Solving Optimal Control Problems by Multigrid Methods with Transforming Smoothers

Volker Schulz
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.