EPSRC Reference: 
EP/R009821/1 
Title: 
Effective preconditioners for linear systems in fractional diffusion 
Principal Investigator: 
Pestana, Dr J 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics and Statistics 
Organisation: 
University of Strathclyde 
Scheme: 
First Grant  Revised 2009 
Starts: 
19 January 2018 
Ends: 
18 January 2020 
Value (£): 
92,739

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 

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Panel History: 

Summary on Grant Application Form 
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 socalled fractional diffusion problems, which occur when the diffusion process involves a number of different flow rates or longrange 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 timeconsuming 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 realworld 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.

Key Findings 
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Potential use in nonacademic contexts 
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Impacts 
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Summary 

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Organisation Website: 
http://www.strath.ac.uk 