Rational Interpolation in the Loewner Framework, Optimal PMOR and Aspects of IRKA

Rational interpolation in the context of system identification and model order reduction plays an important part in the simulation and control of large-scale dynamical systems. A main tool is the Loewner matrix containing divided-differences of a rational function based on measurements. The rank of this matrix is directly related to the degree of the underlying function. This thesis collects and extends a large amount of information regarding the rational interpolation with Loewner matrices which were scattered over different sources. The concept of Loewner matrices can be extended to two-variable rational interpolation, where one variable is an additional parameter like a material property or a system geometry. On the basis of measurements for appropriate frequencies and parameters, the original parametric system can be reduced with respect to both variables at the same time. The second part of this thesis deals with optimal parametric model order reduction, where the H2-norm of the error system should be minimized over a range of parameters. For a given set of points, the IRKA algorithm is used to obtain locally optimal solutions which are interpolated by radial basis functions. The focus lies on clustering, which means a segmentation of the parameter set in order to obtain smooth functions. Since IRKA plays an important part in finding local optima in the context of optimal PMOR, it is analysed in detail. The main tool is the pole-residue formulation of the H2-norm of the error system, since its optima fulfil the Hermite interpolation conditions in the mirror poles of the reduced system. In addition, the loss of rank in the IRKA iterations is considered. In this context, the relationship between the dimension of the rational Krylov space and the Kalman controllability matrix is helpful in order to distinguish between an analytic loss and one which only results from finite-point arithmetics in the numerical scheme.