State-based nested iteration solution of optimal control problems with PDE constraints

TL;DR

This paper introduces a state-based nested iteration method for efficiently solving optimal control problems constrained by PDEs, including elliptic, parabolic, and hyperbolic types, with error analysis and numerical validation.

Contribution

It develops a unified framework for PDE-constrained optimal control problems using nested iteration and finite element discretization, with optimal regularization and error estimates.

Findings

Optimal relation between regularization parameter and mesh size.

Asymptotically optimal complexity in solving algebraic systems.

Numerical results confirm theoretical error estimates.

Abstract

We consider an abstract framework for the numerical solution of optimal control problems (OCPs) subject to partial differential equations (PDEs). Examples include not only the distributed control of elliptic PDEs such as the Poisson equation discussed in this paper in detail but also parabolic and hyperbolic equations. The approach covers the standard $L^2$ setting as well as the more recent energy regularization, also including state and control constraints. We discretize OCPs subject to parabolic or hyperbolic PDEs by means of space-time finite elements similar as in the elliptic case. We discuss regularization and finite element error estimates, and derive an optimal relation between the regularization parameter and the finite element mesh size in order to balance the accuracy, and the energy costs for the corresponding control. Finally, we also discuss the efficient solution of the resulting systems of algebraic equations, and their use in a state-based nested iteration procedure that allows us to compute finite element approximations to the state and the control in asymptotically optimal complexity. The numerical results illustrate the theoretical findings quantitatively.