Computational Optimal Transport

This PhD project focus on the computational optimal transport. Optimal transport (OT) seeks the best way of transforming one probability distribution into another and provides a framework for distances between probability distributions. In recent year, there has been much work in the application of OT to machine learning and imaging problems. For example, in image processing such that one can consider an image as a discrete distribution of pixels in 3D and compare two images by applying the optimal transport methods. More interestingly, there are lots of applications of the continuous OT. For example, in shape analysis one can consider the shape as a continuous uniform distribution in 3D space, and hence compares two different shapes or finds the optimal way to transfer one to the other. Another example is that words can be embedded in some continuous space in text vectorization in natural language processing, and hence one can apply OT techniques to get the best route of transferring two words and hence defined the distance between them.
One of the challenges presented by machine learning and imaging applications is the need of efficient numerical algorithms to deal with large dimensions and to compute the transportation map between continuous distributions from a finite number of samples. While there has been some recent activity in the development of algorithms to meet these challenges, in particular, the development of stochastic algorithms, there remains many challenges to tackle in this area: the efficient representations of the continuous objects related to optimal transport maps is still unclear, moreover, there is still only limited understanding on the theoretical convergence properties of many of the stochastic algorithms proposed.
The aims of this thesis are to investigate techniques for more efficient representations for large scale continuous optimal transport. In particular, we will look into variance reduction techniques in Monte Carlo integration, and also how techniques from convex optimisation can be applied to help accelerate convergence. We will also aim to provide a theoretical understanding of the convergence behaviour of stochastic methods in optimal transport.
This project is a PhD project in the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa) and is funded by Chinese Scholarship Council (CSC). It involves applied and computational mathematics and aims to solve the OT problem numerically. The research methods used in this project align well with the goals of SAMBa - which is to train the new generation of interdisciplinary mathematicians. The funding body CSC provides the chance for Chinese students to study different subjects overseas and students are encouraged to have active academic communications with scholars all over the world. With all the resources provided by SAMBa and especially with the help of my supervisors, I will not only be provided the opportunity to learn more on this specific topic, but also to meet other researchers and to get to know about the latest work in the area.

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.