Automatic Differentiation for All-at-once Systems Arising in Certain PDE-Constrained Optimization Problems

TL;DR

This paper introduces an automated Python-based framework for efficiently solving PDE-constrained optimal control problems, supporting linear and nonlinear cases with advanced preconditioning for large-scale problems.

Contribution

The paper presents a novel automated framework leveraging Firedrake for PDE-constrained optimization, enabling compact problem definition and efficient solution of large-scale problems.

Findings

Effective handling of linear and nonlinear PDE-constrained control problems

Significant speed-up with advanced preconditioning techniques

Successful application to classical control problems with PDEs

Abstract

An automated framework is presented for the numerical solution of optimal control problems with PDEs as constraints, in both the stationary and instationary settings. The associated code can solve both linear and non-linear problems, and examples for incompressible flow equations are considered. The software, which is based on a Python interface to the Firedrake system, allows for a compact definition of the problem considered by providing a few lines of code in a high-level language. The software is provided with efficient iterative linear solvers for optimal control problems with PDEs as constraints. The use of advanced preconditioning techniques results in a significant speed-up of the solution process for large-scale problems. We present numerical examples of the applicability of the software on classical control problems with PDEs as constraints.