Stochastic active flows and interacting particle systems

Mathematical models describing continua have been used very successfully to explain the behaviour of a wide range
of systems beyond the liquid flows that inspired their initial development. For example, the formation of traffic jams
and the collective behaviour of large animal groups. However, these models typically take a purely macroscopic
approach to explain these phenomena. This class of PDE-based models does not capture the effects of noiseinduced
fluctuations arising from finite size effects in 'fluids' composed of discrete moving agents. Compartment
based-methods for the interaction of agents (in reaction diffusion systems for instance) allow for the incorporation of
stochastic effects and quantification of these effects on the system as a whole. This method is less convenient when
interactions between agents are not confined to their local vicinity where only the particles in the same compartment
need to be considered. Microscopic level models allow for the greatest level of accuracy to the underlying rules of
interactions between particles. However, using these models to understand the evolution of the system as a whole is
a difficult challenge for many systems.
The first task of my PhD will be to explicitly link microscopic continuum based models of interacting stochastic agents
with macroscopic kinematic fluid models in one dimension, enabling an analysis of the effects of noise induced by
finite size effects. This will require the classification of the types of macroscopic models which can be understood as
the large particle limit of pair-wise particle interactions. I will then find a stochastic PDE for the evolution of the
generalised system before looking at specific rules of interactions and the SPDEs which result from this. I will
compare simulations of the SPDEs for specific cases of the model with the microscopic simulations, checking for
agreement with the overall density evolution and quantifying the scale of the stochastic effects.
Secondly, I will look into the case of two dimensions. Initially focusing on conservative interactions between particles,
I will then look at the potential for milling pattern formation observed in many swarming behaviours. For both one and
two dimensions I will look for bi-stable systems which could be modelled using pair-wise interaction models of
particles. The mean switching time between these two states, observed in cases such as locust swarming, could then
be quantified from the SPDE for the density evolution.

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.