Optimising a chemical plant or designing a building often requires solving equations so vast that even supercomputers struggle to produce an answer in a useful timeframe. These equations, called partial differential equations (PDEs), describe how physical systems behave—from fluid flow to cell biology—and the goal is to control them for the best outcome, such as maximising yield or minimising turbulence. The core problem is that when these systems are driven by huge amounts of fine-grained data, the resulting mathematical models become computationally intractable. This project introduces randomness into the numerical methods used to solve these optimisation problems. Borrowing techniques from randomised linear algebra, the team will develop algorithms that produce robust solutions in real time while using far less computer memory and enabling parallel computing. They will test these methods on four types of problems, including diffusion-driven processes and systems with uncertain inputs. If successful, the work could make real-time control of large-scale industrial processes feasible—for example, adjusting a chemical reaction on the fly or optimising airflow in a building’s ventilation system. The researchers will collaborate with engineering firms FESTO and Arup to apply the methods to concrete industrial challenges.
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How can a huge-scale chemical process be managed to guarantee that the largest possible yield of a certain substance is produced? How can we ensure that buildings and other infrastructure are optimally designed? How can fluid dynamics processes be impacted so as to minimize turbulence or maximize flow in a particular region? These, and many other important questions from science, engineering, and industry, may be tackled through the optimization and control of problems involving partial differential equations (PDEs). PDEs are used to describe mathematically how real-world physical systems behave: they can model cell biology, chemical reactions, processes in mathematical finance, fluid flow, quantum mechanics, and a vast range of other processes. What we are particularly interested in is the optimization of such problems, where we apply some external forces on the dynamics so that the system will behave in the 'best' possible way. This motivates the main focus of this project: the study of PDE-constrained optimization, including particular problem formulations which are often referred to as infinite-horizon control or model-predictive control problems. The possibilities such formalisms offer is enormous, driving cutting-edge research in engineering, systems biology, chemical processes, imaging, and many other fields. Whereas many such problems can be clearly stated on paper, accurately resolving them on a computer is a very important, and difficult, challenge. Indeed, for many problems with information provided at a very fine level, in particular resulting from processes driven by vast quantities of data, the resulting systems of equations are of such enormous scale that producing accurate numerical solutions can be intractable. This work seeks to resolve this challenge, by bringing to bear modern technologies from the field of numerical linear algebra, in particular through the timely and exciting research area of randomized linear algebra. The exploitation of current methodologies can ensure the generation of robust solutions in real-time, while minimizing computer storage requirements, and often enabling the use of parallel computing. The usage of randomization within solvers for PDEs themselves has been well established, however the development of such solvers is so far an underexplored area for optimization and control problems where the PDEs act as constraints. We will meet this outstanding challenge through four ambitious work packages: (i) using randomization in eigenvalue iteration for parabolic PDEs, an important class of PDEs which describe diffusion-driven processes in particular; (ii) randomized features within iterative methods for modelling processes with nonlinear phenomena; (iii) randomization within model-order reduction for PDE-constrained optimization, where the computational complexity of the model itself is reduced to make feasible a range of numerical algorithms for the solution; (iv) the use of randomized solvers for problems which have uncertain inputs. The final package will bring together all of the previous work, devising an overarching framework for treating optimization problems with randomness built in. Our new algorithms will be analysed theoretically and validated numerically, on a wide variety of huge-scale problems. Interaction with industry and across academic disciplines is a key outcome. Industry impact will be generated in collaboration with project partners FESTO and Arup, with whom we will apply our methodology to optimization and control problems that have a key link to national economic challenges. We will release code libraries, and organise a workshop with academic and industrial invitees, to further enhance the scientific and commercial impact of these new developments.
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