Merge pull request #18 from UCL-ARC/bertrand-projects

Bertrand projects and replaced Cotter project

phd_projects/entries/bertrand1.qmd

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+---
+title: "Stochastic resetting in many-body interacting particle systems"
+department: "Mathematics"
+
+date: "12/12/2024"
+author: 
+    name: "Dr Thibault Bertrand and Prof. Paul Berloff"
+    affiliation: "Imperial"
+institution: "Imperial"
+
+
+---
+## Project Description
+
+Large systems of interacting particles are central to many
+applications across natural and social sciences. In physics, particles
+may represent ions in a plasma, molecules in a passive or active
+fluids, or galaxies in a cosmological model, while in biology, they
+often represent microorganisms like eukaryotic cells or bacteria that
+can exhibit complex behaviours. In economics and social sciences,
+particles typically represent individual agents like investors or
+institutions in a model of financial markets or individuals and
+communities in models of opinion formation. In these systems, robust
+emergent behaviour often arises even from very simple rules of
+interaction. Paradigmatic examples in systems of interacting active
+particles include motility induced phase separation and non-trivial
+swarming behaviour. A major challenge is to reduce the mathematical
+complexity of such systems by studying them at a coarse-grained level
+rather than at the level of single agents.
+
+A classical approach is to derive a macroscopic ic model that provides
+a continuous description of the dynamics in terms of global densities
+evolving according to non-linear partial differential equations. Such
+kinetic formulations date back to the foundations of statistical
+mechanics and the Boltzmann equation of dilute gases interacting via
+direct collisions. This is in general a complicated task and important
+(often uncontrolled) approximations need to be made. In recent years,
+however, much of the focus has been on the mean-field limit of
+particles with long range or collisionless interactions. Two
+paradigmatic examples are interacting Brownian particles in the
+overdamped regime and the Kuramoto model of coupled phase oscillators.
+
+Finally, the concept of stochastic resetting has recently
+emerged. Stochastic resetting is the process in which a system, such
+as a diffusive particle, is intermittently "reset" to an initial
+state, thereby restarting its evolution at stochastic
+times. Stochastic resetting has recently been under intense scrutiny
+because it has been shown to enhance search efficiency, create
+non-equilibrium steady states (NESS), and offer insights into a wide
+range of processes, from chemical reactions to biological foraging
+behaviours in a mathematically tractable framework. However, almost
+all previous studies of stochastic resetting have focused on
+single-particle systems.
+
+
+### Main objectives of the project
+
+The main goal of this project is to use a combination of mean-field
+theory, coarse-graining techniques, dimensional reduction, and
+agent-based numerical simulations to explore the effects of stochastic
+resetting on large-scale interacting particle systems, including both
+Kuramoto-based oscillator networks and systems of passive/active
+particles. Topics of interest include the following:
+
+• Existence of NESS in systems of interacting particles under
+  stochastic resetting – First, we will investigate the existence of a
+  NESS for the population density PDE of an interacting particle
+  system with local resetting and pairwise interactions. We will ask
+  whether the NESS exhibits phase transitions along analogous lines to
+  previous studies of Brownian gases without resetting.
+
+• Exploring differences between local and global resetting – under
+  local resetting each particle is independently reset following its
+  own sequence of times, while in global resetting all particles are
+  simultaneously reset. In the latter case, the resulting PDE for the
+  population density is itself subject to resetting. That is, mean
+  field theory breaks down and statistical correlations between the
+  particles arise even in the absence of interactions. We aim to
+  develop new analytical strategies to derive PDE descriptions of
+  these systems, strategies which will be informed by our large-scale
+  simulations.
+
+• Bridging local and global resetting – in a variety of models,
+  particles can be organized in subsystems (i.e. communities on
+  network-based Kuramoto systems or clusters arising in systems of
+  interacting active Brownian particles). We will introduce the
+  concept of subsystem resetting, in which subsystems can be reset
+  simultaneously leaving the rest of the system unchanged. We will
+  explore the conditions under which subsystem resetting can induce
+  global resetting. Focusing on the Kuramoto model, we will ask
+  whether subsystem resetting can induce system spanning correlation
+  and global synchronization. Using both analytical and numerical
+  methods (like genetic algorithms), we devise strategies to design
+  network topologies which optimize the emergence of synchronization
+  from subsystem resetting.
+
+• Extrinsic vs intrinsic coupling – In large interacting particle
+  systems, the coupling between individual particles can either be
+  “intrinsic” (i.e. direct pairwise interactions) or “extrinsic”
+  (i.e. mediated by a common external medium). An example of extrinsic
+  particle-particle interactions would be the quorum sensing observed
+  in bacterial colonies. We are interested in comparing the emergent
+  collective dynamics observed in the case of systems with intrinsic
+  and extrinsic interactions.
+
+• Passive vs active particles resetting – For passive Brownian
+  particles, the state of each particle is simply defined to be its
+  position. On the other hand, for an active particle it is necessary
+  to specify both its position and velocity state (or at least its
+  orientation). We will explore how the choice of resetting protocol
+  affects the collective behaviour exhibited by these systems.
+
+• Finite-size effects – In all the studies, we will investigate
+  numerically the breakdown of mean field theory as the number N of
+  interacting particles decreases. To do so, we will focus on
+  understanding how macroscopic observables scale with system size.
+
+### Details of Software/Data Deliverables
+
+The success of this project will rely on the development of:
+1.	numerical algorithms for a large-scale computational exploration of a variety of minimal systems in statistical mechanics;
+2.	development of efficient numerical algorithms for agent-based modelling both on networks (in the context of the Kuramoto model) and off-lattice (for simulations of passive and active particles systems);
+3.	purpose-built, scalable and adaptable software implementing advanced numerical solutions to highly nonlinear systems of PDEs and SPDEs; 
+4.	development of genetic algorithms to solve the inverse problem of finding the network structure of our Kuramoto model which optimizes global synchronization from the smallest subsystem resetting.

phd_projects/entries/bertrand2.qmd

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+---
+title: "Accumulation and absorption of active particles at surfaces"
+department: "Mathematics"
+
+date: "12/12/2024"
+author: 
+    name: "Dr Thibault Bertrand and Prof. Paul Berloff"
+    affiliation: "Imperial"
+institution: "Imperial"
+
+
+---
+## Project Description
+
+Active matter provides a powerful quantitative framework for
+understanding complex biological processes by examining the interplay
+between self-organizing, energy-consuming particles and their
+surrounding environment. Systems such as motile bacteria,
+self-propelled colloids, or cytoskeletal filaments exemplify this
+paradigm. Canonical models include run-and-tumble particles (RTPs),
+which change direction through discrete reorientations, and active
+Brownian particles (ABPs), whose motion combines constant propulsion
+speed with rotational diffusion. While the local energy consumption
+puts these systems inherently out-of-equilibrium, in isolation, active
+particles seen at long enough time and large enough distances remain
+diffusive; true nonequilibrium features stem from the interactions of
+active particles with their environment.
+
+For instance, when confined within a channel, active particles tend to
+accumulate at the channel walls, even in the absence of inter-particle
+interactions. This is in clear contradiction with equilibrium
+Boltzmann distributions. Each particle pushes against the wall until a
+tumbling event or rotational diffusion redirects its motion enough
+that they can scatter off; this makes the wall behave like a sticky
+boundary. At the multi-particle level this results in a pressure being
+exerted on the confining walls. This behavior can also be described in
+terms of so-called sticky boundary condition: upon colliding with the
+wall, a particle remains attached for a random time governed by its
+tumbling dynamics. The degree of stickiness is characterized by the
+escape time back into the bulk; it spans from totally reflecting
+boundaries (instantaneous escape) to totally absorbing ones (permanent
+adhesion), with intermediate cases characterized by partial
+retention. Sticky boundary conditions are also relevant in
+understanding biological phenomena such as the dynamics of growing and
+shrinking polymer filaments.
+
+Extending this concept, partially permeable walls introduce another
+layer of complexity. Particles interacting with sticky boundaries may
+either re-enter the bulk or escape permanently, leading to a distinct
+set of behaviors compared to impermeable walls. In this scenario, the
+system lacks a steady-state density for particle position and
+orientation, and attention shifts to dynamic quantities like the mean
+first passage time (MFPT) for permanent absorption and its
+higher-order moments. These features underscore how the interactions
+between active particles and their environments drive nonequilibrium
+phenomena central to active matter systems.
+
+### Main objectives of the project
+
+The main goal of this project is to
+combine nonequilibrium statistical physics, mean field theory, and
+multi-scale computation to investigate the accumulationof
+particles. Recent studies have started to extend the equilibrium
+theory of wetting to systems of active particles showing that the
+stiffness of the wall controls a transition to wetting. We will here
+similarly study the condition of emergence of a wetting transition as
+a function of the absorption behaviour of the wall.
+
+•     First-passage statistics – At the particle level, our study will also focus on determining important first-passage statistics including the mean first-passage time for single-particle absorption at a permeable wall as well as the extremal statistics of absorption in the case of multiple particles, quantifying for instance, first absorption times.
+
+•     Theory of particle-surface interactions – We will develop a microscopic theory of particle-surface interactions and how this affects the accumulation and absorption of particles, including for flexible and active interfaces (modelling for instance a biological membrane).
+
+•     Breakdown of mean-field – Throughout the project, we will compare large-scale particle-based simulations and mean-field analytical arguments. We will then investigate the breakdown of mean field theory due to the absorption and removal of particles from the population.
+
+
+### Details of Software/Data Deliverables
+
+The success of this project will rely on the development of a number
+of advanced numerical simulations:
+
+1. numerical algorithms for a large-scale computational exploration of
+a variety of minimal systems in statistical mechanics including
+efficient sampling techniques to explore rare events, extremal
+statistics and first-passage time statistics;
+
+2. development of efficient numerical algorithms for systems of
+coupled SDEs;
+
+3. purpose-built, scalable and adaptable software implementing
+advanced numerical solutions to highly nonlinear systems of PDEs and
+SPDEs to solve our mean-field models.

phd_projects/entries/cotter_nextgen.qmd

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+---
+title: "Next generation implicit numerics for atmosphere models"
+
+department: "Mathematics"
+
+date: "10/11/2024"
+author: 
+    name: "Prof Colin Cotter"
+    affiliation: "Imperial"
+institution: "Imperial"
+---
+## Project Description
+
+The classical numerical approaches to building atmosphere models rely
+on complicated splitting methods that deal with different parts of the
+model: waves, transport, moisture processes (clouds, evaporation,
+rain, ice etc), radiation, boundary layers, convection, etc. These
+splitting methods lead to highly complicated codes, time schemes that
+are difficult to analyse for stability/accuracy, and occasionally
+numerical artifacts the coupling of fluid dynamics and other physics.
+In this project we are pursuing an alternative goal: to translate as
+much of the system as possible into a single monolithic PDE coupling
+all the variables, and solve it with an implicit Runge-Kutta method.
+This is made possible by recent advances in massively parallel
+iterative methods for solving the implicit systems that come from this
+equation: we shift the complications from the timestepping scheme into
+the iterative solver.
+
+As a first step, we will build an atmosphere model consisting of the
+fluid dynamics component plus moisture processes, in this framework.
+Moisture processes involve switches (e.g., when maximum humidity is
+reached, any surplus water vapour is converted into cloud); we will
+deal with this using advanced "Variational Inequality" Newton solvers
+facilitated using PETSc [1].  The spatial discretisation will be build
+from compatible finite element methods closely related to those being
+implemented in the next generation LFRic modelling system at the Met
+Office. The software will be developed using Firedrake [2], which is a
+system for solving complicated PDEs using advanced finite element
+methods based on domain specific languages and code generation.
+
+The resulting modelling system will be automatically differentiable
+using the py-adjoint system
+(https://github.com/dolfin-adjoint/pyadjoint), making it suitable for
+blending with machine learning tools, towards our goal of hybrid
+physics-based/data-driven modelling approaches.
+
+
+[1] S. Balay, S. Abhyankar, M. Adams, S. Benson, J. Brown, P. Brune,
+K. Buschelman, E. Constantinescu, L. Dalcin, A. Dener, V. Eijkhout,
+J. Faibussowitsch, W. Gropp, V. Hapla, T. Isaac, P. Jolivet,
+D. Karpeyev, D. Kaushik, M. Knepley, F. Kong, S. Kruger, D. May,
+L. Curfman McInnes, R. Mills, L. Mitchell, T. Munson, J. Roman,
+K. Rupp, P. Sanan, J Sarich, B. Smith, H. Suh, S. Zampini, H. Zhang,
+and H. Zhang, J. Zhang, PETSc/TAO Users Manual, ANL-21/39 - Revision
+3.22, 2024. https://doi.org/10.2172/2205494,
+https://petsc.org/release/docs/manual/manual.pdf
+
+[2] David A. Ham, Paul H. J. Kelly, Lawrence Mitchell, Colin
+J. Cotter, Robert C. Kirby, Koki Sagiyama, Nacime Bouziani, Sophia
+Vorderwuelbecke, Thomas J. Gregory, Jack Betteridge, Daniel
+R. Shapero, Reuben W. Nixon-Hill, Connor J. Ward, Patrick E. Farrell,
+Pablo D. Brubeck, India Marsden, Thomas H. Gibson, Miklós Homolya,
+Tianjiao Sun, Andrew T. T. McRae, Fabio Luporini, Alastair Gregory,
+Michael Lange, Simon W. Funke, Florian Rathgeber, Gheorghe-Teodor
+Bercea, and Graham R. Markall. Firedrake User Manual. Imperial College
+London and University of Oxford and Baylor University and University
+of Washington, first edition edition, 5 2023. doi:10.25561/104839.
+
+### Existing background work
+
+We have a body of ten years of research in methods and software
+for atmosphere models, which is summarised in [3] and [4].
+
+[3] Cotter, Colin J. "Compatible finite element methods for
+geophysical fluid dynamics." Acta Numerica 32 (2023): 291-393.
+
+[4] Gibson, Thomas H., Andrew TT McRae, Colin J. Cotter, Lawrence
+Mitchell, and David A. Ham. Compatible Finite Element Methods for
+Geophysical Flows: Automation and Implementation Using
+Firedrake. Springer Nature, 2019.
+
+### Main objectives of the project
+
+This project is available to researchers with a wide variety
+of interests, who might focus on one or more of:
+* designing scalable iterative methods allowing the use of highly parallel
+supercomputers,
+* developing interative solvers that seamlessly incorporate moisture
+processes,
+* developing stabilisation schemes that allow the model to incorporate
+the effects of unresolved turbulent scales,
+* time-parallel algorithms using ParaDiag methods [5],
+* benchmarking the quality of the simulation in challenging testcases
+  such as fronts and storms,
+* exploration of computationally optimal configurations using e.g.
+high order discretisations and emergent Firedrake capability on GPUs.
+
+[5] Hope-Collins, J., Hamdan, A., Bauer, W., Mitchell, L. and Cotter,
+C., 2024. asQ: parallel-in-time finite element simulations using
+ParaDiag for geoscientific models and beyond. arXiv preprint
+arXiv:2409.18792.
+
+
+### Details of Software/Data Deliverables
+
+* The research will contribute to open source software developed
+in Python (with automatically generated high performance C code)