ICA3, University of Stuttgart, Germany

Optimal control problems in PDE are frequently encountered in inverse
modeling, shape optimization or process control problems. Typically these
problems evolve from simulation tasks defining some output states to be
influenced by some input controls. Recent efficient numerical methods
typically rely on the *direct discretization approach*, which treat the
states and controls together as unknowns of a discretized finite dimensional
constrained optimization problem and apply iterative methods of sequential
quadratic programming (SQP) type. Despite the resulting large number of
variables this approach has proven very successful since it enables a
simultaneous solution of the optimization and the simulation problem.

At the core of all SQP type methods lie *Karush-Kuhn-Tucker (KKT)*
systems. They can be considered special saddlepoint problems which, however,
differ from the ones from Stokes or Navier-Stokes discretizations. In this
talk a novel numerical multigrid approach is presented to the numerical
solution of such KKT systems. This approach is based on iterative null-space
methods employing so-called *transforming smoothers*.

A multigrid convergence proof for a model problem as well as numerical results for a practical inverse modeling problem from groundwater flow are given.