Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Santolo Leveque, James R. Maddison, John W. Pearson

TL;DR

An automated framework for the numerical solution of optimal control problems with PDEs as constraints, in both the stationary and instationary settings, and numerical examples of the applicability of the software on classical control problems with PDEs as constraints are presented.

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.

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

TL;DR

An automated framework for the numerical solution of optimal control problems with PDEs as constraints, in both the stationary and instationary settings, and numerical examples of the applicability of the software on classical control problems with PDEs as constraints are presented.

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.
Paper Structure (10 sections, 27 equations, 3 figures, 4 tables)

This paper contains 10 sections, 27 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. The Software
  4. Firedrake
  5. The Control Class
  6. The Preconditioner
  7. Numerical Examples
  8. Poisson Control
  9. Navier--Stokes Control
  10. Conclusions

Figures (3)

  • Figure 3.1: Lines of code required to solve a heat control problem.
  • Figure 4.1: Lines of code required to solve a Poisson control problem.
  • Figure 4.2: Lines of code required to solve a time-dependent incompressible Navier--Stokes control problem, with the trapezoidal rule in time.