Random vortex method and Monte-Carlo simulations for wall-bounded flows

The dynamics of turbulent fluid flows is described by the prominent Navier-Stokes equations. The non-linear structure of these equations accounts for their mathematical complexity and gives rise to these turbulent solutions observable in nature. Turbulent flows are characterised by perceivably irregular changes in velocity which induces computational difficulty of numerical simulations of such flows. For instance, this implies that in the case of direct numerical simulations, one is required to use a mesh of small size to find the solution in finite difference methods leading to high cost in computations.

To overcome the aforementioned difficulty, we develop numerical methods based on Monte-Carlo simulations. Indeed, it might be advantageous as Monte-Carlo schemes are better when dealing with multivariate dynamics. However, to implement this approach, the solution to the incompressible Navier-Stokes equations is required to be explicitly represented in terms of some distributions. This turns out to be possible due to the random vortex method. Using this method, one writes the velocity of the flow in terms of a collection of distributions of Brownian particles following the associated Taylor's diffusion. Thus, one is able to formulate the original incompressible Navier-Stokes equations as an equivalent problem of solving some McKean-Vlasov type stochastic differential equations. Therefore, solving the closure problem for Taylor's diffusion, one derives an integral representation for the velocity field in terms of Taylor's diffusions which are easily simulated as they satisfy some (ordinary) stochastic differential equations. In this case, one can use Monte-Carlo methods to compute the integral representations for the velocity of the flow numerically.

This approach has been recently developed by Z. Qian particularly for flows occupying wall-bounded regions. This case is especially interesting in fluid dynamics and additionally important for the study of turbulence. Indeed, for free fluid flows occupying the whole space without boundary, the phenomenon of turbulence is observable only in the three-dimensional case, however, for wall-bounded regions turbulent motion is seen close to the boundary even in the two-dimensional case. The aim of the project is to develop numerical schemes using the approach we outlined above and conduct computational experiments for simulation of turbulent flows for particular regions with boundary in the two- and three-dimensional cases.

Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors

MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY

The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.

FINANCIAL SERVICES and GOVERNMENT

The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.

DATA SCIENCE:

Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs