Projects: Projects for Investigator 

Reference Number  EP/W035561/1  
Title  EPSRCSFI: Krylov subspace methods for nonsymmetric PDE problems: a deeper understanding and faster convergence  
Status  Started  
Energy Categories  Energy Efficiency(Residential and commercial) 5%; Not Energy Related 95%; 

Research Types  Basic and strategic applied research 100%  
Science and Technology Fields  PHYSICAL SCIENCES AND MATHEMATICS (Applied Mathematics) 85%; PHYSICAL SCIENCES AND MATHEMATICS (Computer Science and Informatics) 15%; 

UKERC Cross Cutting Characterisation  Not Crosscutting 100%  
Principal Investigator 
Dr J Pestana Mathematics University of Strathclyde 

Award Type  Standard  
Funding Source  EPSRC  
Start Date  01 August 2023  
End Date  31 July 2026  
Duration  36 months  
Total Grant Value  £355,812  
Industrial Sectors  No relevance to Underpinning Sectors  
Region  Scotland  
Programme  NC : Maths  
Investigators  Principal Investigator  Dr J Pestana , Mathematics, University of Strathclyde (100.000%) 
Industrial Collaborator  Project Contact , Numerical Algorithms Group Ltd (0.000%) Project Contact , Trinity College Dublin, the University of Dublin, Ireland (0.000%) 

Web Site  
Objectives  
Abstract  Accurate mathematical models of scientific phenomena provide insights into, and solutions to, pressing challenges in e.g., climate change, personalised healthcare, renewable energy, and highvalue manufacturing. Many of these models use groups of interconnected "partial differential" equations (called PDEs) to describe the phenomena. These equations describe the phenomena by abstractly relating relevant quantities of scientific interest, and how they change in space and time, to one another. These equations are humanreadable, but since they are abstract, computers cannot interpret them. As the use of computers is fundamental to effective and accurate modeling, a process of "discretisation" must be undertaken to approximate these equations by something understandable to the computer.Scientific phenomena are generally modelled as occurring in a particular space and during a particular timespan. The process called discretisation samples both the physical space and time at a discrete set of points. Instead of considering the PDEs over the whole space and time, we instead approximate the relationships communicated abstractly by the PDEs only at these discrete points. This transforms abstract, humanreadable PDEs into a set of algebraic equations whose unknowns are approximations of the quantities of interest only at these points. In order that the solution to these equations approximates the solution to the PDEs well enough, the discretisation generally must have a high resolution, meaning there are often hundreds of millions of unknowns or more.These algebra equations are thus largescale and must be treated by efficient computer programs. As the equations themselves are often able to be stored in a compressed manner, iterative methods that do not require direct representation of the equations are often most attractive. These methods produce a sequence of approximate solutions and are stopped when the accuracy is satisfactory for the model in question.The work in this proposal concerns analysing, predicting, and accelerating these iterative methods so they produce a satisfactorily accurate solution more rapidly. It is quite common that the algebraic equations arising from the aforementioned discretisation have an additional structure known as "Toeplitz". A great deal of workhas gone into understanding the behaviour of iterative methods applied to these Toeplitzstructured problems. In this proposal, we will extend this understanding further and develop new accelerated methods to treat these problems. Furthermore, a wider class of structured problems called Generalised Locally Toeplitz (GLT) problems can be used to describe the equations arising from an even larger class of mathematical models. We will extend much of the analysis of Toeplitz problems to the GLT setting. The work in this proposal will lead to faster, more accurate modelling of phenomena with lower energy costs, as they will not require as much timerunning on large supercomputers. Our proposal spans new mathematical developments, the proposal of efficient iterative methods, their application to models of wave propagation and wind turbines, and the production of software for endusers  
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Added to Database  15/02/23 