluid models for dilute polymers such as Oldroyd–B or FENE–P.

These models consist of the standard time-dependent Navier–Stokes equations with the divergence

of an extra stress tensor on the righthand-side, which is coupled to an equation for this extra stress

tensor involving the velocity of the fluid. A formal a priori entropy bound is presented for either

model in [1]. Such an estimate is useful in (i) studying the long-term behaviour of the model,

(ii) proving existence of a global-in-time weak solution to the model and (iii) constructing stable

numerical approximations.

(1) The Use of the Matrix–Logarithm inModelling Complex Fluids

The standard entropy estimate for Oldroyd–B in [1] is only meaningful if one can show that the

extra stress tensor remains symmetric positive definite throughout the evolution. Moreover, it

seems one can only prove existence of a global-in-time weak solution to the Oldroyd–B model in

two space dimensions, and convergence of a numerical approximation, if, in addition, a diffusion

term is added to the stress equation. Although this can be achieved, see e.g. the finite element

approximation constructed in [2], it is not straightforward. This positive definite constraint can be

avoided by rewriting the system using the logarithm transform proposed in [3]. Furthermore, the

formal entropy structure in [1] can be adapted to this transformed system, see [4]. This project will

consider the use of this logarithm transformation from both an analysis and practical viewpoint.

(2) Finite Element Approximation ofthe Oldroyd–B Model

Convergence, as the time step ∆t and spatial discretization parameter h tend to zero, of a finite

element approximation of the Oldroyd–B model in two space dimensions with a diffusion term added

to the stress equation is proved in [2]. A key ingredient is building the finite element approximation

to satisfy a discrete version of the formal entropy bound satisfied by the Oldroyd–B model, see [1].

This numerical approximation at present has three undesirable features: (i) a time step constraint

of the form ∆t ≤ C h2 in order to prove convergence, (ii) at each time level a nonlinear system

involving the approximations of both the fluid velocity and extra stress tensor has to be solved and

(iii) a complicated approximation of the advection term in the extra stress equation in order to

mimic the entropy bound in [1]. This project will consider (a) the use of the characteristic Galerkin

method, see [4], to try to avoid (i) and (iii), and (b) the use of ideas in [5], which are from related

models, to avoid (ii).

Prerequisites

These projects will involve analysis and computation (e.g. Matlab). Knowledge of partial differential

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equations and associated finite element approximations is essential.

References

[1] D. Hu and T. Lelievre, New entropy estimates for Oldroyd-B and related models, Commun.

Math. Sci., 5, (2007), 909–916.

[2] J. W. Barrett and S. Boyaval, Existence and approximation of a (regularized) Oldroyd-B model,

Math. Models Methods Appl. Sci., 21, (2011), 1783–1837.

[3] R. Fattal and R. Kupferman, Constitutive laws for the matrix logarithm of the conformation

tensor, J. Non-Newton. Fluid Mech., 123, (2004), 281–285.

[4] S. Boyaval, T. Lelievre and C. Mangoubi, Free-energy-dissiptive schemes for the Oldroyd-B

model, ESAIM: Math. Model. Numer. Anal., 43, (2009), 523–561.

[5] J. Shen and X. Yang Decoupled, energy stable schemes for phase-field models of two-phase

incompressible flows, SIAM J. Numer. Anal., 53, (2015), 279–296.

Modelling inertial particle transport in the ocean – Dr Berloff

Motivated by the modelling of floating plastic debris on the ocean surface, the project is intended to

study the transport of inertial (i.e., those having finite size and being lighter than water) particles

in the ocean. The goal is to investigate a hierarchy of models for inertial particle movement by

turbulent ocean circulation solutions: modelling particle trajectories on the top of fluid-dynamical

ocean eddies and currents. We’ll consider quasigeostrophic approximation for fluid motion, Ekman

boundary layer corrections, and Maxey-Riley equation and its approximations for the inertial par-

ticles. The project will provide an exciting combination of geophysical fluid dynamics, ODEs and

PDEs, numerical modelling and statistical analyses.

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Simulation and control of agent‐based models

José A. Carrillo & Dante Kalise

In this project, we will study large‐scale multi‐agent systems modelling animal

and/or human collective behaviour. We will introduce control actions over the

system either via leaders, as in a shepherd’s dog controlling a herd of sheep, or

undercover agents, such as bot accounts in social networks. While agent‐based

modelling is a well‐established tool to study social dynamics, we will explore analytic

and computational techniques for the synthesis of external control actions to induce

different dynamical patterns such as consensus, alignment, or swarming.

1. Wolf‐pack (Canis lupus) hunting strategies emerge from simple rules in computational

simulations. C. Muro, R. Escobedo, L. Spector, R.P. Coppinger, Behavioural Processes 88

(2011) 192– 197. 2. Group size, individual role differentiation and effectiveness of cooperation in a

homogeneous group of hunters. C. Muro, R. Escobedo, L. Spector, R.P. Coppinger, J. Roy.

Soc. Interface 11 (2014) 20140204. 3. H. Tompkins and T. Kolokolnikov, Swarm shape and its dynamics in a predator‐swarm

model, to appear, SIAM Undergraduate Research Online. 4. J. A. Carrillo, Y. Huang, S. Martin, Nonlinear stability of flock solutions in second‐order

swarming models, Nonlinear Analysis: Real World Applications 17, 332–343,

2014.40204. 5. Invisible control of self‐organizing agents leaving unknown environments. G. Albi, M.

Bongini, E. Cristiani and D. Kalise, SIAM Journal on Applied Mathematics, 76(4)(2016),

1683‐‐1710. 6. Modeling and control through leadership of a refined flocking system. A. Borzì and S.

Wongkaew, Math. Models Methods Appl. Sci. 25, 255 (2015).

Aggregation patches for a model of laser traps

José A. Carrillo and Matias Delgadino

We will study a model proposed in [3,4] for confinement of atomic gases by an array of laser

beams. These models lead to nonstandard drift‐diffusion like equations with not too well‐

known properties. This equation has similar properties to the Keller‐Segel model for which

aggregation patches have been obtained. The first part of the proposed topic will be to do a

similar existence‐uniqueness results for solutions of these equations in the case without

diffusion like in [1]. Explicit solutions will be sought for both in the repulsive and attractive

cases. The structure of the skeleton obtained for this new equation will be explored by

numerical methods as in [1].

1‐ Bertozzi, T. Laurent, and F. Léger. Aggregation via the Newtonian potential and

aggregation patches. M3AS, vol. 22, Supp. 1, 2012.

2‐ https://cims.nyu.edu/~leger/aggregation.html

3‐ Long‐range one‐dimensional gravitational‐like interaction in a neutral atomic cold gas,

M. Chalony, J. Barré, B. Marcos, A. Olivetti, and D. Wilkowski. Phys. Rev. A 87, 013401

https://hal.archives‐ouvertes.fr/hal‐01141171/document

4‐ Non‐equilibrium Phase Transition with Gravitational‐like Interaction in a cloud of Cold

Atoms, J. Barré, B. Marcos, and D. Wilkowski, Phys. Rev. Lett. 112, 133001.

http://arxiv.org/pdf/1312.2436.pdf;

Geometric integrators for fluid in a hosepipe – Dr Cotter Geometric integrators are numeri-

cal algorithms that are designed to preserve fundamental conservation properties of the ODE/PDE.

Recently, Gay-Balmaz and Putkaradze (Gay-Balmaz, Franois, and Vakhtang Putkaradze. ”Exact

geometric theory for flexible, fluid-conducting tubes.” Comptes Rendus Mcanique 342.2 (2014):

79-84) produced a geometric theory for a fluid moving in a flexible pipe. This describes, for ex-

ample, the motion of a pipe when you switch on a tap, and the pipe flexes all over the place. In

this project, we will take this geometric theory, and build a discretised version of it, using a finite

element discretisation; then we’ll investigate its properties in numerical experiments, facilitated

using the Python code generation library Firedrake (firedrakeproject.org).

Advanced numerical algorithms for the simulation of weather fronts – Dr Cotter The

phenomenon of frontogenesis (the process of front formation in weather systems) was first given

a mathematical description by Hoskins and Bretherton (1972), by proposing a model called the

semigeostrophic equations, and providing a coordinate transformation that exposes the structure

in these equations. In the 1990s, this transformation was interpreted as the solution of an optimal

transport problem in the setting of Kantorovich, and this led to algorithms for solving these equa-

tions which showed the process of front formation. However, these algorithms are very slow and so

have not found their way into use by meteorologists. This is a shame, because these SG solutions

have exposed some issues with operational weather forecasting models which, if solved, could dra-

matically improve the accuracy of weather forecasts. Recently, a faster algorithm has been invented

for these transport problems (Mrigot, Quentin, Jocelyn Meyron, and Boris Thibert. ”An algorithm

for optimal transport between a simplex soup and a point cloud.” arXiv preprint arXiv:1707.01337

(2017)) together with open source software for implementing them. In this project we will adapt

this code to solve the frontogenesis problem.

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Vertical coordinate mappings for numerical

weather prediction

Dr Colin Cotter and Dr Jemma Shipton

November 30, 2017

Finite element methods are attractive for numerical weather prediction andclimate modelling due to the flexibility of the choice of the underlying grid. Inthe horizontal direction, this means that unstructured grids can be used, therebyavoiding the parallel scalability problem encountered when using structuredgrids that inevitably have points clustered in the polar regions. In the verticaldirection, there is freedom to deform the mesh to fit the shape of the Earth’ssurface, i.e. to follow mountain ranges. However, this deformation can lead tounwanted numerical effects such as decreased accuracy in the representation ofthe horizontal pressure gradient and noisy tracer transport. These effects willonly be exacerbated as model resolution increases, enabling representation ofsteeper slopes.

Accurate modelling of waves generated by mountains is important due tothe effect they have on both the local and global dynamics. Local effects in-clude strong downslope winds, enhanced precipitation and clear-air turbulencegenerated in the lee of mountains. Global effects are due to the transport ofmomentum by vertically propagating waves. New numerical schemes for solv-ing the equations used in numerical weather prediction have to prove that theyare capable of modelling these effects. The first step is to introduce a smallmountain and subsequent mesh deformation and observe the effect this has ona simulation of a resting atmosphere. Tests then increase in difficulty, incorpo-rating flow past the mountain in linear and nonlinear regimes, more complexmountain range profiles and more complex vertical stratification profiles.

Recently, in collaboration with the Met Office, we have developed a finiteelement model for numerical weather prediction that exhibits the required con-servation and large scale wave propagation properties for accurate modelling ofthe atmosphere [?]. The schemes developed in this model will provide the basisfor schemes used in the next generation Met Office model. This project wouldinvolve using our model, Gusto, to simulate 2D flow over idealised mountainsusing a variety of different vertical coordinate mappings based on the ideas pre-sented in [?], which suggests using a mapping that smooths coordinate surfaceswith height.

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References

[1] Cotter, Colin J and Shipton, Jemma, Mixed finite elements for numericalweather prediction, Journal of Computational Physics, 231(21): 7076–7091,2012

[2] Klemp, Joseph B, A terrain-following coordinate with smoothed coordinatesurfaces, Monthly weather review, 139(7): 2163–2169, 2011

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MAXWELL’S EQUATIONS AND NEGATIVE INDEX MATERIALS

RICHARD CRASTER AND HARSHA HUTRIDURGA

Many modern-day technological developments have their roots in electromagnetic theory; radio,radar, microwaves amongst many others. Maxwell’s equations for the electric and magnetic fieldsare considered amongst the most influential equations in all of science. These equations are quitechallenging, for instance, the time harmonic Maxwell’s equations in three dimensions take the form

(1)

CurlE = iωµH

CurlH = −iωεEwhere E and H are vectors. Here µ and ε are material parameters representing permeability andpermittivity respectively. The frequency ω ∈ R comes from the time dependence e−iωt in thetime-harmonic wave.

The permittivity differs for different materials and is often complex-valued. It can have negativereal part but non-negative imaginary part. Essentially, the medium is transparent if the real partof ε is positive and is opaque if the real part of ε is negative (which is the case for many metals).

Veselago [Ves68] studied the implications of having both µ and ε negative. Materials withnegative parameters are termed negative index materials. When both µ and ε negative, we obtainsomething called left-handed materials which yield some interesting effects on the wave propagation.

In the last two decades, there has been renewed interest in the investigation of negative indexmaterials. Construction of such materials, both theoretically and experimentally, have been madepossible with the help of meta-materials. Meta-materials are composite materials which are nothingbut the assemblies of smaller components consisting of ordinary materials which behave effectivelyin a way that is not known for the naturally occurring materials. Mathematically speaking, thisquestion of studying effective behaviour of composites falls under the theory of homogenization forpartial differential equations.

The mathematical theory of homogenization [BLP78], loosely speaking, corresponds to the studyof differential equations with highly oscillatory coefficients. For example, take the solution uη(x)to a differential equation

Aηuη(x) = f(x),

where Aη is a differential operator with η-periodic coefficients. A typical result in homogenizationtheory is to show that uη ≈ uhom in the regime η 1, i.e. in the regime where the coefficients inthe operator Aη oscillate more and more. Furthermore, one shows that the approximation uhomsolves the homogenized equation

Ahomuhom(x) = f(x),

with Ahom an effective differential operator whose coefficients are η-independent.In the context of time-harmonic Maxwell’s equations (1), periodic homogenization problem trans-

lates to studying the solution family (Eη, Hη) to the system

(2)

CurlEη = iωµηHη

CurlHη = −iωεηEη1

2 RICHARD CRASTER AND HARSHA HUTRIDURGA

where the permeability-permittivity pair µη, εη are η-periodic functions of the spatial variable. Itso happens that the effective equation in this scenario is again a system similar in structure to (2)with effective permeability-permittivity pair µhom(ω), εhom(ω). Note that the effective coefficientsare frequency dependent.

At first this project will be concerned with the understanding of

• constructing thin wire structures with extreme values to obtain negative effective permit-tivity εhom(ω) (some references of interest are [PHSY96, FB97])• obtaining negative effective permeability µhom(ω) from periodic split ring structures (refer-ences of interest are [PHRS99, BBF2009, BS2010])• combining the construction in wire structures and that in periodic split ring structuresto obtain both the effective parameters µhom(ω), εhom(ω) to be negative simultaneously(references of interest are [SPVNS96, LS2016]).

With a good understanding of the above mentioned constructions, the following finer (and moredifficult) objectives can be set:

• Exploring the range of applicability of the aforementioned constructions in the frequencyvariable (both theoretically and numerically).• Lifting the lossy-layer construction in [LS2016] to arrive at the same results.

Accomplishing the above objectives will require the student to learn some basic notions from thetheory of homogenization – preferably, by working on the electrostatic problem. In fact, studyingthe periodic homogenization problem for (2) is a classic example for homogenization with resonancewhich is quite difficult compared to homogenization problems in electrostatics.

This project is quite challenging as it blends both the mathematical analysis of differentialequations and some associated computational techniques. A successful completion of this projectshould result in a journal publication. This project could involve collaboration with a group fromthe department of Physics and there will be opportunities to interact with them, attend groupmeetings, and work within an active group of PhD students and post-docs. It is an exciting and avery topical area of research with plenty of applications and opportunities.

Pre-requisites: Student taking on this project is required to have some basic understanding ofdifferential equations and functional analysis. It is also preferable to have some working knowledgeof computational environments such as MATLAB (Note that this project might also give the studenta possibility to get acquainted with COMSOL).

References:

[Ves68] v.veselago, The electrodynamics of substances with simultaneously negative values ofε and µ, Soviet Phys. Uspekhi, Vol 10, pp.509–514, 1968.

[BLP78] a.bensoussan, j.-l.lions, g.papanicolaou, Asymptotic analysis of periodic struc-tures, 1978.

[PHSY96] j.b.pendry, a.j.holden, w.j.stewart, i.youngs, Extremely low frequency plas-mons in metallic mesostructures, Phys. Rev. Lett., Vol 76, 1996.

[FB97] d.felbacq, g.bouchitté, Homogenization of a set of parallel fibres, Waves RandomMedia, Vol 7, pp.245–256, 1997.

MAXWELL’S EQUATIONS AND NEGATIVE INDEX MATERIALS 3

[PHRS99] j.b.pendry, a.j.holden, d.j.robbins, w.j.stewart, Magnetism from conductorsand enhanced nonlinear phenomena, IEEE Trans. Microwave Theory Tech., Vol 47, 1999.

[BBF2009] g.bouchitté, c.bourel, d.felbacq, Homogenization of the 3D Maxwell systemnear resonances and artificial magnetism, C. R. Math. Acad. Sci., Vol 347, pp.571–576, 2009.

[BS2010] g.bouchitté, b.schweizer, Homogenization of Maxwell?s equations in a split ringgeometry, Multiscale Model. Simul., Vol 8, pp.717–750 2010.

[SPVNS96] d.r.smith, w.j.padilla, d.c.vier, s.c.nemat-nasser, s.schultz, Compositemedium with simultaneously negative permeability and permittivity, Phys. Rev. Lett., Vol 84, 2000.

[LS2016] a.lamacz, b.schweizer, A negative index meta-material for Maxwell?s equations,SIAM J. Math. Anal., Vol 48, pp.4155–4174 2016.

R.C.: Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom.E-mail address: r.craster@imperial.ac.uk

H.H.: Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom.E-mail address: h.hutridurga-ramaiah@imperial.ac.uk

Characterization of spatially graded metamaterial – Prof Richard Craster and Dr

Andrea Colombi

Elastic metamaterials are an exciting and novel branch of wave physics devoted to the study of

heterogeneous and complex media able to control the propagation of mechanical waves at different

scales and in various applications. Wave control is made possible by a microstructure made of

subwavelength resonators, randomly arranged in the media, leading to extraordinary propagation

properties. The result is that waves can be stopped, slowed down, rerouted around an obstacle

or directed to a target. The potential for applications is high, involving several fields such as

electromagnetism, optics, acoustics and elasticity. On large scales, we can aim at controlling seismic

waves and ground vibrations. In ultrasonic (kHz-MHz) situations, strongly attenuating media can

be developed starting from metamaterials, while in hypersonics (GHz) we aim at developing of very

sensitive nanosensors. Despite the differences in lengthscale, the underlying physics remain very

similar across the applications and scales, with the main complexity given by the accurate modeling

of the resonant microstructure.

This project consists of the characterization of a recently developed type of metamaterial (resonant

metawedge), where the size and properties (i.e. the resonant frequency) of the resonators vary

spatially, giving rise to a material with a spatially graded refractive index. The main investigation

methods involve spectral element simulations and modal analysis (based on finite elements). De-

pending on the student interests, the project may include an analytical study using a simplified

metamaterial model made of masses and springs attached to a plate.

The student will join a multidisciplinary team where mathematicians, engineers and acousticians

work together on wave physics and metamaterials. In the course of the project there will be the

possibility of participating in a laboratory experiment where the spatially graded metamaterial will

be tested.

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Waves in Photonic and Phononic Crystals

Prof. Richard Craster and Dr. Mehul Makwana

November 18, 2017

Project Definition

This project is about extraordinary wave transport properties within contin-uum electronic and acoustic systems, which have various applications to opto-electronic devices. Quite remarkably it is becoming possible to design materialswith properties that are not possible in nature, examples being materials withnegative effective mass, or negative refractive index; these are being used in de-vices. Our aim is to model these exotic materials, design new ones, and interactwith groups that build devices. This can be approached from several differentangles, each could be a project in its own right:

• Engineering mathematics: There are numerous modelling problems involv-ing waves propagating through finite “crystals” of microstructured mate-rials and these can be approached analytically, using asymptotic methodsor numerically.

• Scientific computation: We are developing software to solve these problemssystematically in a general manner. The numerical algorithms are basedon applied mathematical methods and there is scope to generalise themethods and implement them.

• Mathematical modelling: Concepts often seen in a mathematics degree,such as group theory, are very useful when dealing with periodic mediaon a lattice. This does not require a deep Pure mathematics knowledgeof group theory, but is an application of it to an area of physics. Bymanipulating the symmetries of the lattice structure unidirectional edgestates can be produced and their effects amplified. So there is scope for amathematics project that draws upon these ideas and blends it with thephysics application.

• Mathematical physics: The area of topological insulators in solid statephysics and condensed matter theory is vibrant and many exciting ideasare emerging, one of which “topologically protected edge states” has beenvery influential. The ideas behind this are now moving into other areas ofphysics such as the photonic and phononic crystals. There are differenceswhen dealing with the continuum cases and there is scope here for moving

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ideas from quantum mechanics to continua and we want to explore thisaspect.

There will be opportunities to interact with a vibrant research group andattend weekly group meetings. The project could also involve collaboration withthe Physics department or a company that designs these materials.

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Numerical simulations of a pedestrian model including congestion effects – Prof De-

gond

In this project, we are interested in the modelling of pedestrian flows by means of macroscopic

equations (i.e., continuous models). Such models consist of partial differential systems for the mean

density and/or the mean velocity of the pedestrians mainly in two space dimensions and occasionally

also in one-dimensional settings. Many such models have been proposed in the literature but, in

general, they fail to describe the regions where the concentration of pedestrians is high and where

safety is the most at risk.

The model we propose is inspired by the so-called Aw-Rascle (AR) model of road traffic [1], which

considers the mean car density and the mean desired velocity of drivers. The actual velocity of

traffic is less than the desired velocity by a quantity (the velocity offset) which accounts for the

effect of congestion. In the AR model, this velocity offset is taken as a direct function of the local

car density. In the proposed new the velocity offset is taken as the spatial gradient of a congestion

cost function depending on the density and possibly also on the actual speed of pedestrians. This

leads to a nonlinear system of parabolic type that seems at least formally well-posed.

In previous Msc projects [3, 4], the mathematical analysis of the equations were devloped. The

present project aims at the development of numerical simulations and testing of the model in

practical situations. More precisely, the project has two different aspects: (i) The realisation of

numerical simulations in one and two dimensions to compare the behaviour of this model with that

of the AR model and to assess the performances of the model in classical benchmark situations

(circulation in lanes, exit scenario, etc) (ii) The study of the transition between free and congested

traffic.

This project requires knowledge some experience of programmation of scientific computing prob-

lems. The student will be supervised by Prof. Pierre Degond (pdegond@imperial.ac.uk) and

Research Associate Pedro Aceves-Sanchez (p.aceves-sanchez@imperial.ac.uk). He/she will benefit

of the team environment. The simulations will be run on the teams dedicated computer server.

References:

[1] A. Aw and M. Rascle, Resurrection of “Second Order” Models of Traffic Flow, SIAM J. Appl.

Math., 60 (2000) 916938.

[2] N. Brigouleix, A new model for pedestrian traffic, Msc report, 2016.

[3] B. Cavin, A new pedestrian model with congestion, Msc report, 2017

Contact:

Pierre Degond : pdegond@imperial.ac.uk

Pedro Aceves-Sanchez : p.aceves-sanchez@imperial.ac.uk

Numerical simulations of active suspensions in fluids – Prof Degond

19

‘Active particles’ is a generic name for agents producing their own motion, such as micro-organisms.

Active particles immersed in a fluid are ubiquitous in nature from bacteria suspended in water or

sperm. There is considerable effort from the scientific community to produce realistic models of

active particle suspensions in fluids (see review [1]). Generically, the models are either individual-

based models, which consist of differential equations for the motion of each individual particle,

or continuum models, which are partial differential equations for mean quantities such as the lo-

cal density or mean particle velocity. Macroscopic models are obtained by coarse-graining the

individual-based model. Through this procedure, macroscopic observations can inform us on indi-

vidual agents behavior.

The coarse graining of active particle models is difficult but the Imperial College team has realized

it for the Vicsek model, which describes self-propelled particles locally aligning with their neighbors

[2]. The coarse-graining of the Vicsek model leads to compressible fluid-like equations called the

Self-Organized Hydrodynamic model (SOH). However, for suspensions of active particles in a fluid,

it is necessary to include the influence of the fluid. In [3], both the coupling of the Vicsek individual-

based model and that of the continuum SOH model with an incompressible fluid are performed.

In this project we will focus on a description of the fluid through Darcys law, which is valid for

instance in a very shallow fluid on which the bottom exerts a strong friction.

The goal of this project is to develop simulations of the coupled Vicsek-Darcy system on the one

hand, and of the coupled SOH-Darcy system on the other hand, in two dimensions and in a simple

geometry. The goal is to document what differences the coupling with the fluid bring to the system

compared to the case where the particle motion is uncoupled and how these differences translate

at the individual-based model level and at the continuum model level.

The student will benefit from the availability of a library of programmes developed in the team.

In particular, two dimensional codes for the Vicsek model, for the SOH model and for the Darcy

law already exist. The student will have to enrich the current Vicsek and SOH models with the

coupling with the fluid on the one hand, and the current Darcy model with the driving force due

to the presence of the active particles. Among possible applications of this work, a connection

with biologists working on collective sperm cell swimming will be made through the EPSRC grant

modelling sperm-mucus interaction across scales that involves E. Keaveny from the Department.

Research Associates Sara Merino and Pedro Aceves-Sanchez will be involved in the mentoring of

the project.

References:

[1] D. Saintillan and M. J. Shelley, Theory of active suspensions, in Complex Fluids in Biological

Systems, Biological and Medical Physics, Biomedical Engineering, S.E. Spagnolie (ed.), Springer,

2015.

[2] P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction,

Math. Models Methods Appl. Sci., 18, Suppl. (2008), pp. 1193-1215.

[3] P. Degond, S. Merino-Aceituno, F. Vergnet, H. Yu, Coupled Self-Organized Hydrodynamics and

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Stokes models for suspensions of active particles. https://arxiv.org/abs/1706.05666

Contact:

Pierre Degond : pdegond@imperial.ac.uk

Pedro Aceves-Sanchez : p.aceves-sanchez@imperial.ac.uk

Sara Merino-Aceituno : s.merino-aceituno@imperial.ac.uk

PT-symmetric Quantum Mechanics (Dr Christopher Ford)

Background

In both classical and quantum mechanics the potential energy, V , is assumed to be real. In classical

mechanics a complex potential energy would lead to apparently absurd equations of motion and

in quantum mechanics a complex V would lead to complex energy levels and non-unitary time

evolution.

In the 1990s it was realised that the Schrodinger equation with the imaginary potential energy

V (x) = ix3

actually yields real and positive energy levels. Bender and Boettcher considered a more general set

of complex potentials of the form

V (x) = −(ix)N ,

where N is a positive number (not necessarily integer). This potential has the property that it is

PT-symmetric meaning that the potential energy is invariant under a combined reversal of space

and time coordinates. Using this PT-symmetry Bender and Boettcher argued that the quantum

theory based on this complex potential has real energy levels if N ≥ 2.

Remarkably, it is possible to see a hint of this transition in the classical trajectories derived from

the potential. As the potential is complex the solutions of the equation of motion trace curves in

the complex plane. If N ≥ 2 there are closed orbits in the complex plane when the energy, E, is

real. If N < 2 there are no closed orbits for real E.

Objectives

To understand the the quantum theory for integer N . This can be through the complex Schrodinger

equation or alternatives such as the Feynman path integral and resurgence theory.

References

[1] C. Bender, S. Boettcher and P. Meisinger, Journal of Mathematical

Physics, Volume 40, number 5 (1999). This paper and other useful information is available at the

homepage of Carl Bender (University of Washington in St. Louis).

21

Yang-Mills in a Box (Dr Christopher Ford)

Background

Yang Mills theory is a non-linear generalisation of electrodynamics. In the Standard Model of

particle physics interactions between spin 12 particles are mediated by Yang-Mills fields. However,

sixty years after the theory was developed a full understanding of quantum Yang-Mills theory is

lacking. The main difficulty is to extract the infrared or long-distance properties. The problem

is present regardless of the matter content. Indeed, it is still there in the case of pure Yang-Mills

theory which comprises Yang-Mills fields without any additional matter fields.

In the early 1980s Gerard ’t Hooft [1] suggested that one could get a grip on these infrared problems

by quantising the theory on a Euclidean four-torus rather than R4. A four-torus can be viewed

as a four-dimensional box with opposite ‘faces’ identified. ’t Hooft also considered classical Yang-

Mills theory on a four torus. He found some simple solutions with remarkable properties including

constant field strength, self-duality and fractional topological charge.

A four-torus has four periods (or ‘edge-lengths’ when viewing the torus as a box) and classical

solutions are known to exist for arbitrary periods. However, ’t Hooft’s solutions are only valid for

a restricted choice of periods. Morover, the solutions that ’t Hooft wrote down remain the only

known analytical solutions of Yang-Mills theory on a torus.

Objectives

To understand the Lagrangian formulation of pure Yang-Mills theory and the definition and con-

struction of self-dual solutions. To understand how this works for a Euclidean four-torus and study

’t Hooft’s solutions. For the gauge group SU(2), investigate the properties of charge 12 solutions

with arbitrary periods.

Reference

[1] G. ’t Hooft, “Some Twisted Selfdual Solutions for the Yang-Mills Equations on a Hypertorus,”

Commun. Math. Phys. 81, 267 (1981).

Projects in quantum theory (Dr E. M. Graefe)

In my research group we devise and investigate models to describe all sorts of quantum phenomena.

We are in particular interested in quantum dynamics, and in the properties of open quantum

systems (with losses and gains of particles or systems coupled to heat baths). Another focus of

our research is the correspondence between quantum systems and their classical counterparts, in

particular in the realm of chaos.

The projects below offer the opportunity to learn more about quantum mechanics, one of the most

successful theories of modern physics, while applying and developing tools from different branches

22

of mathematics, such as spectral theory, dynamical systems, and geometry. These projects are

“real” research problems linked to other research currently on-going in the group, and offer a great

opportunity to obtain original new results.

While the projects do not require a big amount of background knowledge, a minimum working

knowlegde of quantum mechanics and a willingness to use numerical tools are prerequistes.

Project 1: Quantum rates - Quantum dynamics in a double well potential at finite

temperatures

Background. The understanding of the diffusion properties and mechanisms of hydrogen in metals

is an important scientific challenge with applications in many industries and technologies. The

nature of the problem is such that quantum effects become important even at typical room tem-

peratures. This links the problem to a research topic of a more fundamental type, regarding the

description and simulation of transport phenomena in quantum systems at finite temperature. A

main challenge here is the provision of a quantum theory of transition rates, that counts the typ-

ical number of transitions between different states in activated processes. In classical physics, the

macroscopic transition rate can be connected to the microscopic dynamics. Here we are interested

in formulating a quantum transition rate theory along similar lines, incorporating activated quan-

tum dynamics.

Project. In this project you will focus on a toy-model of a quantum particle in a one-dimensional

double-well potential under the influence of a heat bath, which is effectively modeled by a Lind-

blad equation, or effectively an ensemble of stochastic Schrodinger equation. You will analyse the

behaviour of the system in detail and compare it to its classical counterpart. For this purpose you

will use a combination of numerical and analytical methods.

Project 2: Dissipation and losses in atom-molecule conversion in Bose-Einstein con-

densates

Background. The experimental realisation of Bose-Einstein condensates (BECs) was one of the ma-

jor achievements in physics in the late 20th century. Recent progress in confining and manipulating

BECs in optical potentials has led to a variety of spectacular results. One interesting problem

is the combination of condensed atoms into molecules, which themselves form condensates, and

the inverse process of splitting molecules into atoms. The mathematical description of interacting

atoms and molecules, which can be converted into one another, and possible dynamical schemes

for an effective conversion, is challenging and interesting at the same time. In particular the influ-

ence of noise and particle losses, which is of huge experimental relevance, is theoretically not well

understood.

Project. In this project you will familiarise yourself with different approximations for the descrip-

tion of interacting atoms and molecules in the ground state of an external potential, leading to

classical dynamics on unnusual phase spaces. You will analyse the approximation and the resulting

dynamics in detail, and then explore the influence of noise and dissipation described by Lindblad

equations.

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Project 3: PT-symmetric quantum chaos

Background. The striking difference between quantum and classical behaviour becomes most appar-

ent in the realm of chaos, an extreme sensitivity to initial conditions, which is common in classical

systems but impossible under quantum laws. The investigation of characteristic features of quan-

tum systems whose classical counterparts are chaotic lies at the heart of the flourishing research

area of quantum chaos. One important result in this field is that the statistical fluctuations of the

eigenvalues of quantum systems with classically chaotic counterparts are similar to those of the

eigenvalues of certain Hermitian random matrices (matrices whose elements are random numbers).

The surprising properties of quantum systems with balanced gain and loss (non-Hermitian, but

PT-symmetric systems) have sparked much interest recently, and new experimental areas (involv-

ing for example optical wave guides, cold atoms, and meta materials) are rapidly emerging. Here

we are interested in the hitherto nearly unexplored interplay of chaos and PT-symmetry.

Project a. PT-symmetric quantum maps. In this project you will investigate the quantum and

classical features of PT-symmetric generalisations of so-called quantum maps, described by time-

evolution operators that map the state of the system from one discrete point in time to the next.

The classical counterparts, maps on phase-space, are standard examples in the theory of chaos.

We will investigate the dynamical features of the quantum and classical maps, such as the fixed

points and their stability as well as the occurence of chaos and dissipative attractors. We will fur-

ther investigate the spectral features of the quantum systems and try to identify possible universal

behaviour.

Project b. Non-Hermitian random matrix ensembles. In this project you will investigate the eigen-

value statistics of two types of non-Hermitian random matrices known as the real and complex

Ginibre ensembles. These are very simple in structure, comprising of matrices whose elements

are identically and independently normal distributed complex or real numbers. The properties of

their eigenvalues, however, are still not well understood. You will combine a mixture of numerical

and analytical tools to investigate the eigenvalues of these matrices, with a particular focus on

identifying universal features that might be observed as well in physical model systems.

Projects in automated numerical PDE methods. - Dr David Ham

The numerical solution of PDEs is a key problem of mathematical computing. Many of the largest

supercomputers are dedicated to this task for applications including weather forecasting, engineer-

ing design, and financial instrument pricing. As simulation scale and sophistication has increased,

the combination of numerical analysis, parallel algorithms and complex software engineering re-

quired has frustrated advances in this field.

However, in recent years a radical new approach has emerged. The PDE to be solved, along with its

discretisation, are specified in a high level symbolic mathematics language. High performance paral-

lel implementations are then created using specialised compilers, which combine domain knowledge

24

with cutting edge advances in parallel code generation.

Students choosing these projects have the opportunity to work on complex mathematical problems

while gaining the experience of contributing to professionally engineered open source mathematical

software as an integral part of the Firedrake development team (http://firedrakeproject.org).

The results of their work will be incorporated into released software in production use at institutions

around the world. All of the projects detailed below have the potential, if executed well, to produce

results publishable as journal papers.

Each project combines a core of numerical mathematics with significant programming, so some

level of knowledge of a language such as Python or C is a requirement.

Post-processing techniques for mixed and discontinuous Galerkin methods - withThomas Gibson

This project focuses on studying the effects of elementwise post-processing techniques for discontin-

uous Galerkin and mixed finite element methods. Procedures of this type are able to produce new

approximations, which superconverge at accelerated rates and possess better conservation proper-

ties. We restrict our focus to second order elliptic problems and possibly hyperbolic problems.

Software using the Firedrake project will be a core part of this study. We explore the effects of

different post-processing techniques of model problems and automate its implementation using a

domain-specific language (DSL) for linear algebra. These post-processing techniques may also be

incorporated in preconditioners to accelerate the convergence of iterative solvers as well.

Prerequisites: PDEs, linear algebra, functional analysis (familiarity with basic concepts), program-

ming experience (useful)

Automated differentiation for inverse problems

Inverse problems are pervasive in science and engineering: the forward simulation answers the

question “what happens if?” while the inverse problem ask “what was the cause?”. In fields as

diverse as climate science and financial mathematics, we need to invert simulations to find the

causes of phenomena. In engineering, optimal design requires inverse simulations to design the

system which best produces a desired outcome.

A key requirement in inverse simulation is to differentiate the model. For an automated system

such as Firedrake, this requires the symbolic mathematics code that Firedrake programs are writ-

ten in to be differentiated automatically using techniques from computer algebra. Many parts of

Firedrake are already automatically differentiable, but important holes remain. This project will

enable a student to learn about inverse simulation techniques while contributing new automatic

differentiation capabilities to the Firedrake system.

25

MSc project topics with Professor Darryl D Holm in Geometric Mechanics

The shape of water, metamorphosis and infinite-dimensional geometric mechanics Whenever we say the words ”fluid flows” or ”shape changes” we enter the realm of infinite-dimensional geometric mechanics. Water, for example, flows. In fact, Euler’s fluid equations tell us that water flows a particular way. Namely, it flows to get out of its own way as efficiently as possible. The shape of water changes by smooth invertible maps called diffeos (short for diffeomorphisms). The flow responsible for this optimal change of shape follows the path of shortest length, the geodesic, defined by the metric of kinetic energy. Not just the flow of water, but also the optimal metamorphosis of any shape into another follows one of these optimal paths. This project will study the commonalities between fluid dynamics and shape changes and will use the methods that are most suited to fundamental understanding - the methods of geometric mechanics. In particular, the main approach will use momentum maps and geometric control for steering along the optimal paths from one shape to another. The approach will also use emergent singular solutions of the initial value problem for a nonlinear partial differential equation called EPDiff. The EPDiff equation governs metamorphosis along the geodesic flow of the diffeos. The main application will be in the registration and comparison of Magnetic Resonance Images for clinical diagnosis and medical procedures. Extreme events in potential vorticity gradients that are stirred, but not mixed The evolution at http://www.met.rdg.ac.uk/Data/CurrentWeather/ shows satellite data indicating the development of large gradients of both potential vorticity (PV) and potential temperature (PT) in the Earth’s stratosphere. Recent work of Gibbon and Holm posted at http://arxiv.org/abs/0911.1476 proposes an equation governing the dynamics of this process. The numerical integration of this equation is required in order to investigate this PV and PT stirring dynamics in the stratosphere and to compare it with these observations.

Cell diversity in tumour growth

Supervisor: Professor Henrik Jeldtoft Jensen

As a cancer tumour grows, evolutionary dynamics involving mutations may lead to a divers collec-

tion of cell types. See e.g. [1] The project will use the framework of the spatial Tangled Nature

model of evolutionary ecology, see [2], to investigate the co-evolutionary aspects of tumour het-

erogeneity. Phenomenological insights will be obtained by further development and simulation of

existing C-codes. Some analytical analysis may also be possible at mean field level.

References

[1] https://www.quantamagazine.org/studies-reveal-extreme-diversity-of-cancer-cells-20131113/

[2] D. Lawson and H.J. Jensen, The species-area relationship and evolution. J. Theor. Biol., 241,

590-600 (2006).

Information theoretic analysis of EEG time series

Supervisor: Professor Henrik Jeldtoft Jensen

The project will use information theoretic causal measures to investigate EEG time series from

musicians and possibly also from schizophrenic patients. First the information theoretic formalism

(see e.g. [1]) will be studied, this will be followed by data analysis of various data sets. In particular

we’ll try to relate the analysis to the current suggestion that schizophrenia is related to an increase

in fluctuations of the dynamics of the brain, see e.g. [2].

References

[1] X. Wan, B Cruts and H.J. Jensen, The causal inference of cortical neural networks during music

improvisations. PLoS ONE DOI:10.1371/journal.pone.0112776 December 9, 2014. arXiv:1402.5956.

[2] Front Psychiatry. 2015; 6: 92. Published online 2015 Jun 24. doi: 10.3389/fpsyt.2015.00092

Simulation Based Inference for Molecular Machines – Dr David Rueda (MRC), Dr

Tom Ouldridge, Dr Nick Jones

While nanomachines offer the prospect of technological revolution the characterization of their

dynamics remains a challenge. We will attempt to understand the underlying potential surfaces

associated with high resolution single molecule data gathered in the laboratory of David Rueda.

We will develop a simulation based inference pipeline that allows us, given a belief about an under-

lying energy surface, to simulate real data. This approach draws on tools from stochastic processes,

27

statistical physics and simulation based inference (we will likely use Approximate Bayesian Com-

puting). Deploying this suite of tools will be a unique piece of progress in single molecule data

analysis and so, beyond allowing us to understand the behaviour of a particular nanomachine, the

will thus be of interest to a large scientific community: solid progress is likely to lead to publication

in widely consumed journals. Concrete examples will be drawn from a variety of Biological systems

that play key roles in cellular function and disease, such as polymerases, helicases or deaminanases.

4D Cell Dynamics – Dr Silvia Santos (MRC), Dr Nick Jones

The cell-cycle is the sequence of steps associated with cell division. It is a cannonical process in

q-bio and its malfunction is linked to cancer. The group of Silvia Santos has collected durations of

each of the 4 successive stages of the cell cycle for large sets of individual cells. A population of cells

thus becomes a set of points in this 4D duration space. If cells were strictly identical then each cell

would be mapped to a single point in this space: but in fact there is pronounced variability. What

is the character of this variation? Of what dynamics is it the consequence? Dr Santos has further

collected these 4D point clouds for cancerous cells and stem cells. The student will investigate the

structural differences between the point-clouds associated with each of these distinct cell types and

attempt to construct simple models to account for the structure within the data and between cell

lines. The group of Nick Jones has previously argued that the spread of these point clouds can

be attributed to mitochondrial variability, the student will be given opportunities to explore this.

Dr Santos can also collect data for the cell-cycle-stage durations of a single cell through multiple

divisions: we can thus access dynamics in this 4D space. The data collected in the Santos lab is

unique and yet it addresses a core problem in biology: solid progress is likely to lead to publication

in widely consumed journals.

28

CHARACTERIZING BASINS OF ATTRACTION IN NONLINEAR

DYNAMICS: SIZE, SHAPE AND APPLICATIONS.

DANTE KALISE AND DIEGO OYARZUN

Multistationarity is a hallmark of nonlinear dynamics appearing in all domainsof science. In this project we will develop foundational methods to characterize thebasins of attraction of nonlinear systems. We will apply the results in bacterialgene circuits [1] and ecological models for the Amazonian rain forest [2].

We will begin using deflation-based approaches to compute the multiple equilib-rium points [3]. To determine the size and shape of basins of attraction, we willborrow tools from optimal control theory and dynamic programming through theso-called Zubov method [4]. Further research directions include: a) the develop-ment of high-order semi-Lagrangian methods with accelerated iterative techniques[5], and b) the study of high-dimensional nonlinear systems.

This project will produce efficient numerical methods to compute domains ofattraction applicable across a wide range of scientific disciplines.

Figure 1. Example nonlinear system. The surface above de-scribes the normalized time to reach the origin, which is a stableattractor of x = −x + y , y = 0.1x− 2y − x2 − 0.1x3. The basin ofattraction is the set of all points where f(x, y) < 1.

References

[1] M. Kaufman, C. Soule, R. Thomas. A new necessary condition on interaction graphs formultistationarity, Journal of Theoretical Biology, 248(4), 675–685, 2007.

[2] P. J. Menck, J. Heitzig, N. Marwan, and J. Kurths. How basin stability complements thelinear-stability paradigm, Nature Physics, 9(2), 89–92, 2013.

[3] P. E. Farrell, A. Birkisson, and S. W. Funke. Deflation Techniques for Finding DistinctSolutions of Nonlinear Partial Differential Equations, SIAM J. Sci. Comput. 37(4)(2015),

A-2026–A2045.[4] F. Camilli, L. Grune, and F. Wirth. A Generalization of Zubov’s Method to Perturbed Sys-

tems, SIAM J. Control Optim. 40(2)(2001), 496–515.[5] A. Alla, M. Falcone and D. Kalise. An efficient policy iteration algorithm for dynamic pro-

gramming equations, SIAM J. Sci. Comput., 37(1), 181-200, 2015.

1

COMPUTATIONAL METHODS FOR ACTIVE VIBRATION

CONTROL AND OPTIMAL DAMPING PLACEMENT

DANTE KALISE

Many modern engineering challenges such as the design of long-span bridges,large wind turbine blades, or noise-cancelling headphones [4] rely on effective mech-anisms for vibration mitigation or control [1]. In this project we will develop novelcomputational methods for the control of vibration phenomena [2], and the designof optimal dampers (vibration mitigation structures) [3].

We will begin by studying different models for acoustic and mechanical vibra-tion (wave equations, beams and plate models), its numerical approximation viafinite elements, and the synthesis of control strategies with techniques stemmingfrom optimal control and large-scale numerical optimisation. Further research willaddress the design of dampers in the frame of shape optimisation.

For students with a strong interest in applications, there exists the possibility ofdeveloping an industrial project on acoustic propagation.

Figure 1. Active vibration control in action. Left: the designof dissipation structures carried by means of shape optimisationtechniques cannot be recovered via traditional heuristic methods.Right: optimal feedback control effectively mitigates vibration phe-nomena in real time.

References

[1] C.R. Fuller, S.J. Elliot, P.A. Nelson. Active Control of Vibration, Academic Press, 1997.[2] E. Hernandez, D. Kalise, E. Otarola. Numerical approximation of the LQR problem in a

strongly damped wave equation, Comput. Optim. Appl. 47(1)(2010), 161-178.[3] D. Kalise, K. Sturm and K. Kunisch. Optimal actuator design based on shape calculus,

arXiv:1711.01183 (2017).[4] J. Manaster. How do Active Noise Canceling Headphones Work?, Scientific American,

available at: https://blogs.scientificamerican.com/psi-vid/how-do-active-noise-canceling-headphones-work/

1

Hydrodynamic interactions between shape changing particles – Dr Keaveny

Microorganisms and swimming cells use time-dependent deformations of their bodies to generate

the flows needed to propel themselves from one place to another. This flow then influences the

motion of nearby organisms that are also deforming themselves. This project entails exploring the

non-trivial coupling between the time-dependent flows and shape changes and how it affects the

hydrodynamic interactions between swimming cells. The project will be largely computational and

will begin by exploring the interactions between two simple model swimmers.

Swimming backwards in viscous fluids – Dr Keaveny

There are several rules of thumb regarding propulsion of microorganisms. For flagellated microor-

ganisms, the rule is that they swim in the direction opposite to that of the wave they propagate

along their flagella. This project involves exploring when this rule breaks down and the organism

seemingly swims backwards. This project will be largely computational with the primary goal

being the understanding of the mechanism behind this counterintuitive behaviour.

Fast methods, slow flows – Dr Keaveny

Large-scale simulation of suspensions of microscopic particles in fluid relies heavily upon the ability

to compute the interactions between the particles in a fast manner. The project entails exploring

one such method fast ewald summation for particles in periodic domains, as well as a recent

extension to infinite domains. The project will involve implementing this algorithm, as well as

providing a study of how well it performs with respect to other methods.

31

MSc in Applied Mathematics Dr Robert Nurnberg

MSc Projects 2017-18

1. Image segmention and image restoration with active contours

Two fundamental tasks in image processing are image segmentation and image smoothing.A natural strategy is to combine the two processes in a single step, following the idea ofthe seminal work by [MS]. They introduced the following optimization problem: Find aminimizer (u, S) of the functional

EMS(u, S) = σHd−1(S) + λ

∫

Ω(u− u0)2 dLd +

∫

Ω\S|∇u|2 dLd . (1)

Here, given an image u0 : Rd ⊃ Ω→ R, the task is to find its set of discontinuities S, anda piecewise smooth approximation u : Ω→ R of u0.

As the Mumford–Shah problem is difficult to tackle in its original form, several simplifiedmodels have been proposed. Chief among them are the models by [CV] and [TYW] which,in their simplest forms, assume that S is a closed curve Γ that partitions Ω into tworegions: Ω1 and Ω2. Moreover, u is assumed to be constant or smooth in each of the tworegions. Hence (1) reduces to

E(u,Γ) = σHd−1(Γ) + λ2∑

i=1

∫

Ω(ui − u0)2 dLd +

2∑

i=1

∫

Ωi

|∇ui|2 dLd . (2)

Possible numerical approaches to minimize (2) can be found in e.g. [DMN] and [Ben],where the latter work uses a piecewise constant approximation of u and active contoursbased on [BGN]. The aim of this project is to build on the work in [Ben], but withu = u1XΩ1 + u2XΩ2 being a piecewise smooth approximation.

Prerequisites: Good knowledge of finite differences and finite elements. Programmingskills in C, MATLAB or Python.

References

[Ben] Heike Benninghoff. Parametric Methods for Image Processing Using Actice Contourswith Topology Changes. PhD thesis, University Regensburg, Regensburg, 2015.

[BGN] John W. Barrett, Harald Garcke, and Robert Nurnberg. On the variational approxima-tion of combined second and fourth order geometric evolution equations. SIAM J. Sci.Comput., 29(3):1006–1041, 2007.

[CV] T. F. Chan and L. A. Vese. Active contours without edges. IEEE Trans. Image Process.,10(2):266–277, 2001.

[DMN] Gunay Dogan, Pedro Morin, and Ricardo H. Nochetto. A variational shape optimiza-tion approach for image segmentation with a Mumford-Shah functional. SIAM J. Sci.Comput., 30(6):3028–3049, 2008.

[MS] David Mumford and Jayant Shah. Optimal approximations by piecewise smooth func-tions and associated variational problems. Comm. Pure Appl. Math., 42(5):577–685,1989.

[TYW] A. Tsai, Jr. Yezzi, A., and A. S. Willsky. Curve evolution implementation of theMumford–Shah functional for image segmentation, denoising, interpolation, and mag-nification. IEEE Trans. Image Process., 10(8):1169–1186, 2001.

MSc in Applied Mathematics Dr Robert Nurnberg

2. Cahn–Hilliard discretizations on manifolds

This project will consider phase separation on general surfaces by deriving a finite elementapproximation for the Cahn–Hilliard equation on a given manifold Γ ⊂ Rd, d = 2, 3. Theequation is given by

ut = ∆s (−ε∆s + ε−1 Ψ′(u)) on Γ ,

where ∆s is the Laplace–Beltrami operator on Γ, Ψ is an obstacle double well potentialand ε > 0 is an interfacial parameter.

The project will introduce a mixed finite element approximation using piecewise linearfinite elements and study existence, uniqueness and stability properties of the discretesolutions. A particular emphasis of the project will be on the efficient solution of thesystems of nonlinear equations arising at each time step.

Relevant recent literature is [DJT, ER, OS]. Solution methods for the standard Cahn–Hilliard equation are discussed in e.g. [BNS, BBG, GKS].

Prerequisites: Good knowledge of finite elements. Programming skills in C, C++ orPython.

References

[BBG] Luise Blank, Martin Butz, and Harald Garcke. Solving the Cahn–Hilliard vari-ational inequality with a semi-smooth Newton method. ESAIM Control Optim.Calc. Var., 17(4):931–954, 2011.

[BNS] John W. Barrett, Robert Nurnberg, and Vanessa Styles. Finite element approxi-mation of a phase field model for void electromigration. SIAM J. Numer. Anal.,42(2):738–772, 2004.

[DJT] Qiang Du, Lili Ju, and Li Tian. Finite element approximation of the Cahn-Hilliardequation on surfaces. Comput. Methods Appl. Mech. Engrg., 200(29-32):2458–2470,2011.

[ER] Charles M. Elliott and Thomas Ranner. Evolving surface finite element methodfor the Cahn–Hilliard equation. Numer. Math., 129(3):483–534, 2015.

[GKS] Carsten Graser, Ralf Kornhuber, and Uli Sack. Nonsmooth Schur–Newton methodsfor multicomponent Cahn–Hilliard systems. IMA J. Numer. Anal., 35(2):652–679,2015.

[OS] D. O’Connor and B. Stinner. The Cahn–Hilliard equation on an evolving surface,2016. http://arxiv.org/abs/1607.05627.

Numerical solution of fractional integral and differential equations of rational order

Supervisor: Dr Sheehan Olver

Fractional derivatives and fractional differential equations (FDEs) are becoming increasingly

prevalent in the mathematical modelling of biological and physical processes. For periodic functions

they can be defined naturally in terms of the Fourier coefficients, otherwise, they can be defined in

terms of the (left) fractional integral operator (on [−1, 1]):

Qrxf(x) =∫ x

−1

f(t)

(x− t)1−r dt

via either

Dm+rx f(x) =

dm+1

dxm+1Q1−rx f(x) or Dm+r

x f(x) = Q1−rx

dm+1

dxm+1f(x)

The behaviour of these derivatives is interesting and unintuitive: for example, the half-derivative of

ex and other smooth functions have square root singularities, see left side of Figure 1. The type of

differential equations one might wish to solve include, for example, the fractional advection-diffusion

equation:

∂u

∂t= uxx +Dr

xu.

Typical numerical techniques for computing solutions to fractional differential / integral equa-

tions are finite difference or finite element-based, but these only provide low accuracy due to the

global nature of fractional derivatives: the convergence rate is slow and the discretization matrices

are dense. In recent work Hale and Olver have introduced a spectral method for half-order (r = 1/2)

fractional and differential equations built on the fact that fractional integrals of weighted ultraspher-

ical polynomials, that is, polynomials orthogonal with respect to (1− x2)λ− 12 . This spectral method

results in banded systems: the rate of convergence is fast and the discretization matrices are sparse.

This MSc. project consists of extending this work to general rational orders r = p/q. It will con-

sists of using weighted Jacobi polynomials in place of ultraspherical polynomials, that is, polynomials

orthogonal with respect to (1− x)β(1 + x)α. One difficulty is that Jacobi polynomials of parameters

α, β < −1 arise naturally: part of the project will be to understand these generalizations of Jacobi

polynomials to negative parameters via developments on Sobolev orthogonality utilized recently by

Xu.

Partial differential equations on three-dimensional simplices via multivariate or-thogonal polynomials

Supervisor: Dr Sheehan Olver

In recent work, Olver, Townsend, and Vasil have introduced an approach to solving partial

differential equations on triangles using a hierarchy of multivariate orthogonal polynomials. For

example, on the right triangle T = (x, y) : 0 ≤ x, y ≤ 1, 0 ≤ x+ y ≤ 1, we consider orthogonal

polynomials with respect to the inner product

∫∫

Tf(x, y)g(x, y)xayb(1− x− y)c dA.

1

-1.0 -0.5 0.0 0.5 1.00

1

2

3

4

exp(x)D_x^0.5[exp](x)

Figure 1: Left: The half-derivative of ex. Right: solution to Helmholtz equation in a triangle.

General linear partial differential equations such as the the variable coefficient Helmholtz equation

∆u+ a(x, y)u = f(x, y), u|∂T = 0

become banded-block-banded equations when we use two different choices of a, b, c for the bases used

to represent u and f . An example of a banded-block-banded system would have the zero structure

of the form:

× × ×× × × × ×× × × × × ×

× × × × × ×× × × × × × × ×× × × × × ×

× × × ×× × × × × ×× × × × ×× × ×

u00

u10u11

u20u21u22

u30u31u32u33

=

f00

f10f11

f20f21f22

f30f31f32f33

that is, a block tridiagonal matrix whose the blocks themselves are also tridiagonal. The sparsity

present in this structure allows for extremely high polynomial order approximation methods, with

as many as 100k unknowns. An example solution is given in the right of Figure 1 for the Helmholtz

equation.

The MSc. project consists of extending this methodology to solve PDEs on three-dimensional sim-

plices, e.g., S = (x, y, z) : 0 ≤ x, y, z ≤ 1, 0 ≤ x+ y + z ≤ 1, using multivariate orthogonal polyno-

mials with respect to the inner product

∫∫∫

Sf(x, y, z)g(x, y, z)xaybzc(1− x− y − z)d dV

where choosing (a, b, c, d) differently will reveal an underlying sparsity structure. The project will

involve generalizing the recurrence relationships derived for two-dimensional multivariate orthogonal

polynomials to three-dimensions, using these to construct matrix representations of the operators,

and solving some simple model problems like three-dimensional Poisson equation and Helmholtz

equations.

2

Biomolecular control systems – Dr Diego Oyarzun

We have a number of projects on the analysis and design of biomolecular systems in living cells. Our

projects combine analysis and simulation with ideas from control theory applied to Biotechnology

and Healthcare. You can read a general introduction to our work in this piece (http://goo.gl/tM46k4).

Network methods for precision cancer medicine. A hallmark of cancer cells are systemic

perturbations to their metabolism. In this project you will employ network-theoretic methods to

analyse published models of cancer metabolism. We aim to find signatures in the network topology

for patient-specific treatments with improved prognosis. This is a multidisciplinary project in

collaboration with Prof Mauricio Barahona (Mathematics) and Dr Hector Keun (Department of

Surgery and Cancer).

Reference: Beguerisse, Bosque, Oyarzun, Pico, Barahona. “Flux-dependent graphs for metabolic

networks”. arXiv:1605.01639, 2017.

Next-generation microbial cell factories. Genetic engineers can now design microbes that

function as chemical factories to produce biofuels or therapeutic drugs. In this project we will study

the design of dynamic cell factories that rely on feedback control to improve performance. We aim

to find optimal control architectures that enhance robustness and maximise production, combining

elements from control theory and mixed-integer optimisation.

References: Oyarzun, Stan. “Synthetic gene circuits for metabolic control: design trade-offs and

constraints”. J. R. Soc. Interface, 2013.

Noise-induced phenomena in cellular metabolism. Intracellular fluctuations are an impor-

tant source of noise in microbes. In this project you will combine stochastic simulation and markov

jump processes to identify noisy hotspots in the metabolism of bacterial cells. This project is in

collaboration with Dr Philipp Thomas (Mathematics).

Reference: Oyarzun, Lugagne, Stan. “Noise propagation in synthetic gene circuits for metabolic

control”, ACS Synthetic Biology, 2015.

Size and shape of basins of attraction. Multistability appears in many biochemical networks

involved in cellular decision-making. In this project we will develop efficient methods for computing

the basins of attraction of complex nonlinear systems. The project relies on ideas from dynamic

programming and we will apply the results to systems relevant in biotechnology and cancer. This

project is in collaboration with Dr Dante Kalise (Mathematics).

Reference: Menck, Heitzig, Marwan, Kurths. “How basin stability complements the linear-

stability paradigm”, Nature Physics, 2013.

36

Control of falling liquid films including dispersive effects - Dr. S. Gomes andProf. D.T. Papageorgiou

Project Description. The flow of a thin film down an inclined plane has many industrial applica-tions, including coating and heat transfer. These flows are unstable when the Reynolds number is largerthan a critical value depending on the slope angle. While some applications benefit from a flat film,in many cases, such as heat transfer, one wishes to explore the flows instabilities and drive the systemtowards a non-uniform state.

It is known [3,4] that same-fluid blowing and suction is an effective control strategy to achievea desired interface shape for a hierarchy of models for these flows. This project aims to extend thederivation of the controlled equations for the interface of falling liquid films done in [1] to more realisticconditions such as the presence of dispersion effects, as well as the existence of external effects such aselectric fields, as done in [2] for the uncontrolled case.

After the development of the models, the aim is to study the effect of controls, either in the weaklynonlinear regime (where the equation obtained should be of the Kuramoto-Sivashinsky-Korteweg-deVries type) or in the long-wave regime, where equations should be similar to those in [1,4].

Learning outcomes:

• Mathematical modelling of thin film flows involving physical effects such as dispersion and (ifdesired) electric fields.

• Multi-scale asymptotic analysis applied to nonlinear partial differential equations in order to derivereduced-dimension evolution PDEs.

• Application of Linear Control Theory for fluid dynamics systems.

• Numerical methods appropriate for the evolution PDEs considered, including spectral methodsand Backward Differentiation Formulae schemes.

Relevant courses: The following courses (or equivalent) from the Applied Mathematics M.Sc. pro-gram could prove useful, however not all of them are essential: Fluid Dynamics I, Fluid Dynamics II,Hydrodynamic Stability, Numerical Solution of Ordinary Differential Equations, Computational PartialDifferential Equations , Asymptotic Analysis, Introduction to Partial Differential Equations.

References

[1] Thompson, A.B., Tseluiko, D. and Papageorgiou, D.T., Falling liquid films with blowing and suction,Journal of Fluid Mechanics 787, 292-330, 2016.

[2] Tseluiko, D. and Papageorgiou, D.T., Dynamics of an electrostatically modified Kuramoto-Sivashinsky-Korteweg-deVries equation arising in falling film flows, Physical Review E 82, 016322, 2010.

[3] Gomes, S.N., Pradas, M., Kalliadasis, S., Papageorgiou, D.T. and Pavliotis, G.A., Controlling spa-tiotemporal chaos in active dissipative-dispersive nonlinear systems, Physical Review E 92 , 022912,2015.

[4] Thompson, A.B., Gomes, S.N., Pavliotis, G.A. and Papageorgiou, D.T., Stabilising falling liquidfilm flows using feedback control, Physics of Fluids, 28 012107, 2016.

Transition to chaos in model spatially-developing flows

Supervisors: Prof. D. T. Papageorgiou and Dr. P. Ray

Project Description

Consider the figure below. The image on the left is a bifurcation diagram for the logistic map, xn+1 =rxn(1 − xn); this map was analyzed by Feigenbaum 40 years ago in his foundational study on the period-doubling route to chaos [1]. The image on the right illustrates the spatial development of a fluid boundary layeras it transitions from a steady, laminar flow to fully-developed turbulence. The basic question motivating thisproject is, can the bifurcation diagram on the left at all describe the dynamics depicted in the boundary layeron the right? Boundary layers have great practical significance – consider water flowing through pipes in yourhome, air flow over an aircraft wing, or hurricanes after landfall. While these flows are undoubtedly important,they are also enormously complicated and require simulation and analysis of the 3-D Navier-Stokes equations. Inthis project, we make a pragmatic compromise and focus on the spatially-developing 2-D Kuramoto-Sivashinsky(K-S) equation, ut + (u + c)ux +∇2u +∇4u = 0, which retains many important features of the Navier-Stokesequations but is simpler to analyze and simulate. Transition to chaos in the K-S system has already beeninvestigated for confined dynamics on a periodic domain, x ∈ [0, L) [2], and in this project, we will analyzeopen flows developing along the half line, x ∈ [0,∞). Numerical simulations will be used as a ‘laboratory’for investigating transition scenarios. Statistical methods and nonlinear time series analysis will be applied tosimulation results and connections to insights gained from both chaos theory and linear stability analysis willbe explored and explained.

Learning Outcomes

There will be several learning outcomes emerging from this project:- You will learn about numerical methods for nonlinear PDEs and acquire proficiency in scientific computing- You will also learn about nonlinear time series analysis and statistical analysis of complex spatio-termporaldata- You will learn about linear stability analysis of spatially developing flows and asymptotic methods used toconnect linear theory to observed nonlinear dynamics.

Background: The following courses (or equivalent) from the Applied Mathematics program could proveuseful, however not all are essential: Fluid Dynamics I/II, Hydrodynamic Stability, Asymptotic Analysis,Numerical Solution of ODEs, Computational PDEs. Some programming experience is essential.

References

[1] S.H. Strogatz, Nonlinear dynamics and chaos, 2000.

[2] Y.-S. Smyrlis and D.T. Papageorgiou. Predicting chaos for infinite-dimensional systems: TheKuramoto-Sivashinsky equation, a case study. Proc. Natl. Acad. Sci. USA, 88:11129-11132, 1991.

Dr S. Gomes, Professor G.A. Pavliotis Title: Mean field limits for interacting agents and phase transitions Systems of interacting diffusions arise in many areas of applications, ranging from plasma physics and stellar dynamics to machine learning and mathematical models in the social sciences. Examples of such a modern application are the modelling of opinion formation in societies and the modelling of systemic risk in financial networks. Assuming that the agents are interacting weakly, it is possible to pass to the (thermodynamic) limit of infinitely many agents and to obtain a Fokker-Planck-type equation for the probability distribution function. This equation, the McKean-Vlasov-Fokker-Planck equation, is a nonlinear nonlocal partial differential equation. Depending on the strength of the interaction between the agents, this equation can exhibit more than one stationary distributions. The passage from one to multiple stationary states is related to the presence of phase transitions. Such abrupt changes in the number and structure of stationary distributions have a profound effect on the properties of the interacting agents. The goal of this project is to investigate phase transitions for systems of interacting agents, using a combination of numerical simulations and analytical techniques. Applications to the modelling of opinion formation and to models for systemic risk will also be explored. References Garnier, Josselin; Papanicolaou, George; Yang, Tzu-Wei Large deviations for a mean field model of systemic risk. SIAM J. Financial Math. 4 (2013), no. 1, 151–184. Garnier, Josselin; Papanicolaou, George; Yang, Tzu-Wei Consensus convergence with stochastic effects. Vietnam J. Math. 45 (2017), no. 1-2, 51–75.

Dawson, Donald A. Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys. 31 (1983), no. 1, 29–85. S. N. Gomes, G. A. Pavliotis Mean Field Limits for Interacting Diffusions in a Two-Scale Potential , J. Nonlinear Science, to appear (2017). Available from arXiv:1707.06713 Professor G.A. Pavliotis Machine learning, global optimization and sampling Two problems that arise very frequently in applications are those of sampling from a probability distribution in a high dimensional state space and of finding the minimum of a high dimensional, possibly non-convex function. The standard methodology for sampling from a probability distribution is that of Markov Chain Monte Carlo, which is based on running a Markov chain that is ergodic with respect to the probability distribution from which we want to sample. On the other hand, the stochastic gradient descent algorithms (which can also be interpreted in terms of a Markov process), is the main algorithm that is used for global optimization. The goal of the proposed project is to investigate the connections between Markov Chain Monte Carlo, stochastic gradient descent and machine learning. References: Underdamped Langevin MCMC: A non-asymptotic analysis. X. Cheng, N. Chatterji, P. Bartlett, and M. I. Jordan. arxiv.org/abs/1707.03663, 2017. Gradient descent converges to minimizers. J. Lee, M. Simchowitz, M. I. Jordan, and B. Recht. Proceedings of the

Conference on Computational Learning Theory (COLT), New York, NY, 2016. A Complete Recipe for Stochastic Gradient MCMC Authors: Yi-an Ma, Tianqi Chen and Emily FoxConference: Advances in Neural Information Processing Systems 28Year: 2015Pages: 2899--2907

TURBULENT DIFFUSION: ANALYSIS AND SIMULATIONS

HARSHA HUTRIDURGA AND GRIGORIOS PAVLIOTIS

Transport of physical quantities such as heat or solutes immersed in a fluid flow has received muchattention from the scientific community, thanks to its wide range of applicability in engineering andweather sciences. An advection-diffusion equation for a so-called “passive scalar field” is a simplemathematical model to describe and understand the interplay between molecular diffusion and theadvective fluid field.

This project will consider such transport models with incompressible advective fields. Particularemphasis is to study the effect of heterogeneous fluctuations in the advective fluid field on thepassive scalar field. This project will use some basic mathematical tools from the theory of “ho-mogenization” – both periodic and stochastic – to understand some intricate details in the theoryof turbulent diffusion.

From the mathematical analysis perspective, this project should aim to(i) gather and comprehend relevant literature on the “turbulent moment closure problem”.(ii) perform computations in the spirit of “matched asymptotic” for advection dominated trans-

port equations with velocity fields having multiple scales.(iii) comprehend a recent technique of using non-trivial mean flows in homogenizing strong

advection problems.From the computational perspective, this project should(I) understand the “multiscale finite elements method” to solve some elliptic problems.(II) develop efficient numerical schemes to compute effective diffusion tensor in the non-trivial

mean flow scenarios.(III) use well-known Monte carlo methods for generating stationary random velocity fields.(IV) review some well-known test cases in simulating turbulent diffusion.This project is quite challenging as it blends both the mathematical analysis of differential

equations and some new computational techniques for resolving (numerically) differential equationswith highly heterogeneous coefficients. A successful completion of this project should result in ajournal publication.

Pre-requisites: Student taking on this project is required to have some basic understanding ofdifferential equations and fluid mechanics. It is also preferable to have some knowledge on MonteCarlo simulations.

1

2 HARSHA HUTRIDURGA AND GRIGORIOS PAVLIOTIS

References:

[1] r.h.kraichnan, Diffusion by a random velocity field, The Physics of Fluids, Vol 13, 1970.

[2] a.majda, p.kramer, Simplified models for turbulent diffusion: theory, numerical modellingand physical phenomena, Physics reports, Vol 314, Number 4, pp.237–574, 1999.

[3] r.carmona, f.cérou, Transport by incompressible random velocity fields: Simulations andmathematical conjectures, Stochastic Partial Differential Equations: Six Perspectives, Amer. Math.Soc., Providence, pp.153–181, 1999.

[4] y.efendiev, t.hou, Multiscale finite element methods. Theory and applications, Surveysand Tutorials in the Applied Mathematical Sciences, 4, Springer, New York, 2009.

[5] t.holding, h.hutridurga, j.rauch, Convergence along mean flows, SIAM J. Math. Anal.,Vol 49, Issue 1, pp.222–271, 2017.

H.H.: Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom.E-mail address: h.hutridurga-ramaiah@imperial.ac.uk

G.P.: Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom.E-mail address: g.pavliotis@imperial.ac.uk

Approximation of nonautonomous invariant manifolds – Dr Rasmussen

In nonlinear dynamical systems, invariant manifolds are omnipresent and play a crucial role in a

variety of ways for local as well as global questions: For instance, local stable and unstable manifolds

dictate the saddle point behaviour in the vicinity of hyperbolic solutions (or surfaces), and center

manifolds are a primary tool to simplify given dynamical systems in terms of a reduction of their

state space dimension. Concerning a more global perspective, stable manifolds serve as separatrix

between different domains of attractions. This project aims at computing invariant manifolds for

time-variant discrete dynamical systems numerically, by using and extending results from [1]. Here

a truncation of the Lyapunov-Perron operator, used for the construction of invariant manifolds,

results in a system of nonlinear algebraic equations which can be solved both locally using Newton,

and globally using continuation algorithms, yielding both local and global approximations of the

desired invariant manifold. The project aims in particular at using continuation techniques to study

approximations of one- and two-dimensional invariant manifolds. References: [1] C. Poetzsche

and M. Rasmussen, Computation of nonautonomous invariant and inertial manifolds, Numerische

Mathematik 112 (2009), no. 3, 449-483.

Morse decompositions of nonautonomous set-valued dynamical systems – Dr Ras-

mussen

The global asymptotic behaviour of dynamical systems on compact metric spaces can be described

via Morse decompositions. Their components, the so-called Morse sets, are obtained as intersections

of attractors and repellers of the system. This project aims at generalisations of the classical

theory to dynamical systems that are both nonautonomous and set-valued, by extending results

from [1], [2] and [3]. [1] R.P. McGehee and T. Wiandt, Conley decomposition for closed relations,

Journal of Difference Equations and Applications 12 (2006), no. 1 1-47. [2] M. Rasmussen, Morse

decompositions of nonautonomous dynamical systems, Transactions of the American Mathematical

Society 359 (2007), no. 10, 5091-5115. [3] Yejuan Wang, Desheng Li, Morse Decompositions for

Nonautonomous General Dynamical Systems, Set-Valued and Variational Analysis 22 (2014), 117-

154.

44

Contact networks and disease transmission in pedestrian flows

Supervisor: Dr. Prasun Ray

Project Description

Illustration of a pedstrian (p1) analyzing herpath (taken from [2])

Analysis and optimization of pedestrian flows is fundamen-tally important for both urban design and public health andsafety. ‘Great’ cities facilitate physical in-person contact [1], how-ever large levels of such contact also lead to higher rates of dis-ease transmission. A challenging design problem follows imme-diately – how can we generate efficient, high-density, and safepedestrian flows? This project approaches this problem in threesteps. First, by simulating pedestrian flows in canonical configu-rations such as uni- and bi-directional traffic in a hallway usingthe descision-based model developed in [2]. It is well known thattraffic waves and turbulence can develop under certain conditionsin these flows, and the second step is to analyze the contact net-works that form as waves and turbulence develop. Relevant ques-tions to be considered are: which other walkers does a pedestriancome into contact with? What are the durations of these interac-tions? The third step is to couple the contact network results todisease transmission models in order to characterize the degree towhich simulated traffic facilitates epidemics and the spreading ofinfectious diseases. There is also substantial scope to move beyond this outline and tailor the project to studentinterests.

Learning Outcomes

There will be several learning outcomes emerging from this project:· You will learn about numerical methods for large systems of ODEs and optimzation; you will also acquireproficiency in scientific computing· You will learn about mathematical epidemiology and network science· You will also learn about the dynamics of linear and nonlinear waves in the context of pedestrian traffic

Background: The following courses (or equivalent) may be useful, however not all are essential:Hydrodynamic Stability, Numerical Solution of ODEs, Computational PDEs, Scientific Computing. Someprogramming experience (e.g. Matlab or Python) will be helpful.

References

[1] Sim, A., Yaliraki, S. N., Barahona, M., & Stumpf, M. P. H. (2015). Great cities look small. Journal of TheRoyal Society Interface, 12(109), 20150315.

[2] M. Moussaid, D. Helbing, & G. Theraulaz. How simple rules determine pedestrian behavior and crowddisasters. Proc. Natl. Acad. Sci. USA, 108::6884-6888 2011.

Transition to chaos in model spatially-developing flows

Supervisors: Prof. D. T. Papageorgiou and Dr. P. Ray

Project Description

Consider the figure below. The image on the left is a bifurcation diagram for the logistic map, xn+1 =rxn(1 − xn); this map was analyzed by Feigenbaum 40 years ago in his foundational study on the period-doubling route to chaos [1]. The image on the right illustrates the spatial development of a fluid boundary layeras it transitions from a steady, laminar flow to fully-developed turbulence. The basic question motivating thisproject is, can the bifurcation diagram on the left at all describe the dynamics depicted in the boundary layeron the right? Boundary layers have great practical significance – consider water flowing through pipes in yourhome, air flow over an aircraft wing, or hurricanes after landfall. While these flows are undoubtedly important,they are also enormously complicated and require simulation and analysis of the 3-D Navier-Stokes equations. Inthis project, we make a pragmatic compromise and focus on the spatially-developing 2-D Kuramoto-Sivashinsky(K-S) equation, ut + (u + c)ux +∇2u +∇4u = 0, which retains many important features of the Navier-Stokesequations but is simpler to analyze and simulate. Transition to chaos in the K-S system has already beeninvestigated for confined dynamics on a periodic domain, x ∈ [0, L) [2], and in this project, we will analyzeopen flows developing along the half line, x ∈ [0,∞). Numerical simulations will be used as a ‘laboratory’for investigating transition scenarios. Statistical methods and nonlinear time series analysis will be applied tosimulation results and connections to insights gained from both chaos theory and linear stability analysis willbe explored and explained.

Learning Outcomes

There will be several learning outcomes emerging from this project:- You will learn about numerical methods for nonlinear PDEs and acquire proficiency in scientific computing- You will also learn about nonlinear time series analysis and statistical analysis of complex spatio-termporaldata- You will learn about linear stability analysis of spatially developing flows and asymptotic methods used toconnect linear theory to observed nonlinear dynamics.

Background: The following courses (or equivalent) from the Applied Mathematics program could proveuseful, however not all are essential: Fluid Dynamics I/II, Hydrodynamic Stability, Asymptotic Analysis,Numerical Solution of ODEs, Computational PDEs. Some programming experience is essential.

References

[1] S.H. Strogatz, Nonlinear dynamics and chaos, 2000.

[2] Y.-S. Smyrlis and D.T. Papageorgiou. Predicting chaos for infinite-dimensional systems: TheKuramoto-Sivashinsky equation, a case study. Proc. Natl. Acad. Sci. USA, 88:11129-11132, 1991.

Two problems in stochastic models of cellular size control – Dr Shahrezaei

In these projects, we will make computational stochastic models of cellular size control in bacteria

and yeast. In bacteria, the DNA replication is thought to be coupled to cell size and we will explore

the mechanism and consequence of this for cell size control. Here, we will use agent based models

of cell growth and division and we explore models that produce robust replication and partitioning

of DNA in daughter cells as well as cell size control. In yeast, which is a simple eukaryote, both

nuclear size and gene expression scale with cell size. Also, it is observed, that the size of nucleus

and associated gene expression in multi-nucleated cells is proportional to local cytoplasmic volume.

Here, we explore, models that are based on diffusing factors that are produced from each nucleus

and influence nuclear size to see if they can reproduce the observed relationships. This project

involves doing spatial stochastic simulations of diffusion-reaction systems.

47

MSc projects with Dr Igor Shevchenko

• Large scale low-frequecny variability of the midlatitude ocean circulationUnderstanding origins of the large-scale low-frequency variability (LFV) of the ocean is not onlyone of the central questions in the Earth system modelling and geophysical fluid dynamics, butalso one of the serious challenges in predictive understanding of climate change. The midlat-itude atmosphere and ocean possess significant interannual variability and several large-scalevariability modes on decadal and interdecadal timescales. Physical origins of the LFV modesremain unclear, and it is not even known to what extent these origins are intrinsic atmospheric,intrinsic oceanic, or coupled oceanic-atmospheric. This project focuses on studying the intrinsicoceanic LFV.

• Absorbing boundary conditions for nonlinear wave equationsMany problems in science and engineering are naturally formulated in unbounded domains; typ-ical examples originate from fluid dynamics, solid mechanics, aerodynamics, electrodynamics,acoustics, etc. However, numerical simulations of such problems require a finite computationalregion. This project is aimed to design absorbing boundary conditions for efficient and robustnumerical simulations of nonlinear wave equations in unbounded domains.

• Stochastic parameterisations for ocean modelsStochastic parameterisations of oceanic eddies play an important role in geophysical fluiddynamics because of their ability to represent complex physical processes with relatively simplemodels. In this project we develop parameterisations for the quasi-geostrophic model of wind-driven ocean gyres and analyse their efficiency in modelling unresolved scales.

• Multiscale oceanic energeticsThe goal of this project is to study inter-scale energy transfers in the ocean, examine themulti-scale nature of the forward and backward energy cascade, and how the energy transfersdepend on viscosity.

• Modelling the ocean with primitive equationsModelling the ocean with primitive equations is a vast and active area of research in geophysics.The goal of this project is to simulate and study ocean currents in the North Atlantic withusing the Regional Ocean Modelling System (ROMS).

• Bifurcation analysis of dynamical systems with degenerative solutionsIn this project we consider convection in a porous material saturated with fluid and heated frombelow. This problem belongs to the class of dynamical systems with nontrivial cosymmetry,which gives rise to a hidden parameter in the system and continuous families of infinitely manyequilibria, and leads to non-trivial bifurcations. It is planned to study nonlinear phenomenaresulting from the existence of cosymmetry, describe different non-classical bifurcations, andthe selection scenarios (namely, which of infinitely many equilibria can be realized in physicalexperiments).

1

MSc projects with Ory Schnitzer

1. Asymptotic solutions of the plasmonic eigenvalue problem

Metallic nanoparticles support extraordinary electromagnetic resonances, typically at wave-lengths in the visible range much larger then their own size (Figure a). The resonant frequencies(colours) are to a good approximation scale invariant and, for fixed material properties, dependsolely on particle shape. The phenomenon, known as “localised-surface-plasmon resonance”, iskey to manipulation of light on nanometric scales and below the limitations of traditional opti-cal apparatuses (nanophotonics)1, with numerous applications in nanophotonics and metamaterialsincluding bio-sensing, medical treatment, ultra-sensitive microscopy and nonlinear optics.2 The ob-jective of this project is an asymptotic study of the purely geometric eigenvalue problem governingthe colours of plasmonic nanostructures (Figure b). Geometries that are important in applications3

are typically characterised by multiple scales (Figure c): Nearly touching dimer particles, elongatedparticles, etc. We will use the method of matched asymptotic expansions4 to study the eigenvaluesand eigenmodes of these configurations.5

ϕ = ϕ

∇2ϕ = 0

∇2ϕ = 0λ∂ϕ

∂n=

∂ϕ

∂n

ϕ → 0 as x → ∞

1

2h(b) Plasmonic eigenvalue problem:

For what values of λ are there non-

trivial homogeneous solutions?

(c) Sphere Dimer: Asymptotic

solutions of the eigenvalue problem in

the near-contact limit h→0?

metal

metal

vacuum

metal

vacuum

(a) Resonant scattering from a metallic nanoparticle

1In creating the dichroic glass of the Lycurgus Cup, the ancient Romans had exploited the phenomenon, probablyunknowingly, already in the 4th Century. British Musuem, item G41/dc11; see also I. Freestone et al., Gold Bulletin

40(4) 270-277, 2007.2S. A. Maier, Plasmonics: Fundamentals and applications, Springer Science & Business Media, 20073J. B. Pendry et al, Nature Physics, 9(8), 20134E. J. Hinch, Perturbation methods, Cambridge university press, 19915O. Schnitzer, Physical Review B, 92 23 235428, 2015; O. Schnitzer et al., Physical Review B, 93 4 041409,

2016

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2. Homogenisation of microstructured surfaces exhibiting giant liquid slip

The hydrodynamic resistance of a micro-channel increases rapidly with decreasing cross-sectionarea. The culprit is the no-slip boundary condition satisfied by fluids on solid boundaries. Oneway to overcome this, for liquid flows, is to engineer “superhydrophobic” micro-structured surfacesthat are effectively (macroscopically) slippery. The effective slip is a result of gas being trappedbetween the micro-structure and the liquid above the surface, whereby the liquid flow field satisfiesa spatial combination of no-slip and no-shear boundary conditions. For some designs, the effect isgreatly enhanced in the limit of small solid-to-gas areal fractions; in particular, the theory for postarrays predicts giant effective slip lengths scaling inversely with the square-root of the solid fraction.6

While giant slip lengths have indeed been observed for small solid fractions, theory and experimentsdo not agree quantitatively for pillared surfaces.7 One possible reason for this disagreement is thatthe theory assumes an isolated surface subjected to simple shear flow, whereas in the experimentsthe flow is generated by rotating a solid cone above the micro-structured substrate (the effectiveslip length is inferred from measurements of the torque acting on the cone). The goal of thisproject is to use homogenisation techniques in conjunction with matched asymptotic expansions8

to model small-solid-fraction (singular) pillared surfaces in realistic configurations where the domainis bounded and the surface experiences spatially varying conditions.

Figure 1. Pillared substrates with small solid fractions (from Lee et al. 2008).

Figure 2. Schematic sketch of the experimental setup used in Lee et al. 2008.

6C. Ybert et al., Physics of fluids, 19 123601, 2007; A. M. J Davis and E. Lauga, Journal of Fluid Mechanics,661 402-411, 2010

7C. Lee et al., Physical Review Letters, 101 064501, 20088O. Schnitzer, Physical Review Fluids, 1 052101, 2016; O. Schnitzer, Journal of Fluid Mechanics, 820 580-603,

2017; O. Schnitzer and E. Yariv, Physical Review Fluids, 2 072101(R), 2017

Stochastic dynamics of structured populations – Dr Philipp Thomas and Prof Mauricio

Barahona

Individuals of a population can typically be distinguished by several physiological features such

as age, size, etc. Based on these features individuals reproduce, die, or undergo life-changing

transitions. Biological examples include dividing cell populations and the spread of infections. In

this project, you will investigate the stochastic population dynamics in time-varying environments,

which has important implications for drug treatment. This involves a combination of agent-based

simulation and analysis of partial differential equations.

Investigation of the Instability Properties of Thread-Annular Flow – Dr A. Walton

The prediction of the behaviour of a fluid when it is injected into the body using a needle or

syringe is a challenging problem for theoretical fluid dynamicists. A plastic surgeon may not

only inject fluid into the body but also specially designed surgical threads that consist of synthetic

biocompatible materials. Through modelling the procedure mathematically, a better understanding

can be obtained of the way in which key parameters, such as the speed of injection, affect the fluid

flow characteristics within the syringe. Ultimately, it is hoped that the entire procedure can be

carried out more proficiently, with a minimum of surgical trauma.

A simplified version of the injection process can be modelled mathematically by considering the

axial flow between concentric cylinders with the inner cylindrical core (representing the thread of

material) moving at a constant velocity. The thread itself ideally should be modelled as a form of

compliant surface. A simple model for this would be one which takes into account the elasticity

properties of the thread and the deformation of its surface. This is a new problem that has not

been studied in any detail before. We will assume that any perturbations of the thread surface are

small, so that their effect on the basic flow will be a linear one. The aim of the project is then to

investigate the linear stability properties of the ensuing flow at high Reynolds number and/or at

finite Reynolds number. The former analysis requires a knowledge of asymptotic methods, while

the latter would involve computation with Matlab and require the student to write their own codes.

The procedures will be similar to those for a solid inner cylinder but there is likely to be new and

interesting effects due to the interaction with the compliant surface.

References:

P. G. Drazin: Introduction to Hydrodynamic Stability (C.U.P.)

A. G. Walton: Stability of circular Poiseuille-Couette flow to axisymmetric disturbances Journal

of Fluid Mechanics (2004) Vol. 500, p.169–210.

Useful course to take: M4A30 Hydrodynamic stability.

Roll/streak/wave interaction in shear flows and turbulent spot formation – Dr A.

Walton

In recent times a dynamical systems picture of laminar-turbulent has emerged in which equilibrium

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solutions of the Navier-Stokes equations play a key role in transition and turbulent dynamics. These

equilibrium solutions consist of three crucial components: a roll flow in the cross-stream plane, a

streamwise streak and a three-dimensional wave. These three components interact in a mutually

sustaining manner in which the roll flow drives a spanwise-modulated streak which is itself unstable

to the wave. The wave then self-interacts nonlinearly to reinforce and re-energize the roll flow. At

high Reynolds number this interplay can be expressed in terms of an asymptotic theory known as

vortex-wave interaction. It is possible to formulate and solve the interaction equations for a wide

variety of viscous shear flows. In this project we concentrate on one of the well-known properties

of the solution of such systems: the appearance of solutions which localize as the amplitude of the

motion is increased. This localization is thought to be connected to the experimental observation

that at relatively small disturbance levels viscous travelling waves cause an instability of the flow

which leads to the formation of turbulent spots.

We propose to study the interaction between a roll, streak and viscous wave in an asymptotic

suction boundary layer. It can be shown that the main part of such an interaction is governed

by the nonlinear system:∂v

∂y+

∂w

∂z= 0, v

∂w

∂y+ w

∂w

∂z=

∂2w

∂y2,

subject to the boundary conditions

w = A sin z, v = −1 on y = 0, w → 0 as y →∞.

Here (v, w) are the normal and spanwise components of the roll flow and the condition on w at

the wall represents the wave forcing with amplitude A. It can be shown that this system has a

nonlinear exact solution for small values of the spanwise coordinate z up to a critical amplitude

for A. We propose to carry out numerical calculations of the full equations to investigate how the

localization in the system is related to the singularity in the local exact solution.

References

Deguchi, K., Hall, P. & Walton, A. G. 2013 The emergence of localized vortex-wave inter-

action states in plane Couette flow. J. Fluid Mech. 721, 58–85.

Dempsey, L.,J., Deguchi, K., Hall, P. & Walton, A. G. 2015 Localized vortex/Tollmien-

Schlichting wave interaction states in plane Poiseuille flow. Preprint available on request.

Useful course to take: M4A30 Hydrodynamic stability.

The unsteady Kutta condition and excitation of instability modes in the wake

flow (Prof. X. Wu)

When an object such as an aircraft moves through an ambient fluid, a wake forms downstream.

A wake is usually unstable, which means that a small disturbance to the flow amplifies. The

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amplification leads to interesting and complex dynamics. The wake flow and the development of

perturbations are important because they are related to the drag on the object, and in the case

of aircraft taking off and landing, the wake downstream of one aircraft influences the flight of the

next.

In order to investigate experimentally the instability of a wake, it is customary to produce a sound

wave (using a loudspeaker) to excite an instability wave (mode) in a controlled manner so that the

development of the latter can be studied in detail. This process, which is referred to as receptivity,

is also an important part of aerodynamic noise generation.

Theoretical studies of instability and receptivity are usually based on inviscid approximation. The

solution is however often non-unique, and an unsteady Kutta condition is then imposed in order to

select the physically acceptable solution (Crighton 1985). This condition is supposed to account for

the viscous effects operating in the vicinity of the trailing edge of the object. The appropriateness of

this condition is not apparent, and its justification requires an detailed analysis of the viscous flow

near the trailing edge. Orszag & Crow (1970) used the unsteady Kutta condition in their study of

the effect of the edge on the instability itself, and a corresponding viscous analysis was performed

by Daniels (1978). In these studies, the external forcing (e.g. the incident sound) is absent, that

is, the receptivity was not considered. In the present project, we will investigate the receptivity,

i.e. excitation of instability wave when a sound wave interacts with the edge. Both inviscid and

viscous analyses will be performed using the matched asymptotic expansion as the mathematical

tool, and the asymptotic matching between the inviscid (outer) and viscous (inner) solutions would

define and justify the Kutta conditions to be used. Moreover, we can determine the amplitude

of the instability mode (eigen solution) in terms of the characteristics of the sound wave and the

geometry of the edge.

This project is primarily analytical, involving a systematic analysis using the matched asymptotic

expansion technique, but some numerical calculations are required in order to obtain the results of

relevance for practical applications.

References:

1. D. G. Crighton 1985 The unsteady Kutta condition in unsteady flow. Ann. Rev. Fluid Mech.

17, 411-445.

2. P. G. Daniels 1978 On unsteady Kutta condition. Q. J. Mech. Appl. Math. 31, 49-75.

3. Orszag, S. A. & Crow, S. C. 1970 Instability of a vortex sheet leaving a semi-infinite plate.

Stud. Appl. Math. 49, 167-181.

Effects of transverse suction/injection on shear flows and their instability

(Prof. X. Wu)

Instability of shear flows is known to be extremely sensitive to suction and injection, imposed

transversely to the main flow direction. For example, in the boundary layer subject to a uniform

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suction, the critical Reynolds number for instability increases to 54370 in comparison with 520

for the Blasius boundary layer (Hughes & Reid 1965). This is remarkable given that the suction

velocity is merely 1/54370-th of the main flow. Suction is thus being exploited as an efficient flow

control technique to maintain a laminar flow (Fransson & Alfredsson 2003). Transverse injection

is, on the other hand, proposed as a technique to accelerate transition in other applications where

turbulence is desirable, for example in scramjet (supersonic combustion ramjet) engine, where a

turbulent state is beneficial and indeed crucial for mixing and reactions of the fuel and oxygen.

The mechanism of the extraordinary sensitive effect of suction/injection remains a mystery. In this

project, we investigate, through a combination of asymptotic analysis and numerical computation,

the impact of suction/injection on an otherwise simple shear flow. We will first consider uniform

suction and injection, which influence stability through the transverse velocity as well as through

modifying the distribution of the main velocity. Numerical calculations will be carried out to map

out the stability properties, and these will be supplemented by asymptotic analysis of the stability

in the lower and upper-branch regimes.

In practical applications, it is most likely that a portion of the wall subject to suction/injection

joins a rigid portion without suction/injection. The flow near the junction is of interest in its

own right. Specifically, the flow field upstream the junction starts adjusting before the junction

is reached, exhibiting the so-called ‘elliptic’ character, and this will be analysed using triple-deck

theory.

References:

1. Hughes, T. H. & Reid, W. H. 1965 On the stability of the asymptotic suction boundary layer

profile. J. Fluid Mech. 23, 715-735.

2. Fransson, J. H. & Alfredsson, P. H. 2003 On the disturbance growth in an asymptotic suction

boundary layer. J. Fluid Mech. 482, 51-90.

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