Broadband model order reduction of polynomial matrix equations using single‐point well‐conditioned asymptotic waveform evaluation: derivations and theory

Abstract

SUMMARYTo e ect a model order reduction (MORe) process on a system which has a polynomial matrix equation dependence on the MORe parameter, researchers generally take one of two approaches. The ÿrst is to linearize the system by introducing extra degrees of freedom and then to solve the resulting expanded, linear system with a method such as Lanczos or Arnoldi. The second approach is to work directly with the polynomial system and use a technique such as asymptotic waveform evaluation (AWE). Of course, each approach has advantages and disadvantages. In this paper, a new technique will be presented which has some desirable characteristics from both approaches and which is able to circumvent simultaneously some of their disadvantages. It can be shown that both the Arnoldi and the AWE methods are special cases of this new technique. Finally, numerical results will show the viability of the new method, which will be called the well-conditioned asymptotic waveform evaluation (WCAWE) method.

SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem

“…One pioneering work is due to Su and Craig [22] back to 1991. In recent years, this approach has been repeatedly applied, studied, and improved; for example, see [2,14,19,20]. In particular, the dissertation work of Slone [19] has essentially extended Su and Craig's approach to the model reduction of high-order dynamical systems but is based the popular AWE (asymptotic waveform evaluation) approach as widely known in interconnect analysis of integrated circuits and computational electromagnetics.…”

Structure-Preserving Model Reduction Using a Krylov Subspace Projection Formulation

“…Krylov subspace projection methods are increasingly popular in model reduction owing to their numerical efficiency for very large systems, such as those arising from structure dynamics, control systems, circuit simulations, computational electromagnetics and microelectromechanical systems [13,10,12,5,42,47,48,52]. Recent survey articles [1,4,19] provide in depth review of the subject and comprehensive references.…”

NHAR: A non‐homogeneous Arnoldi method for fast simulation of RCL circuits with a large number of ports

“…moments (Taylor expansion coefficients) of the input signals taken into account. Although the linearization-based method EXPLIN [13] can achieve high accuracy, the high computational cost and memory consumption hamper them from real applications.…”

“…The computational cost of the linearization-based methods is determined by the linearization scheme. In [13], Slone et al proposed an EXPanded LINearization (EXPLIN) scheme for polynomial systems with high-order terms in the RHS, which can be employed to generate the projection matrices for RCL circuits with specified inputs. However, the original system with order N is enlarged to a linearized system with order (N +1)l by the EXPLIN linearization scheme in [13], where l is the number of the 848 X. ZENG ET AL.…”

“…On the other hand, the system with higher-order terms in the RHS can be linearized by an appropriate linearization scheme [13]. Hence, Arnoldi-like orthogonalization methods can be developed to generate the projection matrix, which tackles the numerical instability problem of the incremental orthogonalization algorithm in EKS [11] and IEKS [12].…”