Localization of waves in random resonant media

Introduction and overall objective: This project fall under the theme of trapping and localizing waves. The goal of the
project is to show that waves travelling through an unbounded domain can be trapped if it is a composite (i.e.
heterogeneous) medium with a special interplay between the microgeometry and the material properties. More
precisely, we will consider a space with two key features: (1) on an extended subset, the coefficients of the equation
take large values, and (2) a random landscape outside this subset.
The topic of wave localization is of interest in the physics community because it is contrary to our everyday
experience, for instance, sound (acoustic waves) travel very far in open spaces. For this reason, this phenomenon
received much attention when it first appeared in the quantum physics literature in the 1950s. However, a
mathematically rigorous study of this 'Anderson localization' phenomenon is only available for a few specific models,
so the work is far from complete. Mathematically, we are working with a random environment. The novelty of the
project is to then consider adding in (1), with the expectation that the new setting exhibits 'stronger' wave trapping
behavior.
Methodology and initial aims: Broadly, the subjects of this project include: probability theory, linear analysis,
scattering theory, and mathematical physics. Concretely, this project will study the spectral properties of a random
differential operator. In terms of the setup, we will take one of the models that has been proven to exhibit Anderson
localization behaviour, and modify it to have periodic regions with a coefficient that we will take to infinity. The
modification is to simulate (1). The initial aim is to prove localization for a fixed coefficient in (1) and a convenient
choice of probability measure in (2). In terms of the proof methods, we will look to modify either of the two main
methods used to prove Anderson localization (multi-scale method, fractional moments m ethod.)

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.