A framework for automated PDE-constrained optimisation

TL;DR

This paper introduces a flexible, automated framework based on FEniCS for solving PDE-constrained optimization problems, enabling high-level problem specification and efficient adjoint computation for complex, time-dependent PDEs.

Contribution

It extends a domain-specific language for variational problems to simplify expressing and solving PDE-constrained optimizations with automated adjoint derivation and code generation.

Findings

Efficiently solves complex PDE-constrained optimization problems.

Applicable to steady-state and transient PDEs, linear and nonlinear.

Demonstrates high performance on classical and advanced examples.

Abstract

A generic framework for the solution of PDE-constrained optimisation problems based on the FEniCS system is presented. Its main features are an intuitive mathematical interface, a high degree of automation, and an efficient implementation of the generated adjoint model. The framework is based upon the extension of a domain-specific language for variational problems to cleanly express complex optimisation problems in a compact, high-level syntax. For example, optimisation problems constrained by the time-dependent Navier-Stokes equations can be written in tens of lines of code. Based on this high-level representation, the framework derives the associated adjoint equations in the same domain-specific language, and uses the FEniCS code generation technology to emit parallel optimised low-level C++ code for the solution of the forward and adjoint systems. The functional and gradient information so computed is then passed to the optimisation algorithm to update the parameter values. This approach works both for steady-state as well as transient, and for linear as well as nonlinear governing PDEs and a wide range of functionals and control parameters. We demonstrate the applicability and efficiency of this approach on classical textbook optimisation problems and advanced examples.