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46
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 119 (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 66 (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 65 (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
 Applied and Computational Control, Signals, and Circuits
, 1998
"... In recent years, reducedorder modeling techniques based on Krylovsubspace iterations, especially the Lanczos algorithm and the Arnoldi process, have become popular tools to tackle the largescale timeinvariant linear dynamical systems that arise in the simulation of electronic circuits. This pape ..."
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Cited by 53 (10 self)
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In recent years, reducedorder modeling techniques based on Krylovsubspace iterations, especially the Lanczos algorithm and the Arnoldi process, have become popular tools to tackle the largescale timeinvariant linear dynamical systems that arise in the simulation of electronic circuits. This paper reviews the main ideas of reducedorder modeling techniques based on Krylov subspaces and describes the use of reducedorder modeling in circuit simulation. 1 Introduction Krylovsubspace methods, most notably the Lanczos algorithm [81, 82] and the Arnoldi process [5], have long been recognized as powerful tools for largescale matrix computations. Matrices that occur in largescale computations usually have some special structures that allow to compute matrixvector products with such a matrix (or its transpose) much more efficiently than for a dense, unstructured matrix. The most common structure is sparsity, i.e., only few of the matrix entries are nonzero. Computing a matrixvector pr...
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 50 (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
, 1999
"... The simulation of electronic circuits involves the numerical solution of very largescale, sparse, in general nonlinear, systems of differentialalgebraic equations. Often, the size of these systems can be reduced considerably by replacing the equations corresponding to linear subcircuits by approxim ..."
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Cited by 43 (9 self)
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The simulation of electronic circuits involves the numerical solution of very largescale, sparse, in general nonlinear, systems of differentialalgebraic equations. Often, the size of these systems can be reduced considerably by replacing the equations corresponding to linear subcircuits by approximate models of much smaller statespace dimension. In this paper, we describe the use of Krylovsubspace methods for generating such reducedorder models of linear subcircuits. Particular emphasis is on reducedorder modeling techniques that preserve the passivity of linear RLC subcircuits. Key words: Lanczos algorithm; Arnoldi process; Linear dynamical system; VLSI interconnect; Transfer function; Pad'e approximation; Stability; Passivity; Positive real function 1 Introduction Today's integrated electronic circuits are extremely complex, with up to tens of millions of devices. Prototyping of such circuits is no longer possible, and instead, computational methods are used to simulate and ...
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 35 (5 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 35 (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
Equivalent Elmore Delay for RLC Trees
 Proceedings of the ACM/IEEE Design Automation Conference
, 2000
"... Abstract—Closedform solutions for the 50 % delay, rise time, overshoots, and settling time of signals in an tree are presented. These solutions have the same accuracy characteristics of the Elmore delay for trees and preserves the simplicity and recursive characteristics of the Elmore delay. Specif ..."
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Cited by 30 (8 self)
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Abstract—Closedform solutions for the 50 % delay, rise time, overshoots, and settling time of signals in an tree are presented. These solutions have the same accuracy characteristics of the Elmore delay for trees and preserves the simplicity and recursive characteristics of the Elmore delay. Specifically, the complexity of calculating the time domain responses at all the nodes of an tree is linearly proportional to the number of branches in the tree and the solutions are always stable. The closedform expressions introduced here consider all damping conditions of an circuit including the underdamped response, which is not considered by the Elmore delay due to the nonmonotone nature of the response. The continuous analytical nature of the solutions makes these expressions suitable for design methodologies and optimization techniques. Also, the solutions have significantly improved accuracy as compared to the Elmore delay for an overdamped response. The solutions introduced here for trees can be practically used for the same purposes that the Elmore delay is used for trees.