A huge number of important and challenging applications in operational research are governed by optimization problems. One crucial class of these problems, which has significant applicability to real-world processes, is that of partial differential equation (PDE)-constrained optimization, where an optimization problem is solved with PDEs acting as constraints. To provide one illustration, such formulations arise widely in image processing applications: this produces a crucial link to scientific and technological challenges from far-and-wide, for example determining the health of complex human organs such as the brain, exploring underground geological structures, and enabling Google cars to function without a human driver by assessing traffic situations. The possibilities offered by PDE-constrained optimization problems are immense, and consequently they have recently attracted tremendous interest from researchers in mathematics, as well as applied scientists more widely. These formulations may also be used to describe processes in fields as wide-ranging as fluid dynamics, chemical and biological mechanisms, other image processing problems such as medical imaging, weather forecasting, problems in financial markets and option pricing, electromagnetic inverse problems, and many other applications of importance. The study of these problems is therefore a cutting-edge research area, and one which can forge a huge advance in the fields of operational research and optimization. There has been much theoretical work undertaken on these problems, however the construction of strategies for solving these optimization problems numerically is a relatively recent development. In this project I wish to build fast and effective solvers for the matrix systems involved (these systems contain all of the equations which arise from the problem). The solvers are coupled with the development of a powerful 'preconditioner' (the idea of which is to approximate the corresponding matrix accurately in some sense, but in a way that is cheap to apply on a computer). Carrying this out is a highly non-trivial challenge for many reasons, specifically that it is often infeasible to store the matrix in its entirety at any one time, it is very difficult to build an approximation that captures the properties of the matrix in an effective way and is also cheap to apply, it is frequently necessary to build solvers which are parallelizable (meaning that computations may be carried out on many different computers at one time), and one is often required to carry out the expensive process of re-computing many different matrices. The aim of this project is to build powerful solvers, which counteract the above issues, for PDE-constrained optimization problems of significant real-world and industrial value. I will consider four specific applications: optimal control problems arising from medical imaging applications, PDE-constrained optimization formulations of image processing problems, models for the optimal control of fluid flow, and control problems arising in chemical and biological processes. I will consider problem statements that have the maximum practical potential, and generate viable, fast and effective solution strategies for these problems.

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The soil that makes up the earth's substrate is an ever-changing assemblage of organic matter, minerals, organisms, gases, and liquids. Plants, animals, and humans in a host of ways mine this substrate to make life possible and use it as a container for things both precious and toxic. In this central function of earthly survival, soil's composition is both an archive of past life as well as the planet's largest carbon sink, capturing within it emissions from centuries ago. It is, in its very essence, a historical record that shapes our collective future. "Stories from the Substrate" reflects on the historical composition of soil by using it as a medium for engaging with and narrating East African history and as a point of view for considering the epoch of the Anthropocene. This project begins with both archival and fieldwork in East Africa (Tanzania, Uganda, and Kenya) that will form the basis of a collaborative, interactive soil map website and scholarly monograph that highlights how East Africans have relied on and shaped soil with the help of plants and animals. Previous scholarship on soil in East Africa has focused almost exclusively on colonial development projects and campaigns against erosion, this project takes a multispecies approach and centres East African epistemologies of soil to view it as not just a medium of agriculture, but as a building material, a source for mining mineral salts, the haven of "good" and "bad" microbial life, a sink for pollutants, a metaphor of home and belonging, and a valuable commodity of both local and global trade. Chronicling a history of the region from the soil seeks to set aside the prevailing themes of extant regional histories that narrate from political events and the ethnic and colonial logics of the archive. By considering the region through its substrate, this project hews close to the everyday and the often unseen or overlooked ways that people, animals, and their environments make one another and shape history. While each case study takes a particular place as its starting point, several themes will run throughout, assembling a picture of landscape, ecology, human health and labor in the region. In bringing together a variety of attachments to the soil in East Africa, this project aims to contribute to Black Ecologies, an interdisciplinary field seeking to examine the relationship between Black people and their environments. This work highlights the harm and toxicity that comprise many of these environments due to centuries of oppression and dispossession, but it also examines modes of knowing and being with nature that have gone under-explored. From this initial research, the project then shifts to collaborate with a wider research community to "Think with the Substrate". Bringing together scholars who work on histories of substrate materials, we will consider how humans have variably remade and relied on this middle layer between the subterranean and terrestrial. This inter-disciplinary project, in the spirit of other work on the Anthropocene such as the recent Plantationocene project, aims to engage with an unwieldy chunk of human history through a unifying object of inquiry. While the Anthropocene has been defined as the epoch when human impact has been the most consequential force shaping the earth, many scholars have critiqued the totalizing concept for eliding the fact that all societies have not had equal impact. To think about the Anthropocene "with the substrate" is not to assemble a list of different uses and interventions into the ground. It is a project of using historical engagements with substrate matter to reveal variable social relations, conceptions of nature, and engagements with nonhuman animals. These stories will no doubt chronicle the rise of global capitalism, but it will also capture the many alternative ontological and epistemological worlds that are sometimes forgotten and overshadowed in blunt narratives of capitalism and the Anthropocene.

The understanding of non-equilibrium quantum systems is one of the greatest challenges of modern science, as recognised by the EPSRC Grand Challenges Programme. Its development will have profound effects across different research areas including quantum computation and information, quantum optics, and biology. Theoretical understanding of these systems will form the basis for future development of new generation fast and energy efficient microchips and instruments for precise measurements. The occurrence of non-equilibrium behaviour is very common in Nature. The simplest example of this can be found when two objects with different temperatures come into contact. Other systems can show various levels of complexity from the physical process that leads to emission of a laser beam to the ultimate case of living organisms. The common characteristic property of these systems is the absence of uniform thermodynamic quantities such as temperature. Some of the state-of-the-art experiments in this field are made with semiconductor nano-structures in high magnetic fields and very low temperatures. In these systems electrons move in a coherent way similar to photons in a laser beam. Remarkably, because of strong interactions, the electrons in these systems form new strongly-correlated emergent states which exhibit quasi-particles with only a fraction of the electron charge. Similar quasi-particles also occur in quantum magnetic materials in the so-called spin liquid states. In the future it is hoped that these particles will be used as fundamental building blocks of topological quantum computers. The problem of quantum motion of a large number of quasi-particles is in the class of non-equilibrium quantum problems, whose study constitutes one of the main aims of this research programme. Interestingly, many of these systems show non-equilibrium steady states. Take a piece of metal and connect it on opposite sides to a heater and a refrigerator, a configuration which will result in a steady heat flow. A similar situation occurs in a system of interacting electrons in a quantum wire connected to a battery. The important differences with the former arise from the fact that the motion of particles in the wire obeys the laws of quantum mechanics, which lead to unusual quantum states. Recently it became possible to study these states in experiments, which resulted in a number of unexpected observations e.g. PRL 96, 016804 (2006); PRL 105, 056803 (2010). Next generation experiments will build quantum devices that use and explore the physics of non-equilibrium states based on the new theoretical and experimental insights. The project is aimed at theoretical understanding of quantum systems which are driven far from equilibrium by, for example, applied voltage or fast switching of external fields. In this setting many physical systems with examples ranging from semiconductor nano-structures and superconductors to quantum magnets and ultra-cold atomic gases show remarkable emergent behaviour (see for example PRL 105, 056803 (2010), arXiv:1308.4336, Science 331, 189 (2011), Nature Physics 8, 325 (2012) etc). This comes as a result of intricate quantum entanglement which occurs in these systems due to motion of interacting particles under non-equilibrium conditions. The properties of these systems cannot be explained using standard theoretical framework, and it is the one of the central tasks of this project to develop this theoretical description.

The structure-property relation of semiconducting polymers is poorly understood and there is no guidance for the design of new materials. We have noted an important mismatch between the assumption used to model charge transport in polymers (phenomenological theories) and the results of electronic structure calculation of realistic polymers (atomistic theories). The latter show that, as carriers are promoted to higher energy, they access more delocalized states characterized by longer range electron transfer and smaller polaronic effects. No model of transport currently takes this effect into account. This proposal will build new foundations of the phenomenological theories based on a more detailed knowledge of the electronic properties of few selected systems. Existing results on amorphous low mobility polymers (like PPV) and semicrystalline polymers (line P3HT and PBTTT) will be combined with new simulations of amorphous high mobility polymers (of the DPP class) to achieve a detailed description of representatives from each main class of semiconducting polymers. A new, more general and more accurate, expression for the rate of change hopping between sites will be introduced. A model Hamiltonian will be built to reproduce the main features of the electronic structure of realistic polymers. In essence, we will build a connection between detailed models of the chemistry of the system and the more simplified models needed to study charge transport. With the new methodology we will determine the actual number of parameters that affects the mobility in polymeric semiconductors considering the recent experimental observation by Paul Blom's group that the incredible diversity in semiconducting polymers may be actually reducible into a single effective parameter per material. Moreover, the methodology lends itself to making predictions on new chemical structures of high mobility polymers. This proposal benefits from the collaboration of Paul Blom (Eindhoven), David Haddleton (Warwick). A PhD student already in the group will contribute to some of the tasks.