Subject: Efficient Iterative Solvers for Saddle Point Systems arising in PDE-constrained 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)

elliptic desired state

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 PDE-constraint, namely the distributed optimal control of elliptic equations, the distributed optimal control of multiharmonic-parabolic 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 multiharmonic-parabolic 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 Moreau-Yosida regularized constraints on the state.

These additional constraints render the resulting first-order optimality system nonlinear. In order to cope with this nonlinearity, a primal-dual 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 parameter-dependence 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 parameter-dependencies 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.

Stokes pressure Stokes force
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