Efficient smoothers for all-at-once multigrid methods for Poisson and Stokes control problems

TL;DR

This paper develops efficient all-at-once multigrid smoothers for Poisson and Stokes control problems, demonstrating improved convergence and speed, especially with a Gauss-Seidel like variant, and providing theoretical and numerical validation.

Contribution

It introduces a Gauss-Seidel like smoother for multigrid methods in optimal control problems, enhancing convergence speed and robustness over existing normal equation smoothers.

Findings

Multigrid solver convergence is robust to grid size and regularization parameter.

Gauss-Seidel like smoother speeds up the solver by about a factor of two.

Proposed methods are competitive with Vanka type methods.

Abstract

In the present paper we concentrate on an important issue in constructing a good multigrid solver: the choice of an efficient smoother. We will introduce all-at-once multigrid solvers for optimal control problems which show robust convergence in the grid size and in the regularization parameter. We will refer to recent publications that guarantee such a convergence behavior. These publications do not pay much attention to the construction of the smoother and suggest to use a normal equation smoother. We will see that using a Gauss Seidel like variant of this smoother, the overall multigrid solver is speeded up by a factor of about two with no additional work. The author will give a proof which indicates that also the Gauss Seidel like variant of the smoother is covered by the convergence theory. Numerical experiments suggest that the proposed method are competitive with Vanka type methods.