Development of time-parallel numerical integration algorithms using probabilistic methods with applications to magnetic fusion plasma.

The context of the research - In many branches of science, from astrophysics to epidemiology, mathematical and computational modelling is vital and often involves the numerical integration of large systems of differential equations. Many simulations are limited by bottlenecks that arise when attempting to numerically integrate over long time periods sequentially, causing them to become computationally intractable in real time. By distributing a calculation across the many processors present in modern supercomputers, the computational load can be reduced drastically. Whilst spatially-parallel integration methods have been well explored in the literature, much less attention has been devoted to parallelisation in the time dimension.
The aims and objectives of the research - The purpose of this PhD project will be to develop new (or improve existing) time-parallel integration algorithms, using stochastic and statistical methods, in order to reduce simulation times for large-scale systems of differential equations. The interplay between numerics and statistics (known as uncertainty quantification) is currently a very open field and the algorithm(s) being developed will need to be general enough to adapt to the complexity of the problems being solved.
The novelty of the research methodology - The majority of currently known time-parallel algorithms have been formulated, studied and analysed using deterministic methods. The algorithm(s) developed in this project will incorporate a measure of stochasticity in order to exploit the statistical differences between deterministically generated solutions. In doing this, statistical and machine learning approaches may be exploited for numerical gain.
The potential impact, applications, and benefits - A successfully designed algorithm could be used by UKAEA in order to significantly reduce simulation run times and hence integrate over previously (computationally) intractable ranges of time for magnetic fusion plasmas. These methods are not strictly limited to numerical fusion problems either. The same benefits could be realised in applications to other large-scale integration problems in climate modelling, astrophysics, drug research and many more.
How the research relates to the remit - By developing statistically-based time-parallel algorithms with applications to magnetic fusion plasma, the proposed area of research contributes to multiple themes under the EPSRC remit. Namely the 'Numerical analysis' and 'UK Magnetic Fusion Research Programme' themes.
Research area; Energy, Mathematical Sciences
External Partner - Culham Centre for Fusion Energy (part of the UK Atomic Energy Authority)

In the 2018 Government Office for Science report, 'Computational Modelling: Technological Futures', Greg Clarke, the Secretary of State for Business Energy and Industrial Strategy, wrote "Computational modelling is essential to our future productivity and competitiveness, for businesses of all sizes and across all sectors of the economy". With its focus on computational models, the mathematics that underpin them, and their integration with complex data, the MathSys II CDT will generate diverse impacts beyond academia. This includes impacts on skills, on the economy, on policy and on society.

Impacts on skills.
MathSys II will produce a minimum of 50 PhD graduates to support the growing national demand for advanced mathematical modelling and data analysis skills. The CDT will provide each of them with broad core skills in the MSc, a deep knowledge of their chosen research specialisation in the PhD and a complementary qualification in transferable skills integrated throughout. Graduates will thus acquire the profiles needed to form the next generation of leaders in business, government and academia. They will be supported by an integrated pastoral support framework, including a diverse group of accessible leadership role models. The cohort based environment of the CDT provides a multiplier effect by encouraging cohorts to forge long-lasting professional networks whose value and influence will long outlast the CDT itself. MathSys II will seek to maximise the influence of these networks by providing topical training in Responsible Research and Innovation, by maintaining a robust Equality, Diversity & Inclusion policy, and by integration with Warwick's global network of international partnerships.

Economic impacts.
The research outputs from many MathSys II PhD projects will be of direct economic value to commercial, public sector and charitable external partners. Engagement with CDT partners will facilitate these impacts. This includes co-supervision of PhD and MSc projects, co-creation of Research Study Groups, and a strong commitment to provide placements/internships for CDT students. When commercial innovations or IP are generated, we will work with Warwick Ventures, the commercial arm of the University of Warwick, to commercialise/license IP where appropriate. Economic impact may also come from the creation of new companies by CDT graduates. MathSys II will present entrepreneurship as a viable career option to students. One external partner, Spectra Analytics, was founded by graduates of the preceding Complexity Science CDT, thus providing accessible role models. We will also provide in-house entrepreneurship training via Warwick Ventures and host events by external start-up accelerator Entrepreneur First.

Impacts on policy.
The CDT will influence policy at the national and international level by working with external partners operating in policy. UK examples include Department of Health, Public Health England and DEFRA. International examples include World Health Organisation (WHO) and the European Commission for the Control of Foot-and-mouth Disease (EuFMD). MathSys students will also utilise the recently announced UKRI policy internships scheme.

Impacts on society.
Public engagement will allow CDT students to promote the value of their research to society at large. Aside from social media, suitable local events include DataBeers, Cafe Scientifique, and the Big Bang Fair. MathSys will also promote a socially-oriented ethos of technology for the common good. Concretely, this includes the creation of open-source software, integration of software and data carpentry into our computational and data driven research training and championing open-access to research. We will also contribute to the 'innovation culture and science' strand of Coventry's 2021 City of Culture programme.