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213
PRIMA: Passive Reducedorder Interconnect Macromodeling Algorithm
, 1997
"... This paper describes PRIMA, an algorithm for generating provably passive reduced order Nport models for RLC interconnect circuits. It is demonstrated that, in addition to requiring macromodel stability, macromodel passivity is needed to guarantee the overall circuit stability once the active and pa ..."
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Cited by 429 (10 self)
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This paper describes PRIMA, an algorithm for generating provably passive reduced order Nport models for RLC interconnect circuits. It is demonstrated that, in addition to requiring macromodel stability, macromodel passivity is needed to guarantee the overall circuit stability once the active and passive driver/load models are connected. PRIMA extends the block Arnoldi technique to include guaranteed passivity. Moreover, it is empirically observed that the accuracy is superior to existing block Arnoldi methods. While the same passivity extension is not possible for MPVL, we observed comparable accuracy in the frequency domain for all examples considered. Additionally, a path tracing algorithm is used to calculate the reduced order macromodel with the utmost efficiency for generalized RLC interconnects.
The Quadratic Eigenvalue Problem
, 2001
"... . We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skewHermitian) and t ..."
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Cited by 260 (21 self)
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. We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skewHermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software. Key words. quadratic eigenvalue problem, eigenvalue, eigenvector, matrix, matrix polynomial, secondorder differential equation, vibration, Millennium footbridge, overdamped system, gyroscopic system, linearization, backward error, pseudospectrum, condition number, Krylov methods, Arnoldi method, Lanczos method, JacobiDavidson method AMS subject classifications. 65F30 Contents 1 Introduction 2 2 Applications of QEPs 4 2.1 Secondorder differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Vibration analysis of structural systems ...
A trajectory piecewiselinear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices
 in Proc. Int. Conf. ComputerAided Design
"... Abstract—In this paper, we present an approach to nonlinear model reduction based on representing a nonlinear system with a piecewiselinear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components as piecewise linear and then ..."
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Cited by 144 (8 self)
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Abstract—In this paper, we present an approach to nonlinear model reduction based on representing a nonlinear system with a piecewiselinear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components as piecewise linear and then composing hundreds of components to make a system with exponentially many different linear regions, we instead generate a small set of linearizations about the state trajectory which is the response to a “training input. ” Computational results and performance data are presented for an example of a micromachined switch and selected nonlinear circuits. These examples demonstrate that the macromodels obtained with the proposed reduction algorithm are significantly more accurate than models obtained with linear or recently developed quadratic reduction techniques. Also, we propose a procedure for a posteriori estimation of the simulation error, which may be used to determine the accuracy of the extracted trajectory piecewiselinear reducedorder models. Finally, it is shown that the proposed model order reduction technique is computationally inexpensive, and that the models can be constructed “on the fly, ” to accelerate simulation of the system response. Index Terms—Microelectromechanical systems (MEMS), model order reduction, nonlinear analog circuits, nonlinear dynamical systems, piecewiselinear models. I.
Low rank solutions of Lyapunov equations
 SIAM Journal Matrix Anal. Appl
, 2002
"... Abstract. This paper presents the Cholesky factor–alternating direction implicit (CF–ADI) algorithm, which generates a low rank approximation to the solution X of the Lyapunov equation AX +XAT = −BBT. The coefficient matrix A is assumed to be large, and the rank of the righthand side −BBT is assume ..."
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Cited by 106 (4 self)
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Abstract. This paper presents the Cholesky factor–alternating direction implicit (CF–ADI) algorithm, which generates a low rank approximation to the solution X of the Lyapunov equation AX +XAT = −BBT. The coefficient matrix A is assumed to be large, and the rank of the righthand side −BBT is assumed to be much smaller than the size of A. The CF–ADI algorithm requires only matrixvector products and matrixvector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A. This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF–ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X. Key words. Lyapunov equation, alternating direction implicit iteration, low rank approxima
A multiparameter moment matching model reduction approach for generating geometrically parametrized interconnect performance models
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 2004
"... this paper we describe an approach for generating geometricallyparameterized integratedcircuit interconnect models that are efficient enough for use in interconnect synthesis. The model generation approach presented is automatic, and is based on a multiparameter modelreduction algorithm. The eff ..."
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Cited by 97 (8 self)
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this paper we describe an approach for generating geometricallyparameterized integratedcircuit interconnect models that are efficient enough for use in interconnect synthesis. The model generation approach presented is automatic, and is based on a multiparameter modelreduction algorithm. The effectiveness of the technique is tested using a multiline bus example, where both wire spacing and wire width are considered as geometric parameters. Experimental results demonstrate that the generated models accurately predict both delay and crosstalk effects over a wide range of spacing and width variation.
A survey of model reduction by balanced truncation and some new results
 International Journal of Control
"... Abstract Balanced truncation is one of the most common model reduction schemes. In this note, we present a survey of balancing related model reduction methods and their corresponding error norms, and also introduce some new results. Five balancing methods are studied: (1) Lyapunov balancing, (2) St ..."
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Cited by 94 (2 self)
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Abstract Balanced truncation is one of the most common model reduction schemes. In this note, we present a survey of balancing related model reduction methods and their corresponding error norms, and also introduce some new results. Five balancing methods are studied: (1) Lyapunov balancing, (2) Stochastic balancing (3) Bounded real balancing, (4) Positive real balancing and (5) Frequency weighted balancing. For positive real balancing, we introduce a multiplicativetype error bound. Moreover, for a certain subclass of positive real systems, a modi£ed positivereal balancing scheme with an absolute error bound is proposed. We also develop a new frequencyweighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians. Two numerical examples are illustrated to verify the ef£ciency of the proposed methods.
A CoordinateTransformed Arnoldi Algorithm for Generating Guaranteed Stable ReducedOrder Models of RLC Circuits
, 1996
"... Since the first papers on asymptotic waveform evaluation (AWE), Padébased reducedorder models have become standard for improving coupled circuitinterconnect simulation efficiency. Such models can be accurately computed using biorthogonalization algorithms like Padé via Lanczos (PVL), but the res ..."
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Cited by 93 (20 self)
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Since the first papers on asymptotic waveform evaluation (AWE), Padébased reducedorder models have become standard for improving coupled circuitinterconnect simulation efficiency. Such models can be accurately computed using biorthogonalization algorithms like Padé via Lanczos (PVL), but the resulting Padé approximates can still be unstable even when generated from 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 reducedorder models. In this paper we present a computationally efficient modelorder reduction technique, the coordinatetransformed Arnoldi algorithm, and show that this method generates arbitrarily accurate and guaranteed stable reducedorder models for RLC circuits. Examples are presented which demonstrates the enhanced stability and efficiency of the new method.
Krylov Subspace Techniques for ReducedOrder Modeling of Nonlinear Dynamical Systems
 Appl. Numer. Math
, 2002
"... Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of largescale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bilinearization method, which extends Kry ..."
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Cited by 93 (5 self)
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Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of largescale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bilinearization 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 reducedorder bilinear system is constructed in such a way that it matches certain number of multimoments corresponding to the first few kernels of the VolterraWiener representation of the bilinear system. It is shown that the twosided Krylov subspace technique matches significant more number of multimoments than the corresponding oneside technique.
A survey of model reduction methods for largescale systems
 Contemporary Mathematics
, 2001
"... 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 freque ..."
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Cited by 91 (9 self)
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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
A collection of benchmark examples for model reduction of linear time invariant dynamical systems. SLICOT Working Note 20022. Available from
"... Summary. We present a benchmark collection containing some useful real world examples, which can be used to test and compare numerical methods for model reduction. All systems can be downloaded from the web and we describe here the relevant characteristics of the benchmark examples. 1 ..."
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Cited by 73 (10 self)
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Summary. We present a benchmark collection containing some useful real world examples, which can be used to test and compare numerical methods for model reduction. All systems can be downloaded from the web and we describe here the relevant characteristics of the benchmark examples. 1