Iterative Methods for PDE-Constrained Optimization

Nonlinear optimization is a fundamental branch of numerical analysis andis ubiquitous in science and engineering. It allows one, for example, todetermine an optimal flight plan under specified circumstances, to findthe shortest paths between fixed positions on a given manifold(geodesics), to predict the shape and layout of telescope arrays thatmaximize light intensity, or to find an optimal design among infinitelymany possible elastic structures. Recent progress in large-scalenonlinear optimization and linear algebra opens new perspectives on thesolution of optimization problems whose constraints are defined bydiscretized sets of partial-differential equations (PDEs). Such problemsare challenging both because of the structural properties present in thecontinuous equations and because of their sheer intimidating size. Theyare important because of their wide-ranging application; a particularlyvital instance is the Navier-Stokes equations, the basic equations offluid mechanics. Challenging as such equations may be, theirdiscretization by the finite-element method leads to a consistentstructure that can be exploited by numerical algorithms.Our proposed research is to design and analyse numerical methodsfor solving PDE-constrained optimization problems, and to buildhigh-quality software to allow others to use our work. We shallparticularly be concerned with methods which are guaranteed toconverge to a locally-best solution, and which ultimately convergerapidly to this solution fast. We shall address the issues ofadditional side constraints that might be imposed physically andof natural rank-deficiencies that arise from the differential equations.

Who will benefit: PDE-constrained problems are one of the most challenging classes of optimization problems today. Being able to solve instances with a specific structure not only leads to scientific advances in optimization and related fields but also to advances in industry. The potential impact of advances is thus significant in regards to science and industry but also for wealth creation. The main beneficiaries will be users of optimization, particularly those whose interests lie in simulation-based optimization. There are UK academic research groups (e.g., in Birmingham, Edinburgh, Oxford and Southampton) with strong interests in PDE-constrained optimization. Outside of academia, industries such as aerospace (optimal design of aeroplane components subject to flow constraints), utilities (power generation subject to Kirchhoff or similar continuous mass-balance laws) and finance (parameter fitting to Black-Scholes or similar models) are obvious target audiences. Other application areas are numerous and include oil extraction from porous media, tyre and heart pump design, medical imaging, wake vortex minimization, weather forecasting, optimal structure design and free-material optimization. How will they benefit: By definition, optimization must have a lasting impact as it aims to improve over what is currently acceptable. So, for the above applications, lighter components or reduced drag will lead to savings in fuel, optimized power scheduling to cheaper operating costs, and careful parameter fitting to more faithful prediction of future financial markets. But more importantly, since generic optimization is ``application independent'', gains in knowledge have immediate beneficial consequences for any ``customers'' of optimization. What will be done to ensure that they benefit. Our research is aimed at analysing which optimization methods have convergence guarantees, which have general applicability and which are robust. However, since our aim is to understand the generic rather than the specific, translating good algorithms into good software and thereafter tuning this for specific applications is a long-term enterprise. Our experience with both the GALAHAD and HSL software libraries is that reliable, well-documented code based on sound numerical methods takes months, sometimes years to build, but also that this investment pays off because of its general applicability - literally thousands of people have downloaded our software for their applications. We find it far more cost effective to design generic software, and leave the end users to tune this to their applications than to consider applications case by case. We believe that the high quality of our Group's software and its documentation makes this model feasible for any well-motivated end user. The software from this project will ultimately be available, without cost for academics and at reasonable cost to industry, as part of GALAHAD. It is, frankly, unlikely that polished software will result over the short timescale of this grant, more that the theoretical underpinning resulting from the collaboration will allow the PI/ReCoI and visiting researcher to develop such software in the succeeding months - we have an excellent track record going back almost ten years of successful collaborations. The software will be advertised, as are all of our products, by our Group's newly-appointed software manager.