Subject: Multigrid Solvers for Saddle Point Problems in PDE-Constrained Optimization
treated by: René Simon
Supervisor: Prof. Dr. Walter Zulehner
External Referee: Prof. Dr. Ronald H.W. Hoppe (University of Houston, University of Augsburg)

This works deals with efficient solution methods for solving large scale discretized optimization problems with constraints in form of partial differential equations (PDEs). Using the simultaneous approach, i.e. the state constraint is not eliminated, one has to solve the corresponding optimality system, also called the Karush-Kuhn-Tucker (KKT) system. This leads to large scale saddle point problems.

A special class of efficient iterative solvers are multigrid methods. In this work the so-called one-shot multigrid-strategy is chosen, i.e. the multigrid method is directly applied to the whole discretized problem and not only as a preconditioner for the involved PDE and the Schur complement of the KKT system. One of the most important ingredients of a multigrid iteration is an appropriate smoother. We consider here the construction of additive Schwarz-type patch smoothers, where the computational domain is divided into small patches. On each patch local problems are solved one-by-one in a Jacobi-type manner. Under suitable conditions on the choice of the patches and the local problems, a complete multigrid convergence analysis is given. Numerical experiments are shown which confirm the theoretical results.

The theoretical and numerical results are presented for a model problem of a typical class of elliptic optimal control problems. However, the method itself carries over to more general problems, since the construction of the smoother does not rely on structural information of the problem.

In a second approach the special structure of the optimal control problem with distributed control is used to reduce the KKT system. For this reduced linear system the construction of a similar patch smoother is presented, which leads to a highly efficient multigrid method with convergence rates comparable to convergence rates known from classical scalar elliptic problems. Again a rigorous convergence analysis is given.

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