Accurate mathematical models of scientific phenomena provide insights into, and solutions to, pressing challenges in e.g., climate change, personalised healthcare, renewable energy, and high-value 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 human-readable, 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 time-span. 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, human-readable 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 large-scale 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 work
has gone into understanding the behaviour of iterative methods applied to these Toeplitz-structured 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 time running 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 end-users.