Can we make long-term predictions?

The project will consist of proving uniform in time error bounds on approximations to the solutions of Stochastic differential equations, beginning with those obtained through multiscale methods. In order to do this, common tools in Markov processes, Markov semigroup theory, stochastic analysis, numerical methods, and probability will be utilised. Multiscale methods are particularly relevant when modelling biological processes and multiscale methods. Understanding the error bounds that we can obtain on our approximations will allow better quantification of uncertainty, as well as tighter predictions. The bounds being uniform in time will allow an understanding of when we can expect our approximations to hold well for all time, and when they will deteriorate over time.

Current error bounds often hold for finite-time windows. That is, we have proven for many processes and models that the approximation is a good one up until some time T. The novel nature of the project is in obtaining bounds that are independent of time, to allow for long term approximations and greater certainty of predictions.

The overarching goal of this project is to produce a novel theory, including practical criteria, to understand when a given random dynamics can be approximated - either via numerical schemes or via other procedures - with an error which does not increase in time. This project contains two "sub-projects". One considering approximations produced via numerical schemes and one considering approximations produced via other procedures; in the latter case it will in concentrate on averaging or homogenization procedures.

The project will not be focussing on applications specifically but these problems are inspired by applications to mathematical biology, swarming in particular, with a number of applications in engineering, material science, physics etc.

MAC-MIGS develops computational modelling and its application to a range of economic sectors, including high-value manufacturing, energy, finance and healthcare. These fields contribute over £500 billion to the UK economy. The CDT involves collaborations with more than a dozen companies and organisations, including large corporations (AkzoNobel, IBM, Dassault, P&G, Aberdeen Standard Investments, Intel), mid-size firms, particularly in the engineering and power sectors (NM Group, which provides monitoring services to power grid operators in 30 countries, Artemis Intelligent Power, the world leader in digital displacement hydraulics, Leonardo, a provider of defense, security and aerospace services, and Oliver Wymans, a management consultancy firm) and startups such as Brainnwave, which develops data-modelling solutions, and Opengosim which designs state-of-the-art and massively parallel software for subsurface reservoir simulation. Government and other agencies involved will include the British Geological Survey, Forestry Commission, James Hutton Institute, and Scottish National Heritage. Engagement will be via internships, short projects and PhD projects. BIS has stated that "Organisations using computer generated modelling and simulations and Big Data analytics create better products, get greater insights, and gain competitive advantage over traditional development processes". Our partners share this vision and are keen to develop deeper collaborations with us over the duration of the CDT.

Our CDT will achieve the following:

- Produce 76 highly skilled mathematical scientists and professionals, ready to take up positions in academia or in companies such as our partners. The students will have exposure to projects, modelling camps and high-level international collaborations.

- Deliver economic and societal benefits through student research projects developed in close collaboration with our partners in industry, business and government and other agencies.

- Create pathways for impact on computer science, chemistry, physics and engineering by involving interdisciplinary partners from Heriot-Watt and Edinburgh Universities in the supervision and training of our students.

- Organise a large number of lectures and seminars which will be open to staff and students of the two universities. Such lectures will inform the wide university communities about the state-of-the-art in computational and mathematical modelling.

- Work with other CDTs both in Edinburgh and beyond to organise a series of workshops for undergraduates, intended to foster an increased uptake of PhD studentship places in technical areas by female students and those from ethnic minorities, with potential impact on the broader UK CDT landscape.

- Organise industrial sandpits and modelling camps which offer the possibility for our partners to present a challenge arising in their work, and to explore innovative ways to tackle that challenge, fully involving the CDT students. This will kick-start a change in the corporate mindset by exposing the relevant staff to new approaches.

- Develop a new course, "Entrepreneurship for Doctoral Students in the Mathematical Sciences" in conjunction with Converge Challenge (Scotland's largest entrepreneurial training programme) and UoE's School of Business. This and other support measures will develop an innovation culture and facilitate the translation of our students' ideas into commercial activities.