Hybrid asymptotic-numerical schemes for exponentially small selection mechanisms

[1] There are a number of problems in fluid mechanics and the wider physical sciences that involve the asymptotic analysis of nonlinear differential equations studied in some singular limit, where an underlying selection mechanism determines a sequence of discrete countably infinite eigenvalues. In certain challenging cases, this mechanism is governed by exponentially small terms beyond-all-orders and the resultant analysis demands specialised techniques.
[2] One classic problem for which this occurs is in the context of Saffman-Taylor viscous fingering where the selection of the finger width is determined by terms exponentially small in the surface tension parameter. The resolution of the Saffman-Taylor problem pioneered modern methods in exponential asymptotics. However, the problem contains certain niceties that render the analysis tractable. In particular, the leading-order surface-tension-free solution is known in closed form and this turns out to be a crucial component in the application of exponential asymptotics. Similar selection mechanisms governing viscous fingering are expected to apply in generalisations to time-dependent flows or flows in complex geometries, but it remains an open challenge to adopt the asymptotic techniques to these extensions.
[3] This thesis will focus on problems where the asymptotic or perturbative solutions cannot be determined in closed form, even at leading order. In such cases, it is essential to consider the extension of classical exponential asymptotic methodologies to hybrid schemes where numerical methods are used in conjunction with analytical theory. These numerical methods involve, for example, solutions of ordinary or partial differential equations and subsequent analytic continuation of real-valued solutions to higher-dimensional complex-valued spaces. The study of analytic continuation then yields key properties of solutions near singular points, which is then encoded into the asymptotic schemes.
[4] The PhD will be focused on the development of these analytical and numerical methods, and their applications to several open problems in continuum and fluid mechanics. Applications will include some or all of the following: (i) the study of jet separation in a two-dimensional nozzle; (ii) jet separation in a three-dimensional or axi-symmetric nozzle; and (iii) Saffman-Taylor viscous fingering in a wedge.

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.