#
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

### Abstract:

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.