### Refine

This chapter contains three advanced topics in model order reduction (MOR): nonlinear MOR, MOR for multi-terminals (or multi-ports) and finally an application in deriving a nonlinear macromodel covering phase shift when coupling oscillators. The sections are offered in a preferred order for reading, but can be read independently.

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 5.1, written by Jorge Fernández 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.

Model Order Reduction
(2015)

This chapter offers an introduction to Model Order Reduction (MOR). It gives an overview on the methods that are mostly used. It also describes the main concepts behind the methods and the properties that are aimed to be preserved. The sections are in a prefered order for reading, but can be read independentlty. Section 4.1, written by Michael Striebel, E. Jan W. ter Maten, Kasra Mohaghegh and Roland Pulch, overviews the basic material for MOR and its use in circuit simulation. Issues like Stability, Passivity, Structure preservation, Realizability are discussed. Projection based MOR methods include Krylov-space methods (like PRIMA and SPRIM) and POD-methods. Truncation based MOR includes Balanced Truncation, Poor Man’s TBR and Modal Truncation.Section 4.2, written by Joost Rommes and Nelson Martins, focuses on Modal Truncation. Here eigenvalues are the starting point. The eigenvalue problems related to large-scale dynamical systems are usually too large to be solved completely. The algorithms described in this section are efficient and effective methods for the computation of a few specific dominant eigenvalues of these large-scale systems. It is shown how these algorithms can be used for computing reduced-order models with modal approximation and Krylov-based methods.Section 4.3, written by Maryam Saadvandi and Joost Rommes, concerns passivity preserving model order reduction using the spectral zero method. It detailedly discusses two algorithms, one by Antoulas and one by Sorenson. These two approaches are based on a projection method by selecting spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros. The reduced model preserves passivity.Section 4.4, written by Roxana Ionutiu, Joost Rommes and Athanasios C. Antoulas, refines the spectral zero MOR method to dominant spectral zeros. The new model reduction method for circuit simulation preserves passivity by interpolating dominant spectral zeros. These are computed as poles of an associated Hamiltonian system, using an iterative solver: the subspace accelerated dominant pole algorithm (SADPA). Based on a dominance criterion, SADPA finds relevant spectral zeros and the associated invariant subspaces, which are used to construct the passivity preserving projection. RLC netlist equivalents for the reduced models are provided.Section 4.5, written by Roxana Ionutiu and Joost Rommes, deals with synthesis of a reduced model: reformulate it as a netlist for a circuit. A framework for model reduction and synthesis is presented, which greatly enlarges the options for the re-use of reduced order models in circuit simulation by simulators of choice. Especially when model reduction exploits structure preservation, we show that using the model as a current-driven element is possible, and allows for synthesis without controlled sources. Two synthesis techniques are considered: (1) by means of realizing the reduced transfer function into a netlist and (2) by unstamping the reduced system matrices into a circuit representation. The presented framework serves as a basis for reduction of large parasitic R/RC/RCL networks.