Automating optimisation subject to partial differential equations on high-performance computers.

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Optimisation problems appear across all areas of engineering. Optimisation consists of maximising the performance or minimising the cost of a system, subject to some constraints. For example, an aeronautical engineer will want to choose the best shape for a wing to maximise its efficiency, subject to the constraint that the wing will lift the aircraft, while a civil engineer will want to design the cheapest bridge that will support its load. An important class of optimisation problems is where the constraint is given by the laws of physics, such as the physical laws for fluids (in the wing case) and structures (in the bridge case). These problems can be very hard, and usually require massive supercomputers to solve them. A significant amount of mathematical research has gone into investigating techniques for solving them.Engineers currently face a major practical difficulty when trying to solve new kinds of such optimisation problems. The software required to solve these is very intricate, and often takes months or years to develop. This poses a very formidable barrier. This matters a lot, because these problems appear everywhere in engineering, and if we could solve them then we could design many things in a better way.I propose to do this by developing a software framework to generate optimisation codes, rather than have engineers develop them by hand. While the optimisation software is very complex, it has a compact mathematical structure: I propose to generate the optimisation software from a simple high-level description of this mathematical structure. By generating the necessary software, engineers can spend their time on using it to solve real problems. This framework will provide engineers with the necessary optimisaton software in days or weeks instead of months or years.Generating the optimisation codes from simple high-level input has another major advantage. The high-performance supercomputers necessary to solve these optimisation problems are extremely difficult to program efficiently, and are changing rapidly. Code must be tailored for a particular hardware architecture. As each new kind of computing platform comes out, an engineer must adapt the code. Instead, with my new approach, the engineer can simply re-generate the code from the same mathematical input, and the framework will specialise the code to best exploit the different platform. By updating the framework once, many engineers working on many different codes in many different areas can benefit quickly from advances in computational hardware.I will apply the software developed to two important engineering problems. The first engineering problem is found in the design of marine turbine farms for renewable energy. Marine renewable energy is very important to the UK. The government predicts that the industry will be worth £76 billion to the UK economy by 2050. A major problem facing the industry is how to position the turbines to extract the maximum possible energy from the tide. Choosing the best design is very important, as it can greatly change the efficiency. Solving this problem will directly contribute to the UK's energy security and carbon reduction goals.The second engineering problem is identifying regions of the heart that are damaged (ischaemic). Ischaemic heart disease is the most common cause of death in Western countries. When a doctor suspects that a patient has ischaemia, it would be very beneficial to know its location and extent. One possible approach to rapidly identify ischaemia is to extract information from electrocardiograms (ECGs). The optimisation problem is to identify the ischaemia that best explains the ECG measured from the patient. Solving this problem will directly contribute to better healthcare decisions, reducing the mortality rate and improving the long-term prognosis of survivors.