# PhD Defence: All-at-once Multigrid Methods for Optimality Systems Arising from Optimal Control Problems

**All-at-once Multigrid Methods for Optimality Systems Arising from Optimal Control Problems**

In this thesis we construct and analyze multigrid methods for solving
the optimality system characterizing the solution of an optimal control
problem. Originally multigrid methods were constructed for elliptic
problems. However, the (discretized) optimality system is not elliptic.
We make use of the fact that the matrix is a block-matrix with saddle
point structure: on the one hand we have two blocks of variables
representing state and control. These two blocks are already part of the
optimal control problem itself. On the other hand, the conversion to the
optimality system requires the introduction of Lagrange multipliers,
which form the third block of variables.

There are several possibilities to use multigrid methods for
constructing solvers for such saddle point problems. One approach to
solve such problems is to apply multigrid methods in every step of an
overall block-structured iterative method to equations in just one of
these blocks of variables. Another approach, which we will follow here,
is to apply the multigrid idea directly to the (reduced or not reduced)
optimality system, which is called an all-at-once approach.

The choice of an appropriate smoother is the key issue in constructing
such a multigrid method. The other part of the method -- the coarse grid
correction -- can be set up in a canonical way because we will use a
framework of conforming geometric multigrid method. In this framework
the smoother will be constructed such that the convergence rates are
independent of the grid level. This leads to an overall computational
complexity that is linear in the number of unknowns which is called an
optimal convergence.

For a sub-class of the class of problems we will introduce in a first
point, we will go one step further and construct methods where the
convergence rates are also robust in a certain regularization or cost
parameter which is part of the problem. Moreover, we will show this
fact. Methods that do not take this into account typically show quite
poor convergence rates if this parameter attains small values.

For the analysis, we will introduce a general framework that is based on
Hackbusch's splitting of the analysis into smoothing and approximation
property. This allows to give general convergence theorems for the
methods under investigation. A second point we consider is local Fourier
analysis which allows to compute sharp bounds of the convergence rates.