This paper presents a class of algorithms suitable for model reduction of distributed systems. Distributed systems are not suitable for treatment by standard model-reduction algorithms such as PRIMA, PVL, and the Arnoldi schemes because they generate matrices that are dependent on frequency (or other parameters) and cannot be put in a lumped or state-space form. Our algorithms build on well-known projection-based reduction techniques, and so require only matrix-vector product operations and are

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... thus suitable for operation in conjunction with electromagnetic analysis codes that use iterative solution methods and fast-multipole acceleration techniques. Under the condition that the starting systems satisfy system-theoretic properties required of physical systems, the reduced systems can be guaranteed to be passive. For distributed systems, we argue that causality of the underlying representation is as important a consideration as passivity has become. H¥ s¦ u¥ s¦ , where u¥ s¦ and y¥ s¦ are the Laplace-domain representations of inputs u¥ t ¦ and outputs y¥ t ¦ . Hence, H¥ s¦ is an immittance function: either an admittance matrix Y¥ s¦ , or an impedance matrix Z¥ s¦ . Let us introduce two inner products in X , the standard inner product¨u© y ¡ ∞ ∞ y¥ t ¦ T u¥ t ¦ d t © and a product which acts on truncated signals u© y τ If u and y are port current/voltage pairs,¨u© y τ is the total energy dissipated by the system up to time τ. We will generally work in the space of signals x X