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74
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 156 (17 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 ...
Krylov Projection Methods For Model Reduction
, 1997
"... This dissertation focuses on efficiently forming reducedorder models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov p ..."
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Cited by 124 (3 self)
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This dissertation focuses on efficiently forming reducedorder models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczosbased methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete modelreduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multipleinput multipleoutput systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees.
Interconnect design for deep submicron ICs
 IN PROC. INT. CONF. ON COMPUTER AIDED DESIGN
, 1997
"... Interconnect has become the dominating factor in determining circuit performance and reliability in deep submicron designs. In this embedded tutorial, we first discuss the trends and challenges of interconnect design as the technology feature size rapidly decreases towards below 0.1 micron. Then, we ..."
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Cited by 68 (22 self)
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Interconnect has become the dominating factor in determining circuit performance and reliability in deep submicron designs. In this embedded tutorial, we first discuss the trends and challenges of interconnect design as the technology feature size rapidly decreases towards below 0.1 micron. Then, we present commonly used interconnect models and a set of interconnect design and optimization techniques for improving interconnect performance and reliability. Finally, we present comparisons of different optimization techniques in terms of their efficiency and optimization results, and show the impact of these optimization techniques on interconnect performance in each technology generation from the 0.35µm to 0.07µm projected in the National Technology Roadmap for Semiconductors.
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 67 (14 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 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 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.
Reducedorder modeling techniques based on Krylov subspaces and their use in circuit simulation
, 1998
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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 51 (3 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.
Krylovsubspace methods for reducedorder modeling in circuit simulation
 J. Comput. Appl. Math
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ReducedOrder modeling of large passive linear circuits by means of the SyPVL algorithm
 in Tech. Dig. 1996 IEEE/ACM International Conference on ComputerAided Design
, 1996
"... This paper discusses the analysis of large linear electrical networks consisting of passive components, such as resistors, capacitors, inductors, and transformers. Such networks admit a symmetric formulation of their circuit equations. We introduce SyPVL, an eficient and numerically stable algorit ..."
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Cited by 42 (14 self)
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This paper discusses the analysis of large linear electrical networks consisting of passive components, such as resistors, capacitors, inductors, and transformers. Such networks admit a symmetric formulation of their circuit equations. We introduce SyPVL, an eficient and numerically stable algorithm for the computation of reducedorder models of large, linear, passive networks. SyPVL represents the specialization of the more general PVL algorithm, to symmetric problems. Besides the gain in eficiency over PVL, SyPVL also preserves the symmetry of the problem, and, as a consequence, can often guarantee the stability of the resulting reducedorder models. Moreover, these reducedorder models can be synthesized as actual physical circuits, thus facilitating compatibility with existing analysis tools. The application of SyPVL is illustrated with two interconnectanalysis examples. 1
Reducedorder modeling of timevarying systems
 Circuits and Systems II: Analog and Digital Signal Processing
, 1999
"... We present a theory for reducedorder modelling of linear timevarying systems, together with efficient numerical methods for application to large systems. The technique, called TVP (TimeVarying Padk), is applicable to deterministic as well as noise analysis of many types of communication subsystem ..."
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Cited by 40 (6 self)
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We present a theory for reducedorder modelling of linear timevarying systems, together with efficient numerical methods for application to large systems. The technique, called TVP (TimeVarying Padk), is applicable to deterministic as well as noise analysis of many types of communication subsystems, such as mixers and switchedcapacitor filters, for which existing model reduction techniques cannot be used. TVP is therefore suitable for hierarchical verification of entire communication systems. We present practical applications in which TVP generates macromodels which are more than two orders of magnitude smaller, but still replicate the inputoutput behaviour of the original systems accurately. The size reduction results in a speedup of more than 500. 1
Stable and efficient reduction of substrate model networks using congruence transforms
 in ICCAD
, 1995
"... Parasitic analogdigital noise coupling has been identified as a key issue facing designers of mixedsignal integrated circuits. In particular, signal cross talk through the common chip substrate has become increasingly problematic. This paper demonstrates a new methodology for developing simulation ..."
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Cited by 36 (0 self)
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Parasitic analogdigital noise coupling has been identified as a key issue facing designers of mixedsignal integrated circuits. In particular, signal cross talk through the common chip substrate has become increasingly problematic. This paper demonstrates a new methodology for developing simulation, synthesis, and verification models to analyze the global electrical behavior of the nonideal semiconductor substrate. RC substrate network models, which are generated via a triangular discretization method, are accurately approximated for subsequent analysis by an efficient reduction algorithm. This algorithm utilizes the wellconditioned Lanczos process to formulate Padé approximations of the network port admittance. Congruence transformations are employed to ensure stability, and to create reduced networks which are easily realizable with SPICEcompatible RC elements. For validation, the strategy has been successfully applied to several mixedsignal circuit examples. 1