We present a full multigrid (FMG) half-space analysis for one-shot multigrid methods for time-dependent boundary control problems. The control problem is written as an optimization problem consisting of a linear parabolic PDE constraint and a quadratic tracking objective. More precisely, we want to find the optimal control which steers the state of the system as close as possible to a given, predetermined state. In addition we regularize the problem with a Tikhonov regularization term given by the squared L2-norm of the control, which can be interpreted as the cost of the control.
Numerical experiments reveal strong divergence of classical multigrid methods for small weights of the regularization term. Here classical is interpreted as using well known modifications of standard multigrid components near the boundary, e.g. restriction, prolongation, boundary condition discretization, etc. For the interior equations well established multigrid components were choosen, which illustrate that the divergence is induced by the boundary control and not the interior equations.
In order to analyse our multigrid method classical tools as for example Fourier analysis are insufficient. They fail to incorporate the effect of the boundary condition on the multigrid method. More precisely, they only analyse the interior equations on an infinite or periodic grid, i.e. they analyse the control problem without the boundary control. A more complicated FMG half-space analysis allows us to analyse the effect of the boundary. In our talk we present this FMG half-space analysis for a selection of boundary control problems, as well as the insights obtained from it. In addition we propose a modified multigrid method to solve the given boundary control problem.