The automation of PDE-constrained optimisation and its applications

Abstract

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.