This Chapter introduces parameterized, or parametric, Model Order Reduction (pMOR). The Sections are offered in a prefered order for reading, but can be read independently. Section 4.1, written by Jorge Fernandez Villena, L. Miguel Silveira, Wil H.A. Schilders, Gabriela Ciuprina, Daniel Ioan and Sebastian Kula, overviews the basic principles for pMOR. Due to higher integration and increasing frequency-based effects, large, full Electromagnetic Models (EM) are needed for accurate prediction of the real behavior of integrated passives and interconnects. Furthermore, these structures are subject to parametric effects due to small variations of the geometric and physical properties of the inherent materials and manufacturing process. Accuracy requirements lead to huge models, which are expensive to simulate and this cost is increased when parameters and their effects are taken into account. This Section introduces the framework of pMOR, which aims at generating reduced models for systems depending on a set of parameters.
Section 4.2, written by Gabriela Ciuprina, Alexandra Stefanescu, Sebastian Kula and Daniel Ioan, provides robust procedures for pMOR. This Section proposes a robust specialized technique to extract reduced parametric compact models, described as parametric SPICE-like netlists, for long interconnects modeled as transmission lines with several field effects such as skin effect and substrate losses. The technique uses an EM formulation based on partial differential equations (PDE), which is discretized to obtain a finite state space model. Next, a variability analysis of the geometrical data is carried out. Finally, a method to extract an equivalent parametric circuit is proposed.
Section 4.3, written by Michael Striebel, Roland Pulch, E. Jan W. ter Maten, Zoran Ilievski, and Wil H.A. Schilders, covers ways to efficiently determine sensitivity of output with respect to parameters. First direct and adjoint techniques are considered with forward and backward time integration, respectively. Here also the use of MOR via POD (Proper Orthogonal Decomposition) is discussed. Next, techniques in Uncertainty Quantification are described. Here pMOR techniques can be used efficiently.
Section 4.4, written by Kasra Mohaghegh, Roland Pulch and E. Jan W. ter Maten, provides a novel way in extending MOR to Differential-Algebraic Systems. Here new MOR techniques for reducing semi-explicit system of DAEs are introduced. These techniques are extendable to all linear DAEs. Especially pMOR techniques are exploited for singularly perturbed systems.