Since the first papers on asymptotic waveform evaluation (AWE), Padé-based reduced-order models have become standard for improving coupled circuit-interconnect simulation efficiency. Such models can be accurately computed using bi-orthogonalization algorithms like Padé via Lanczos (PVL), but the resulting Padé approximates can still be unstable even when generatedfrom stable RLC circuits. For certain classes of RC circuits it has been shown that congruence transforms, like the Arnoldi algorithm, can generate guaranteed stable and passive reduced-order models. In this paper we present a computationally efficient model-order reduction technique, the coordinate-transformed Arnoldi algorithm, and show that this method generates arbitrarily accurate and guaranteed stable reduced-order models for RLC circuits. Examples are presented which demonstrates the enhanced stability and efficiency of the new method.

Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of large-scale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bi-linearization method, which extends Krylov subspace techniques for linear systems. In this approach, the nonlinear system is first approximated by a bilinear system through Carleman bilinearization. Then a reduced-order bilinear system is constructed in such a way that it matches certain number of multimoments corresponding to the first few kernels of the Volterra-Wiener representation of the bilinear system. It is shown that the two-sided Krylov subspace technique matches significant more number of multimoments than the corresponding one-side technique.

An overview of model reduction methods and a comparison of the resulting algorithms is presented. These approaches are divided into two broad categories, namely SVD based and moment matching based methods. It turns out that the approximation error in the former case behaves better globally in frequency while in the latter case the local behavior is better. 1 Introduction and problem statement Direct numerical simulation of dynamical systems has been an extremely successful means for studying complex physical phenomena. However, as more detail is included, the dimensionality of such simulations may increase to unmanageable levels of storage and computational requirements. One approach to overcoming this is through model reduction. The goal is to produce a low dimensional system that has

In this paper we review the state of affairs in the area of approximation of large-scale systems. We distinguish among three basic categories, namely the SVD-based, the Krylov-based and the SVD-Krylov-based approximation methods. The first two were developed independently of each other and have distinct sets of attributes and drawbacks. The third approach seeks to combine the best attributes of the first two. Contents 1 Introduction and problem statement 1 2 Motivating Examples 3 3 Approximation methods 4 3.1 SVD-based approximation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.1 The Singular value decomposition: SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.2 Proper Orthogonal Decomposition (POD) methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.3 Approximation by balanced truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

The major concerns in state-of-the-art model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteed-passive models, for systems with special internal structure, using numerically stable and efficient Krylov-subspace iterations. Truncated Balanced Realization (TBR) algorithms, as used to date in the design automation community, can achieve smaller models with better error control, but do not necessarily preserve passivity. In this paper we show how to construct TBR-like methods that guarantee passive reduced models and in addition are applicable to state-space systems with arbitrary internal structure.

The purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems. The resulting approach is completely automatic once an error tolerance is specified and also yields an error bound. This is to be contrasted with existing projection methods, namely PVL (Pad via Lanczos) and rationa Krylov, which do not satisfy these properties. Our approach is based on the computation and approximation of the cross gramtan of the system. The cross gramtan is the solution of a Sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights.

In this paper we present a generalization of popular linear model reduction methods, such as Lanczos- and Arnoldi-based algorithms based on rational approximation, to systems whose response to interesting external inputs can be described by a few terms in a functional series expansion such as a Volterra series. The approach allows automatic generation of macromodels that include frequency-dependent nonlinear effects.

Abstract. Krylov subspace based moment matching algorithms, such as PVL (Padé approximation Via the Lanczos process), have emerged as popular tools for efficient analyses of the impulse response in a large linear circuit. In this work, a new derivation of the PVL algorithm is presented from the matrix point of view. This approach simplifies the mathematical theory and derivation of the algorithm. Moreover, an explicit formulation of the approximation error of the PVL algorithm is given. With this error expression, one may implement the PVL algorithm that adaptively determines the number of Lanczos steps required to satisfy a prescribed error tolerance. A number of implementation issues of the PVL algorithm and its error estimation are also addressed in this paper. A generalization to a multiple-input-multiple-output circuit system via a block Lanczos process is also given.

As VLSI circuit speeds have increased, reliable chip and system design can no longer be performed without accurate threedimensional interconnect models. In this paper, we describe an integral equation approach to modeling the impedance of interconnect structures accounting for both the charge accumulation on the surface of conductors and the current traveling in their interior. Our formulation, based on a combination of nodal and mesh analysis, has the required properties to be combined with Model Order Reduction techniques to generate accurate and guaranteed passive low order interconnect models for efficient inclusion in standard circuit simulators. Furthermore, the formulation is shown to be more flexible and efficient than previously reported methods.

This dissertation concerns the model reduction of large, linear, time-invariant systems. A new method called the Dominant Gramian Eigenspaces method, which utilizes low rank approximations to the exact system gramians, is proposed for such system. The Cholesky Factor