I am a Reader in the Department of Mathematics at the University of Surrey. My research interests are in localised pattern formation and mathematical modelling. I am also co-founder and director of the Surrey Centre for Criminology at the University of Surrey.
Please see my personal webpage for more details.
Spots and localised patches of cellular hexagons have been observed in a variety of experiments from magnetic fluids to vertical vibrated media. Research here is focused on understanding two- and higher-dimensional localised structures in pattern forming systems.
Collaborators: Drs. Daniele Avitabile (Surrey), John Burke (Boston) and Profs. Jurgen Knobloch (Ilmenau), Edgar Knobloch (UC, Berkeley), Bjorn Sandstede (Brown), Sergey Zelik (Surrey), Reinhard Richter (Bayreuth)
PhD Students: Tasos Rossides, Jacob Brooks, Daniel Hill
Analysis of models of burglary hotspots and Data Assimilation issues
PhD Student: Daniel Ennis
Indicators of esteem
Runner-up - DSWeb 2018 Software Contest: EMBER (Emergent and Macroscopic Behaviour ExtRaction)
AUTO Tutorial for localised patterns
Help on installing AUTO under various platforms can be found here.
EMBER (Emergent and Macroscopic Behaviour ExtRaction)
Stochastic continuation toolbox written in Java:
Runner-up - DSWeb 2018 Software Contest with Dr. Spencer Thomas (NPL) and Prof. Anne Skeldon (Surrey)
2D Localised Pattern Codes for the Swift-Hohenberg equation
These codes (tgz) were created to produce all the figures in the paper:
Localized hexagon patterns in the planar Swift-Hohenberg equation, DJB Lloyd, B Sandstede, D Avitabile and AR Champneys, SIAM J. Appl. Dyn. Sys. 7(3) 1049-1100, 2008. pdf.
and may be downloaded from the SIAM J. Appl. Dyn. Sys. webpage. The list of programs in localised_pattern_codes.tgz are given below:
Requirements: Matlab with optimization toolbox (tested on version 2007b) and AUTO07p.
(FSOLVE in the optimization toolbox is used to solve the BVPS. However, BVPS have been set-up
so that any globalised Newton solver will work.)
To untar files use: tar xvzf localised_pattern_codes.tgz
Note: All sub-directories have README files to allow immediate running of all codes.
Matlab codes: 1D BVP solvers:
- solves 1D quadratic/cubic Swift-Hohenberg equation BVP on Half line. Finds a localised pulse and computes its stability with respect to perturbations on the full line. Uses Fourier differentiation matrices.
- solves 1D quadratic/cubic Swift-Hohenberg equation BVP on Half line. Finds a localised pulse and computes its stability with respect to perturbations on the full line. Uses finite differences and sparse matrices to speed up computations.
Matlab codes: 2D BVP solvers:
- Solves 2D quadratic/cubic Swift-Hohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised planar <10> hexagon pulse and plots the solution.
- Solves 2D quadratic/cubic Swift-Hohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised planar <11> hexagon pulse and plots the solution.
- Solves 2D quadratic/cubic Swift-Hohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised hexagon patch and plots the solution.
- Solves 2D quadratic/cubic Swift-Hohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised rhomboid patch and plots the solution.
- Gets initial data and continues Hexagon Maxwell curve in two parameters of the quadratic/cubic Swift-Hohenberg equation. Calls compute_Maxwell.m and SH2DBVPFOUR.m
- solves quadratic/cubic radial Swift-Hohenberg equation BVP on [0,L] with Neumann bcs at r=L. Uses L'Hopitals rule for r=0 boundary conditions. Finds a localised pulse and computes its stability with respect to perturbations on the half line. Uses finite differences and sparse matrices to speed up computations.
Matlab codes: 2D IVP solvers
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code computes figure 1(a) of "Localised Hexagon patterns in the planar Swift-Hohenberg equation" by Lloyd, Sandstede, Avitabile and Champneys, SIADS 2008.
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds a hexagon patch.
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds <10> hexagon pulse in the Swift-Hohenberg equation.
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds <11> hexagon pulse in the Swift-Hohenberg equation.
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds a localised ring in the Swift-Hohenberg equation.
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code starts from a localised random patch.
Note: All codes are tested on AUTO07p. Initial data is supplied for immediate running. Conversion scripts and Matlab codes for data handling (procurement of initial data and post-processing of AUTO output), are supplied. README files in each tar file for instruction on immediate running and data handling.
- Code computes pulses on a finite cylinder of the quadratic/cubic Swift-Hohenberg equation using a Fourier-cosine projection in the circumference direction. Code computes <10> and <11> hexagon pulses on the half line/cylinder.
- Continues periodic solutions, Maxwell curves and localised pulses of the 1D Swift-Hohenberg equation with periodic boundary conditions.
- Continues hexagon patches in the 2D quadratic/cubic Swift-Hohenberg equation using a Fourier-cosine projections in the angular direction as described in section 4.4 of "Localised Hexagon patterns in the planar Swift-Hohenberg equation" by Lloyd, Sandstede, Avitabile and Champneys, SIADS 2008.
- Continues radial localised pulses in the radial quadratic/cubic Swift-Hohenberg equation.
- Continues pulses in the 1D quadratic/cubic Swift-Hohenberg equation and the leading eigenfunction.
- Continues radial pulses and hexagon eigenfunction in the quadratic/cubic Swift-Hohenberg equation. Code traces out the hexagon pitchfork locus in the linear and quadratic bifurcation parameters.
- Converts AUTO output files b.foo and s.foo to matlab readable files. To use type $autox to_matlab.autox foo convertedfoo
Postgraduate research supervision
- 2019-present: Steven Falconer (co-supervised with Drs. N. Santitissadeekorn (Maths), Nadia Smith and Spencer Thomas (National Physical Laboratory))
Project: Data analysis and modeling of the Royal College of GPs, Research and Surveillance data
- 2019-Present: Daniel Ennis (co-supervised with Dr. N. Santitissadeekorn (Maths), Prof. I. Brunton-Smith (Sociology))
Project: Modeling and data analysis of crime
- 2017-Present: Daniel Hill (co-supervised with Dr. Matt Turner)
Project: Localised Ferrofluid Patterns
- 2016-Present: Jacob Brooks (co-supervised with Prof. Gianne Derks (Maths)).
Project: Nonlinear Wave Equations
- 2012-2016: Craig Shenton (co-supervised with Prof. Angela Druckman (Centre for Environmental Strategy)).
Project: Food Security
- 2011-2014: Tasos Rossides (co-supervised with Prof. Sergy Zelik (Maths)).
Thesis: Computing multi-localised structures for some parabolic PDE systems
- 2010-2014: Gary Chaffey (co-supervised with Drs. Norman Kirkby (Civil Eng) and Anne Skeldon (Maths)).
Thesis: Modelling the Cell Cycle
- 2010-2014: Jessica Rowden (co-supervised with Prof. Nigel Gilbert (Sociology)).
Thesis: Application of Two Mathematical Modelling Approaches for Real World Systems
- 2007-2011. Jeremy Chamard
Thesis: Mountain Pass Algorithms and Applications
with application to urban crime data, Computational Statistics and Data Analysis 128 pp. 163-183 Elsevier
? have been used to understand how crime rates evolve in time and/or
space. Within the context of these models and actual crime data, parameters
are often estimated using maximum likelihood estimation (MLE) on batch
data, but this approach has several limitations such as limited tracking in
real-time and uncertainty quantification. For practical purposes, it would be
desirable to move beyond batch data estimation to sequential data assimilation.
A novel and general Bayesian sequential data assimilation algorithm is
developed for joint state-parameter estimation for an inhomogeneous Poisson
process by deriving an approximating Poisson-Gamma ?Kalman? filter
that allows for uncertainty quantification. The ensemble-based implementation
of the filter is developed in a similar approach to the ensemble Kalman
filter, making the filter applicable to large-scale real world applications unlike
nonlinear filters such as the particle filter. The filter has the advantage
that it is independent of the underlying model for the process intensity,
and can therefore be used for many different crime models, as well as other
application domains. The performance of the filter is demonstrated on synthetic
data and real Los Angeles gang crime data and compared against a
very large sample-size particle filter, showing its effectiveness in practice. In addition the forecast skill of the Hawkes model is investigated for a forecast
system using the Receiver Operating Characteristic (ROC) to provide a useful
indicator for when predictive policing software for a crime type is likely
to be useful. The ROC and Brier scores are used to compare and analyse
the forecast skill of sequential data assimilation and MLE. It is found that
sequential data assimilation produces improved probabilistic forecasts over
a spatially dependent scaling of the sine-Gordon potential term. The uncoupled inhomogeneous
sine-Gordon equation has stable stationary front solutions that persist in the coupled system.
Carrying out a numerical investigation it is found that these inhomogeneous sine-Gordon fronts
loose stability, provided the coupling between the two inhomogeneous sine-Gordon equations is
strong enough, with new stable fronts bifurcating. In order to analytically study the bifurcating
fronts, we first approximate the smooth spatial inhomogeneity by a piecewise constant function.
With this approximation, we prove analytically the existence of a pitchfork bifurcation. To
complete the argument, we prove that transverse fronts for a piecewise constant inhomogeneity
persist for the smooth ?hat-like? spatial inhomogeneity by introducing a fast-slow structure and
using geometric singular perturbation theory.
We investigate stationary spatially localized hexagon patterns of the two-dimensional (2D) Swift-Hohenberg equation in the parameter region where the trivial state and regular hexagon patterns are both stable. Using numerical continuation techniques, we trace out the existence regions of fully localized hexagon patches and of planar pulses which consist of a strip filled with hexagons that is embedded in the trivial state. We find that these patterns exhibit snaking: for each parameter value in the snaking region, an infinite number of patterns exist that are connected in parameter space and whose width increases without bound. Our computations also indicate a relation between the limits of the snaking regions of planar hexagon pulses with different orientations and of the fully localized hexagon patches. To investigate which hexagons among the one-parameter family of hexagons are selected in a hexagon pulse or front, we derive a conserved quantity of the spatial dynamical system that describes planar patterns which are periodic in the transverse direction and use it to calculate the Maxwell curves along which the selected hexagons have the same energy as the trivial state. We find that the Maxwell curve lies within the snaking region, as expected from heuristic arguments.
state) for general magnetic permeabilities. Using umerical continuation techniques we investigate the existence of localized hexagons in the Young-Laplace equation coupled to the Maxwell equations. We find cellular hexagons possess a Maxwell point providing an energetic explanation for the multitude of measured hexagon patches. Furthermore,it is found that planar hexagon fronts and hexagon patches undergo homoclinic snaking corroborating the experimentally detected intervals. Besides making a contribution to the specific area of ferrofluids, our work paves the ground for a deeper understanding of homoclinic snaking of 2D localized patches of cellular patterns in many physical systems.
a process that is known as 'homoclinic snaking' where patterns are added to the stripe
core as a bifurcation parameter is varied. The parameter region where homoclinic snaking takes
place usually occupies a small region in the bistability region between the stripes and quiescent
state. Outside the homoclinic snaking region, the localised patterns invade or retreat where
stripes are either added or removed from the core forming depinning fronts. It remains an
open problem to carry out a numerical bifurcation analysis of depinning fronts. In this paper,
we carry out numerically bifurcation analysis of depinning of fronts near the homoclinic
snaking region, involving a spatial stripe cellular pattern embedded in a quiescent state, in
the two-dimensional Swift-Hohenberg equation with either a quadratic-cubic or cubic-quintic
nonlinearity. We focus on depinning fronts involving stripes that are orientated either parallel,
oblique and perpendicular to the front interface, and almost planar depinning fronts. We show
that invading parallel depinning fronts select both a far-field wavenumber and a propagation
wavespeed whereas retreating parallel depinning fronts come in families where the wavespeed is
a function of the far-field wavenumber. Employing a far-field core decomposition, we propose
a boundary value problem for the invading depinning fronts which we numerically solve and
use path-following routines to trace out bifurcation diagrams. We then carry out a thorough
numerical investigation of the parallel, oblique, perpendicular stripe, and almost planar invasion
fronts. We find that almost planar invasion fronts in the cubic-quintic Swift-Hohenberg equation
bifurcate off parallel invasion fronts and co-exist close to the homoclinic snaking region.
Sufficiently far from the 1D homoclinic snaking region, no almost planar invasion fronts exist
and we find that parallel invasion stripe fronts may regain transverse stability if they propagate
above a critical speed. Finally, we show that depinning fronts shed light on the time simulations
of fully localised patches of stripes on the plane. The numerical algorithms detailed have wider
application to general modulated fronts and reaction-diffusion systems.