  The work is organized as follows. First an introduction is given in Chapter
1. In Chapter 2 we introduce the POD method in finite and infinite-dimensional
Hilbert spaces and discuss various applications. Chapter 3 is devoted to to
POD-based Galerkin schemes for evolution problems. Mainly, we study linear
problems taking different discretization methods into account. We provide a
certified a-priori and a-posteriori error analysis. Furthermore, the numerical
realizations are explained and illustrated by test examples. Quadratic
programming problems governed by liner evolution problems are investigated in
Chapter 4. As in Chapter 3 we present a certified a-priori and a-posteriori
error analysis. Moreover, we discuss basis update strategies. In Chapter 5 we
give an outlook to further directions in reduced-order modeling in optimal
control and optimization. More precisely, a nonlinear optimal control problem
is studied. Moreover, state-constrained optimization problems are solved by a
tailored combination of primal-dual active set methods and POD-aesed
reduced-order modeling. Furthermore, POD Galerkin methods for multiobjective
optimal control problems are investigated. Finally, some required theoretical
foundations are summarized in the appendix.
