PDE-constrained optimisation: why is it so challenging and some methods to overcome these challenges

Sue Thorne
Computational Science & Engineering Department
Rutherford Appleton Laboratory
Chilton, Oxfordshire, UK


PDE-constrained optimisation is a hot topic with, for example, large European and German Science Foundation programmes aimed at tackling these problems. In this talk we will mainly focus on distributed control and boundary control problems:

Consider a PDE of the form L(u) = f on domain Omega, with Dirichlet or Neumann boundary conditions g. In distributed control, we are given g and a target uhat, and we wish to calculate f such that u approximates uhat over some domain. In boundary control, we are given f and a target uhat, and we wish to calculate g such that u approximates uhat over some domain. For example, we may wish to heat a room such that the temperature in certain parts of the room are close to a target value.

After discretisation of the above problems, we are left with a linear system that must be solved: this is a saddle-point system. The discretised PDE forms part of this linear system and we will show that this overall system has some very unfavourable properties. We will then consider the distributed control problems and how these systems may be efficiently solved with iterative methods. We will develop a (constraint) preconditioner that does not require us to perform accurate solves with the discretised PDE and show that we obtain mesh size independent convergence.

Finally, we will discuss Neumann boundary control problems and show that the method used for Distributed control cannot be immediately applied to these boundary control problems. Hence, we will require more exotic methods to solve these problems: one idea will be briefly presented.