Model order reduction and approximation on general domains in the complex plane

In this thesis we develop model-order reduction methodologies for the approximation of systems with transfer functions analytic in general domains in the complex plane. We use results from H2-optimal model-order reduction and balanced truncation to derive new theoretical tools for the construction of a framework for model reduction in general domains. We then show that such a generalization is numerically achievable with some minor changes of commonly used model reduction algorithms. In more detail, based on the concept of Hardy spaces in general domains, we derive two optimal approximation frameworks along with necessary optimality conditions. In the first framework we choose a special structure of the approximant, resulting in first-order optimality conditions that resemble optimal H2 interpolation conditions for discrete-time systems. With this in mind, we then connect this method to model reduction of discrete-time time-invariant delay systems with particular emphasis on discretized linear systems. We then develop a data-driven algorithm for the computation of (locally) optimal approximants. In the second framework, we use transfer functions of linear dynamical systems as reduced models and derive structured optimality conditions. We show that, with additional assumptions, we can simplify such conditions to make them suitable for an algorithm based on the iterative rational Krylov algorithm (IRKA). For both frameworks we develop and discuss some key topics. First, we study a connection between the derived interpolatory optimality conditions and Schwarz functions. Second, we generalize the concept of Gramians to our setting and demonstrate that these are unique solutions to Stein and Lyapunov equations. Third, we derive matrix-equation-based optimality conditions. Finally, we develop a balanced-truncation-based algorithm for the second framework and demonstrate how the resulting reduced model preserves algebraic stability. In the last part of this thesis we propose a data-driven match-based algorithm to solve parametric eigenvalue problems where the data is provided by a Loewner-based contour-integral eigensolver. To develop this method we introduce matching strategies to handle critical scenarios, in particular eigenvalues crossing the contour and bifurcation phenomena. We conclude with an adaptive strategy for increased accuracy. For all the proposed methods we provide numerical examples to demonstrate the efficiency of the corresponding algorithms.