Effective preconditioners for linear systems in fractional diffusion

Mathematical models of the diffusion of signals and particles are important for gaining insights into many key challenges facing the UK today. These problems include understanding how fluid flows through the ground, which helps to ensure that we have safe drinking water; characterising the propagation of electrical impulses through the heart, which aids our understanding of heart disease; and developing accurate models of financial processes, which improve our economy by providing better predictions of financial markets. This project focuses on so-called fractional diffusion problems, which occur when the diffusion process involves a number of different flow rates or long-range effects. Fractional diffusion occurs in many applications, including the groundwater flow, cardiac electrical propagation, and finance problems listed above.

Solving mathematical models of fractional diffusion is challenging, and typically requires a numerical method, i.e. a computer simulation. Usually, the most time-consuming part of this simulation is solving thousands, or even millions, of interdependent linear equations on a computer. Indeed, the time required to solve this system of equations may be so large that we are prevented from simulating fractional diffusion problems that capture the true complexity of real-world applications. Reducing this solve time is thus crucial if we are to generate new scientific insights in important applications involving fractional diffusion.

This project will develop new methods for solving these huge systems of equations that are guaranteed to be fast. We will focus on iterative solvers, which are well suited to the class of numerical methods (computer simulations) on which we focus. Iterative solvers of systems of equations compute a new approximation to the solution at each step, and so are fast if a good approximation is found after only a few iterations. However, this is generally only possible if we apply a convergence accelerator, called a preconditioner, which captures the 'essence' of the linear system, but is cheap to use.

For many fractional diffusion problems, this preconditioner is currently chosen heuristically, i.e. without theoretical justification. Consequently, the preconditioner may fail to reduce the (very large) computation time needed to solve the linear system. The goal of this project is to propose new preconditioners and iterative methods that are theoretically justified, and hence guaranteed to converge quickly, for a range of fractional diffusion problems. We will develop new software that will enable people with fractional diffusion problems to easily use our improved solvers. Additionally we will apply these fast preconditioners and iterative solvers in a fractional diffusion model of groundwater flow of an important UK aquifer. Solving this model quickly will enable us to better track our drinking water, and identify possible sources of contamination.

The outcome of the proposed research will be state-of-the-art methods for the numerical solution of fractional diffusion problems. Since fractional diffusion equations arise in models with heterogenous or anomalous diffusion rates, they are applicable to a number of important physical processes in, e.g. hydrology, finance and biology. Companies in these fields looking to improve their understanding of these diffusion processes may, therefore, benefit from our research.

One specific application we have identified is groundwater flow in the presence of rock fractures, which often occurs in aquifers. We will work with the British Geological Society to implement our solvers in a fractional diffusion model of fluid transport through a UK aquifer. Since quality issues affect almost 50% of public supply groundwater in the UK, it is clear that improved models of groundwater flow, that give insight into where and why these quality issues arise, could have real economic and societal benefits.

Other external beneficiaries will be identified by working with Strathclyde University's Statistics and Mathematics Advice, Research and Training (SMART) consultancy unit, and the university's Technology and Innovation Centre, both of which connect academics to external partners. Additionally, a two-day workshop will bring together academics and industrial partners with an interest in fractional diffusion problems. The aim of this workshop will be to foster collaborations and develop new pathways to impact.

Since easy-to-use, well documented software is critical to the uptake of our work, we will provide Python and Matlab codes on a dedicated website with examples of use. The site will also discuss the different numerical methods available to solve fractional diffusion problems and will provide links to other researchers in this field. We will additionally discuss our results with the Numerical Algorithms Group, who provide the largest commercially available collection of numerical and statistical algorithms in the world, and will aim to produce preconditioners of value to their collection.

Public engagement will also form part of our strategy to maximise impact. Specific events we will target are the Engage with Strathclyde week, the University's flagship knowledge exchange event; the Glasgow Science Festival, which attracts more than 50 000 members of the public annually; and the "meet the scientist" scheme at the University of Glasgow. To ensure these activities are successful, the Principal Investigator will also undertake public engagement training.