What accurately describes such realworld processes as fluid flow mechanisms, or chemical reactions for the manufacture of industrial products? What mathematical formalism enables practitioners to guarantee a specific physical behaviour or motion of a fluid, or to maximise the yield of a particular substance? The answer lies in the important scientific field of PDEconstrained optimisation.
PDEs are mathematical tools called partial differential equations. They enable us to model and predict the behaviour of a wide range of realworld physical systems. From the optimisation pointofview, a particularly important set of such problems are those in which the dynamics may be controlled in some desirable way, for instance by applying forces to a domain in which fluid flow takes place, or inserting chemical reactants at certain rates. By influencing a system in this way, we are able to generate an optimised outcome of a realworld process. It is hence essential to study and understand PDEconstrained optimisation problems.
The possibilities offered by such problems are immense, influencing groundbreaking research in applied mathematics, engineering, and the experimental sciences. Crucial realworld applications for such problems arise in fluid dynamics, chemical and biological mechanisms, weather forecasting, image processing including medical imaging, financial markets and option pricing, and many others. Although a great deal of theoretical work has been undertaken for such problems, it has only been in the past decade or so that a focus has been placed on solving them accurately and robustly on a computer, by tackling the matrix systems of equations which result. Much of the research underpinning this proposal involves constructing powerful iterative methods accelerated by 'preconditioners', which are built by approximating the relevant matrix in an accurate way, such that the preconditioner is much cheaper to apply than solving the matrix system itself. Applying our methodology can then open the door to scientific challenges which were previously out of reach, by only storing and working with matrices that are tiny compared to the systems being solved overall.
Recently, PDEconstrained optimisation problems have found crucial applicability to problems from data analysis. This is due to the vast computing power that is available today, meaning that there exists the potential to store and work with hugescale datasets arising from commercial records, online news sites, or health databases, for example. In turn, this has led to a number of applications of datadriven processes being successfully modelled by optimisation problems constrained by PDEs. It is essential that algorithms for solving problems from these applications of data science can keep pace with the explosion of data which arises from realworld processes. Our novel numerical methods for solving the resulting hugescale matrix systems aim to do exactly this.
In this project, we will examine PDEconstrained optimisation problems under the presence of uncertain data, image processing problems, bioinformatics applications, and deep learning processes. For each problem, we will devise stateoftheart mathematical models to describe the process, for which we will then construct potent iterative solvers and preconditioners to tackle the resulting matrix systems. Our new algorithms will be validated theoretically and numerically, whereupon we will then release an open source code library to maximise their applicability and impact on modern optimisation and data science problems.
