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.

This project aims to understand how development projects and biodiversity conservation interact, focusing on the case of Gola Forest in West Africa.

Tree planting is at the forefront of the current environmental agenda to mitigate climate change and the biodiversity crisis. For instance, the Scottish Government has a current target of planting 12,000 ha of trees per year, increasing to 15,000 ha from 2024. Agricultural land covers over 70% of the UK, so it is likely that a large proportion of tree cover expansion will take place on farmland. Given the many demands we place on rural landscapes, alongside the semi-permanency of tree cover, it is crucial to determine the spatial arrangements that maximise the biodiversity and wider environmental benefits derived from trees. Trees can be planted in a range of spatial configurations along a gradient of land-sparing (e.g. creating woodland patches) to land-sharing (e.g. in agroforestry systems that integrate trees with crops or livestock). Whilst both these strategies have the potential to deliver a range of biodiversity benefits, they are unlikely to be equivalent. The aim of this studentship is to assess the relative merits of separating versus integrating trees with food production to provide the greatest benefits for biodiversity (with a focus on invertebrate communities because of their diverse ecological roles which underpin ecosystem functions and services vital to food production). Specific research questions to address include: 1) How are invertebrate communities in agricultural landscapes influenced by trees in various spatial configurations, representing a gradient of land-sparing to land-sharing management approaches? 2) How do invertebrates with varying life histories and functional traits respond to tree planting along the land sharing-sparing continuum, and what implications does this have for ecosystem service provision? 3) Are land managers more amenable to planting trees in certain spatial configurations than in others, and what implications does this have for the implementation of tree planting strategies? To address these questions, fieldwork will be conducted in agricultural sites where trees have been planted across a range of spatial configurations encompassing a gradient of land-sparing to land-sharing gradient. Data collection will focus on ground-dwelling predators (e.g. carabid beetles and spiders), with the potential to also include insect pollinators (e.g. hoverflies, bees and butterflies) to encompass a wider range of functional traits and ecosystem services. The student will have access to an existing dataset of beetles (30,000 individuals of 130 species) and spiders (4,000 individuals of 103 species) which have been recorded at 60 woodlands planted on former agricultural land (part of the WrEN project , and to a beetle functional traits database compiled by SRUC.

Doctoral Training Partnerships: a range of postgraduate training is funded by the Research Councils. For information on current funding routes, see the common terminology at https://www.ukri.org/apply-for-funding/how-we-fund-studentships/. Training grants may be to one organisation or to a consortia of research organisations. This portal will show the lead organisation only.

Terrestrial ecosystems currently absorb one quarter of the carbon dioxide that Humankind releases into the atmosphere, thus reducing the rate of climate change. In this context, Amazon rainforest is extremely important, absorbing more than half a billion tonnes of carbon per year. This represents more than the combined emissions from the USA and China. However, we have limited understanding of how the productivity of Amazon forests is controlled, and this reduces our ability to predict what will happen in the future as atmospheric CO2 concentrations continue to rise and the climate changes. One of the main paradigms in ecology is that the productivity of tropical ecosystems, which occur on old, highly-weathered soils, is limited by the availability of phosphorus. This contrasts with more temperate ecosystems whose productivity has been shown to be limited by nitrogen availability. However, the phosphorus paradigm has not been tested in detail as there have been very few nutrient manipulation studies in tropical forests, and no large-scale study has been carried out in Amazon forest. This is a major issue because soil nutrient availability in most of Amazonia is substantially lower than in Panama, the location of the only ongoing fertilisation experiment in tropical lowland rainforest. Thus, the Panama findings may not be representative of large areas of Amazonia, and, therefore, our understanding of the role soil fertility plays in controlling tropical forest productivity is incomplete. Testing the phosphorus paradigm in Amazonia is critical for two reasons. Firstly, eastern and central Amazonia, the area which contains the lowest fertility soils, is considered to be at risk from the adverse effects of climate change, with widespread dieback predicted by some scientists. The resilience of these forests is considered to be highly dependent on whether trees are able to increase their growth in response to rising atmospheric CO2 concentrations, and this ability is likely to depend on the extent to which their growth is currently limited by soil nutrient availability. Secondly, there is growing evidence that the response of ecosystems to global change may differ depending on which nutrient limits their productivity. Therefore, establishing the first large-scale nutrient manipulation study in Amazonia should represent one of greatest priorities for ecosystem and climate change research. We will do just that, manipulating nitrogen, phosphorus and cation availability in central Amazon forest, at a site representative of the most common soil type in the Basin, and will quantify the response of key forest processes. We will determine the impacts on photosynthesis, plant respiration, biomass production and turnover, and decomposition, ultimately allowing us to take a full-ecosystem approach to establish how carbon storage has been affected. The new knowledge and understanding which we generate will be used to improve Amazon process representation in the Joint UK Land Environment Simulator (JULES). This will be the first time that multi-nutrient control of tropical forest function has been included in a dynamic global vegetation model, allowing for more realistic simulation of the response of the Amazon carbon cycle to environmental change. This will improve our ability to predict how the Amazon rainforest will change during the 21st century and what the implications will be for rates of regional and global climate change. In summary, our project will address a fundamental ecological question and will improve greatly our understanding of an issue that contributes substantially to uncertainty in predictions of rates of 21st century climate change; namely, how the productivity of one of the most important natural carbon sinks on the planet, the Amazon rainforest, is controlled.