M13: Solvers for optimal control of time-dependent PDEs

Background

Partial differential equations (PDEs) are used throughout science and engineering in order to describe varied physical processes. Specifically, many engineering problems require numerical methods to optimise a particular problem with respect to the underlying physics and this leads to problems of PDE-constrained optimisation. For such problems the computation of gradients required for the optimisation leads to the need to solve the adjoint PDE as well as the original PDE.

For steady state PDEs much has been achieved in recent years and now there is a relatively well-understood theory and several capable methods. However, the success of these methods depends largely on whether the underlying linear algebraic equations are of small enough dimension to be solved.

Techniques and Challenges

This problem has been significant in the case of steady PDEs. However it has been regarded as an overwhelming one for time-dependent  PDEs. The aim of this work will be to develop and understand techniques to solve these types of systems.

The Future

In very recent work, a radical and apparently feasible way to solve such time-dependent PDE-constrained optimisation problems has been identified and this project will involve furthering these ideas.

References

Elman H., Silvester D., Wathen A.: Finite Elements and Fast Iterative Solvers: with applications in incompressible fluid dynamics, Oxford University Press, 2005

Rees, T., Dollar, S., Wathen, A.J.: Optimal Solvers for PDE-Constrained Optimization,  SIAM J. Sci. Comput. (1), pp. 271-298, 2010