Subject:  Efficient Iterative Solvers for Saddle Point Systems arising in PDEconstrained Optimization Problems with Inequality Constraints 
Author:  Dipl.Ing. Markus Kollmann 
Supervision:  A.Univ.Prof. Dipl. Ing. Dr. Walter Zulehner 
External referee:  Prof. Dr. Roland Herzog (TU Chemnitz) 
This thesis deals with the construction and analysis of efficient solution methods for a class of optimization problems with constraints in terms of partial differential equations (PDEs). In detail, we consider the following three optimal control problems with a quadratic cost functional and a linear PDEconstraint, namely the distributed optimal control of elliptic equations, the distributed optimal control of multiharmonicparabolic equations and the distributed optimal control of the Stokes equations. Those three problems appear in various applications in practice: the optimal control of elliptic equations arises in the field of optimal stationary heating, the optimal control of multiharmonicparabolic equations arises in the field of control of eddy current problems in electromagnetics and the optimal control of the Stokes equations arises in the field of velocity tracking in flow control. Their efficient and fast solution is of prime importance. Usually, in practical problems the control variable and the state variable have to fulfill various additional conditions. In this thesis we focus on pointwise inequality constraints on the control and MoreauYosida regularized constraints on the state. These additional constraints render the resulting firstorder optimality system nonlinear. In order to cope with this nonlinearity, a primaldual active set method is applied. It turns out that, after discretization, the resulting linear system to be solved in each step of this linearization method is a large scale saddle point system that depends on various model and discretization parameters. This parameterdependence badly influences the convergence of iterative methods if directly applied to those systems. Therefore, in order to obtain fast solution methods, appropriate preconditioners are needed, that improve the spectral properties of the saddle point systems with respect to the parameterdependencies and are efficiently realizable. The main focus of this thesis is the construction and analysis of such efficient preconditioners for the three mentioned problem classes. For each of the three model problems, we propose preconditioners and compare them with other preconditioners available in literature. 
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