##### Abstract

This work is concerned with the construction of
preconditioners for indefinite linear systems.
The systems under investigation arise in the
numerical solution of quadratic programming
problems, for example in the form of
Karush--Kuhn--Tucker (KKT) optimality
conditions or in interior--point methods.
Therefore, the system matrix is referred to as a
KKT matrix. It is not the purpose of this thesis
to investigate systems arising from general
quadratic programming problems, but to study
systems arising in linear quadratic control
problems governed by partial differential
equations. The KKT matrix is symmetric,
nonsingular, and indefinite. For the solution of
the linear systems generalizations of the
conjugate gradient method, MINRES and
SYMMLQ, are used. The performance of
these iterative solution methods depends on the
eigenvalue distribution of the matrix and of the
cost of the multiplication of the system matrix
with a vector. To increase the performance of
these methods, one tries to transform the
system to favorably change its eigenvalue
distribution. This is called preconditioning and
the nonsingular transformation matrices are
called preconditioners. Since the overall
performance of the iterative methods also
depends on the cost of matrix--vector
multiplications, the preconditioner has to be
constructed so that it can be applied efficiently.
The preconditioners designed in this thesis are
positive definite and they maintain the symmetry
of the system. For the construction of the
preconditioners we strongly exploit the
structure of the underlying system. The
preconditioners are composed of
preconditioners for the submatrices in the KKT
system. Therefore, known efficient
preconditioners can be readily adapted to this
context. The derivation of the preconditioners is
motivated by the properties of the KKT
matrices arising in optimal control problems. An
analysis of the preconditioners is given and
various cases which are important for interior
point methods are treated separately. The
preconditioners are tested on a typical problem,
a Neumann boundary control for an elliptic
equation. In many important situations the
preconditioners substantially reduce the number
of iterations needed by the solvers. In some
cases, it can even be shown that the number of
iterations for the preconditioned system is
independent of the refinement of the
discretization of the partial differential equation.