Robust finite element discretization and solvers for distributed elliptic optimal control problems

TL;DR

This paper develops robust finite element discretization methods and efficient solvers for distributed elliptic optimal control problems, achieving optimal error bounds and computational complexity with simple preconditioned iterative methods.

Contribution

It introduces a novel choice of regularization parameter and preconditioned iterative solvers that ensure asymptotically optimal complexity for solving discretized elliptic control problems.

Findings

Optimal error estimate with regularization parameter $ ho=h^4$.

Efficient Jacobi-like preconditioned MINRES and Bramble-Pasciak CG methods.

Validated results on benchmark problems with various target regularities.

Abstract

We consider standard tracking-type, distributed elliptic optimal control problems with $L^2$ regularization, and their finite element discretization. We are investigating the $L^2$ error between the finite element approximation $u_{\varrho h}$ of the state $u_\varrho$ and the desired state (target) $\bar{u}$ in terms of the regularization parameter $\varrho$ and the mesh size $h$ that leads to the optimal choice $\varrho = h^4$. It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble-Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.