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Das Koppelmatrix-Modell ist ein häufig verwendetes Werkzeug zur Optimierung und Analyse von Filterstrukturen. Diese Dissertation zeigt einen neuen Weg für eine häufig auftretende Aufgabe bei der Verwendung dieses Modells auf: Der Umformung der Koppelmatrix von einer gegebenen Starttopologie in eine frei wählbare Zieltopologie. Die Eigenschaften des zugrunde liegenden inversen Eigenwertproblems werden dabei ausgenutzt um mit geringem durchschnittlichem Zeitaufwand eine Lösung zu finden. Basierend auf diesem Algorithmus wird ein Verfahren vorgestellt, mit dessen Hilfe aus gemessenen Streuparametern eines Filters Rückschlüsse auf dessen Koppelstruktur möglich sind. Die Anwendung dieses Verfahrens wird im Kontext verschiedener praktischer Aufgabenstellungen des Entwurfs und der Analyse von Filtern demonstriert.

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Compared to multi-core processors, GPUs typically offer a higher memory bandwidth, which makes them attractive for memory-bounded codes like sparse linear and eigenvalue solvers. The fundamental performance issue we encounter when implementing such methods for modern GPUs is that the ratio between memory bandwidth and memory capacity is significantly higher than for CPUs. When solving large-scale problems one therefore has to use more compute nodes and is quickly forced into the strong scaling limit. In this paper we consider an advanced eigensolver (the block Jacobi-Davidson QR method [1]), implemented in the PHIST software (https://bitbucket.org/essex/phist/). We aim to provide a blueprint and a framework for implementing other iterative solvers like Krylov subspace methods for modern architectures that have relatively small high-bandwidth memory. The techniques we explore to reduce the memory footprint of our solver include mixed precision arithmetic and recalculating quantities `on-the-fly'. We use performance models to back our results theoretically and ensure performance portability.

The present dissertation deals with the oscillation behavior of linear Hamiltonian systems and related eigenvalue problems with general, linear independent self-adjoint boundary conditions. The main new aspect of this dissertation is the fact that we do not require controllability, strong observability or strong normality of the system. In view of this generalization it is necessary to introduce a new notion of "proper" eigenvalues and their multiplicities of the related eigenvalue problem. We show that the "proper" eigenvalues of the related eigenvalue problems are always isolated. Furthermore we introduce a new notion of the multiplicity of a "proper" focal point of so-called conjoined bases of the differential system. We derive oscillation theorems which give a formula for the number of all "proper" eigenvalues (including multiplicities) smaller or equal than a certain constant with respect to the number of all "proper" focal points (including multiplicities) of a certain conjoined basis of the Hamiltonian system. Due to our generalization we are able to treat more general Sturm-Liouville eigenvalue problems as in the existing literature.