Fast Solvers for Real-World PDE-Constrained Optimization

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.

PDE-constrained optimization is a rich subject area, with a great number of application areas. As such, there is enormous potential for this project to generate impact amongst scientists working in these operational research, optimization and applied science fields. In particular practitioners working on the following four application areas, which form the focus of the project, will greatly benefit from the work undertaken:
- Medical imaging applications, such as modelling glioma growth in brain tumours.
- Image processing, in particular models for optimal image denoising.
- Fluid dynamics, specifically researchers interested in the optimal control of fluid flow.
- Practical and computational chemistry and biology, in particular the modelling of chemical reactions and pattern formation processes within mathematical biology.

More generally, the research communities in operational research and applied mathematics will benefit from the new possibilities offered by this exciting field. I will attempt to maximize this benefit by submitting high-quality research articles to journals across a range of fields, as well as a review-type article on cutting-edge work in this area, writing publicly available software on solving real-world PDE-constrained optimization problems, and supervising a number of MSc students on related projects.

Another major priority of mine during this Fellowship is to establish greater links between researchers from academia and industry working on these problems, which I will do by conversing and working with industrial partners to discover problems of significant real-world applicability, and providing them with effective solution techniques for these problems. I will also hold two 2-day workshops during the Fellowship, with the first based explicitly on industrial applications of PDE-constrained optimization, and the second more academically based though with invitations again extended to industrial partners. I intend to invite many figures from relevant companies to give plenary talks at these impact-generating workshops.

I believe that the United Kingdom's research base in operational research and optimization will benefit greatly from the UK taking a more prominent role in the PDE-constrained optimization community. At the present time there are relatively few research institutions in the UK that are actively researching these important problems, when compared with continental Europe (in particular Germany and Austria), and I hope to play a role in changing this in a positive and significant way during this Fellowship.

In summary, I will maximize the impact of this project in the following ways:
- Publish a number of papers in esteemed journals, including a review-type article aimed at unifying state-of-the-art research in this subject area. The journals will include those within the mathematical sciences, as well as journals in other physical and chemical sciences.
- Present my findings at well-regarded scientific conferences across a range of disciplines.
- Collaborate and communicate extensively with industrial partners interested in application areas of PDE-constrained optimization, in order to adapt our models to reflect scientific processes in a potent and meaningful way, and aid the understanding of these real-world processes for scientists in the UK and elsewhere.
- Organize two workshops during my Fellowship, with an explicit aim of promoting interaction between academia and industry.
- Write publicly available software for solving a range of real-world PDE-constrained optimization problems, using previous experience that I have in writing and contributing to software projects.
- Supervise 4-6 Operational Research MSc students on topics related to this project, to facilitate Knowledge Transfer and enhance the scope of the work.