This thesis is concerned with the automation of solving optimisation problems
constrained by partial differential equations (PDEs). Gradient-based
optimisation algorithms are the key to solve optimisation problems of practical
interest. The required derivatives can be efficiently computed with
the adjoint approach. However, current methods for the development of
adjoint models often require a significant amount of effort and expertise, in
particular for non-linear time-dependent problems.
This work presents a new high-level reinterpretation of algorithmic differentiation
to develop adjoint models. This reinterpretation considers the
discrete system as a sequence of equation solves. Applying this approach
to a general finite-element framework results in an automatic and robust
way of deriving and solving adjoint models. This drastically reduces the
development effort compared to traditional methods.
Based on this result, a new framework for rapidly defining and solving
optimisation problems constrained by PDEs is developed. The user specifies the discrete optimisation problem in a compact high-level language
that resembles the mathematical structure of the underlying system. All
remaining steps, including parameter updates, PDE solves and derivative
computations, are performed without user intervention. The framework
can be applied to a wide range of governing PDEs, and interfaces to various
gradient-free and gradient-based optimisation algorithms.
The capabilities of this framework are demonstrated through the application
to two PDE-constrained optimisation problems. The first is concerned
with the optimal layout of turbines in tidal stream farms; this optimisation
problem is one of the main challenges facing the marine renewable energy industry. The second application applies data assimilation to reconstruct
the profile of tsunami waves based on inundation observations. This provides
the first step towards the general reconstruction of tsunami signals
from satellite information.