PDE-constrained optimization in scientific processes

This project concerns the fast and accurate numerical solution of optimization problems with partial differential equation (PDE) constraints.

A vast number of important and challenging applications in mathematics and engineering are governed by optimization problems. One crucial class of these problems, which has significant applicability to real-world processes, including those of fluid flow, chemical and biological mechanisms, medical imaging, and others, is that of PDE-constrained optimization. However, whereas such problems can typically be written in a precise form, generating accurate numerical solutions on the discrete level is a highly non-trivial task, due to the dimension and complexity of the matrix systems involved. In order to tackle practical problems, it is essential to devise strategies for storing and working with systems of huge dimensions, which result from fine discretizations of the PDEs in space and time variables.

The project supervisor Dr John Pearson has demonstrated the viability of solving a range of problems of this form, using "all-at-once" solvers coupled with appropriate preconditioning techniques. These preconditioners may be embedded within suitable iterative methods to greatly accelerate the convergence of the solver, in such a way that one may frequently prove the expected convergence rates in a robust way.

This project has the following goals:

1. Investigate the potency of this approach for the application areas described above, by devising new preconditioners and storage strategies for the discretizations of the optimization problems.

2. Explore a recently developed methodology for solving such formulations with additional bound constraints, using the well established class of interior point methods.

3. Devise methods for each problem that have the potential to be applied in parallel over many computational units.

4. Produce high quality software 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/M018857/1, ’Fast solvers for real-world PDE-constrained optimization problems’. Please see http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/M018857/1 for further details.

Contact [Email Address Removed] for informal enquires.

References
[1] John W. Pearson and Andrew J. Wathen, A New Approximation of the Schur Complement in Preconditioners for PDE-Constrained Optimization, Numerical Linear Algebra with Applications, 19(5), pp. 816-829, 2012.

[2] John W. Pearson and Martin Stoll, Fast Iterative Solution of Reaction-Diffusion Control Problems Arising from Chemical Processes, SIAM Journal on Scientific Computing, 35, pp. B987-B1009, 2013.

[3] John W. Pearson, On the Development of Parameter-Robust Preconditioners and Commutator Arguments for Solving Stokes Control Problems, Electronic Transactions on Numerical Analysis, 44, pp. 53-72, 2015.

[4] John W. Pearson and Jacek Gondzio, Fast Interior Point Solution of Quadratic Programming Problems Arising from PDE-Constrained Optimization, to appear in Numerische Mathematik.