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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 96 (10 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
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 89 (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.
H2 model reduction for largescale linear dynamical systems
 SIAM J. Matrix Anal. Appl
"... Abstract. The optimal H2 model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focusing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, w ..."
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Cited by 69 (27 self)
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Abstract. The optimal H2 model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focusing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, without any apparent connection between the two approaches. In this paper, we develop a new unifying framework for the optimal H2 approximation problem using best approximation properties in the underlying Hilbert space. This new framework leads to a new set of local optimality conditions taking the form of a structured orthogonality condition. We show that the existing Lyapunovand interpolationbased conditions are each equivalent to our conditions and so are equivalent to each other. Also, we provide a new elementary proof of the interpolationbased condition that clarifies the importance of the mirror images of the reduced system poles. Based on the interpolation framework, we describe an iteratively corrected rational Krylov algorithm for H2 model reduction. The formulation is based on finding a reduced order model that satisfies interpolationbased firstorder necessary conditions for H2 optimality and results in a method that is numerically effective and suited for largescale problems. We illustrate the performance of the method with a variety of numerical experiments and comparisons with existing methods.
PRIMO: probability interpretation of moments for delay calculation
 Proc. of Design Automation Conference
, 1998
"... Moments of the impulse response are widely used for interconnect delay analysis, from the explicit Elmore delay (first moment of the impulse response) expression, to moment matching methods which create reduced order transimpedance and transfer function approximations. However, the Elmore delay is f ..."
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Cited by 33 (0 self)
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Moments of the impulse response are widely used for interconnect delay analysis, from the explicit Elmore delay (first moment of the impulse response) expression, to moment matching methods which create reduced order transimpedance and transfer function approximations. However, the Elmore delay is fast becoming ineffective for deep submicron technologies, and reduced order transfer function delays are impractical for use as earlyphase design metrics or as design optimization cost functions. This paper describes an approach for fitting moments of the impulse response to probability density functions so that delays can be estimated from probability tables. For RC trees it is demonstrated that the incomplete gamma function provides a provably stable approximation. The step response delay is obtained from a onedimensional table lookup. 1
Padé Approximation Of LargeScale Dynamic Systems With Lanczos Methods
, 1994
"... The utility of Lanczos methods for the approximation of largescale dynamical systems is considered. In particular, it is shown that the Lanczos method is a technique for yielding Pad'e approximants which has several advantages over more traditional explicit moment matching approaches. An exten ..."
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Cited by 30 (1 self)
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The utility of Lanczos methods for the approximation of largescale dynamical systems is considered. In particular, it is shown that the Lanczos method is a technique for yielding Pad'e approximants which has several advantages over more traditional explicit moment matching approaches. An extension of the Lanczos algorithm is developed for computing multipoint Pad'e approximations of descriptor systems. Keywords: Dynamic system, Pad'e approximation, Lanczos algorithm, model reduction. 1. Introduction This paper explores the use of Lanczos techniques for the reducedorder modeling and simulation of largescale, SISO dynamical systems. One can define such a system through the set of state space equations ae E x(t) = Ax(t) + bu(t) y(t) = cx(t) + du(t): (1) The scalar functions u(t) and y(t) are the system's input and output while x(t) is the state vector of dimension n. For simplicity, the directcoupling term, d, will be assumed to be zero. The system matrix, A 2 R n\Thetan ...
HiPRIME: hierarchical and passivity preserved interconnect macromodeling engine for RLKC power delivery
 IEEE Trans. ComputerAided Design
, 2005
"... Abstract—This paper proposes a general hierarchical analysis methodology, HiPRIME, to efficiently analyze RLKC power delivery systems. After partitioning the circuits into blocks, we develop and apply the IEKS (Improved Extended Krylov Subspace) method to build the multiport Norton equivalent circ ..."
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Cited by 19 (3 self)
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Abstract—This paper proposes a general hierarchical analysis methodology, HiPRIME, to efficiently analyze RLKC power delivery systems. After partitioning the circuits into blocks, we develop and apply the IEKS (Improved Extended Krylov Subspace) method to build the multiport Norton equivalent circuits which transform all the internal sources to Norton current sources at ports. Since there are no active elements inside the Norton circuits, passive or realizable model order reduction techniques such as PRIMA can be applied. The significant speed improvement, 700 times faster than Spice with less than 0.2 % error and 7 times faster than a stateoftheart solver, InductWise, is observed. To further reduce the toplevel hierarchy runtime, we develop a secondlevel model reduction algorithm and prove its passivity. Index Terms—Model order reduction, power distribution, power grid, signal integrity. I.
A method for reducedorder modeling and simulation of large interconnect circuits and its application to PEEC models with retardation
 IEEE Trans. Circuits Syst. II
, 2000
"... Abstract—The continuous improvement in the performance and the increases in the sizes of VLSI systems make electrical interconnect and package (EIP) design and modeling increasingly more important. Special software tools must be used for the design of highperformance VLSI systems. Furthermore, larg ..."
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Cited by 18 (1 self)
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Abstract—The continuous improvement in the performance and the increases in the sizes of VLSI systems make electrical interconnect and package (EIP) design and modeling increasingly more important. Special software tools must be used for the design of highperformance VLSI systems. Furthermore, larger and faster systems require larger and more accurate circuit models. The partial element equivalent circuit (PEEC) technique is used for modeling such systems with threedimensional full wave models. In this paper, we present a practical, readily parallelizable procedure for generating reducedorder frequencydomain models from general full wave PEEC systems. We use multiple expansion points, and piecemeal construction of poleresidue approximations to transfer functions of the PEEC systems, as was used in the complex frequency hopping algorithms. We consider general, multipleinput/multipleoutput PEEC systems. Our block procedure consists of an outer loop of local approximations to the PEEC system, coupled with an inner loop where an iterative modelreduction method is applied to the local approximations. We systematically divide the complex frequency region of interest into small regions and construct local approximations to the PEEC system in each subregion. The local approximations are constructed so that the matrix factorizations associated with each of them are the size of the original system and independent of the order of the approximation. Results of computations on these local systems are combined to obtain a reducedorder model for the original PEEC system. We demonstrate the usefulness of our approach with three interesting examples. Index Terms—Arnoldi, electronic interconnect package (EIP), interconnect circuits, iterative methods, Lanczos, MIMO systems, model reduction, reduced models, reducedorder systems, PEEC systems, time delays, transfer function. I.
Noiseaware Repeater Insertion and Wire Sizing for Onchip Interconnect Using Hierarchical MomentMatching
, 1999
"... Recently, several algorithms for interconnect optimization via repeater insertion and wire sizing have appeared based on the Elmore delay model. Using the Devgan noise metric [6] a noiseaware repeater insertion technique has also been proposed recently. Recognizing the conservatism of these delay an ..."
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Cited by 17 (0 self)
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Recently, several algorithms for interconnect optimization via repeater insertion and wire sizing have appeared based on the Elmore delay model. Using the Devgan noise metric [6] a noiseaware repeater insertion technique has also been proposed recently. Recognizing the conservatism of these delay and noise models, we propose a momentmatching based technique to interconnect optimization that allows for much higher accuracy while preserving the hierarchical nature of Elmoredelaybased techniques. We also present a novel approach to noise computation that accurately captures the effect of several attackers in linear time with respect to the number of attackers and wire segments. Our practical experiments with industrial nets indicate that the corresponding reduction in error afforded by these more accurate models justifies this increase in runtime for aggressive designs which is our targeted domain. Our algorithm yields delay and noise estimates within 5% of circuit simulation results.
Krylov Type Subspace Methods for Matrix Polynomials
, 2002
"... We consider solving eigenvalue problems or model reduction problems for a quadratic with large and sparse A and B.Weproposenew Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with ..."
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Cited by 12 (4 self)
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We consider solving eigenvalue problems or model reduction problems for a quadratic with large and sparse A and B.Weproposenew Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coe#cient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear inputoutput system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.