Modern Numerical Linear Algebra for Huge-Scale Optimization Problems

The fast and efficient solution of huge-scale optimization problems on modern computing architectures is ubiquitous in areas such as linear and quadratic programming, machine learning, data science, semidefinite programming, optimal control, design engineering, and many more. It is therefore essential to construct state-of-the-art numerical algorithms with the goal of solving the large and structured matrix systems that reveal the solution of the optimization problem as a whole.

The project supervisory team have previously demonstrated the viability of solving a number of problems of this form, using a bespoke class of mathematical solvers coupled with appropriate preconditioners. These may be embedded within suitable iterative methods to greatly accelerate the convergence of the solver, in such a way that one may solve large-scale problems that were previously beyond the capability of existing computing technologies. The aim of this project is to discover new methods for examining problems of wide scientific interest, thus opening up vast new avenues of research in this area.

During the course of study, the successful candidate will be tasked with accomplishing the following:

1. Investigating the potency of this iterative solution approach to a number of the application areas described above, by devising new preconditioners and computer storage strategies for these optimization problems.

2. Devising methods for each problem that have the potential to be applied in parallel over many computational units, using both mathematical theory and practical high performance computing.

3. Producing high quality software, in MATLAB, C++ or a similar programming language, which can be made publicly available, and therefore be readily used by experts from academia and industry.

This project is associated with the EPSRC Fellowship EP/M018857/2, ’Fast solvers for real-world PDE-constrained optimization problems’. Please see http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/M018857/2 for further details.