Title: Optimisation with PDE constraints using automated consistent adjoints of finite element models
Speaker: Simon Funke
Affiliation: Applied Modelling and Computation Group at Imperial College
Location: CPSE seminar room (C615 Roderic Hill)
Time: 2:00pm

Abstract. Optimisation with partial differential equations (PDE) as constraints arise in many research areas from science engineering to finance. Typical examples are data assimilation for weather forecasting or ocean modelling and shape optimisation for wing designs in which PDEs enforce the laws of physics. The resulting optimisation problems are constrained by these nonlinear, time dependent differential equations which can be computationally extremely demanding to solve. Therefore the usage of gradient based optimisation algorithms is usually essential to reduce the number of optimisation iterations.
In this talk we present work towards a new framework for solving PDE constrained optimisation problems that aims to automate many of the steps involved in solving these kind of problems. Given a differentiable PDE model, the parameter set and a functional of interest, it applies gradient based optimisation to solve the optimisation problem. The key feature of this framework is the efficient gradient computation by automatically deriving and solving the associated the adjoint equation. The framework is demonstrated on examples for steady and unsteady optimal control problems.
One of the major difficulties in practice is the derivation and implementation of the adjoint system to efficiently compute gradient information; Naumann (2011) described it as “[…] one of the great open challenges in the field of High-Performance Scientific Computing” for large scale simulation code. There are two current approaches to derive the adjoint equation each of which suffer from their own limitations. Algorithmic differentiation (AD) derives the adjoint model directly from the model source code. In practice this technique has strong restrictions, and requires a major initial and ongoing investment to prepare the code for automatic adjoint generation. An alternative is to formulate the adjoint PDE and to discretise this separately. This approach, known as the continuous adjoint has the disadvantage that two different model code bases must be maintained and manually kept synchronised as the model develops.
In this talk, we present an alternative approach where the PDE is formulated in a high level language that resembles the matematical notation. The model is automatically generated using code generation (using the FEniCS project). In this approach it is the high level code specification which is differentiated, a task very similar to the formulation of the continuous adjoint. However since the forward and adjoint models are generated automatically, the difficulty of maintaining them vanishes and the software engineering process is therefore robust. The scheduling and execution of the adjoint model, including the application of an appropriate checkpointing strategy is managed by a library called libadjoint. In contrast to the conventional algorithmic differentiation description of considering a model as a series of primitive mathematical operations, libadjoint employs a new abstraction of considering the model as a sequence of discrete equations which are assembled and solved. It is the coupling of the respective abstractions employed by libadjoint and the FEniCS project which produces the adjoint model automatically, without further intervention from the model developer.

About the speaker. Simon is a PhD student in the Applied Modelling and Computation Group at Imperial College.