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48
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...
Efficient ReducedOrder Modeling of FrequencyDependent Coupling Inductances associated with 3D Interconnect Structures
, 1994
"... Reducedorder modeling techniques are now commonly used to efficiently simulate circuits combined with interconnect, but generating reducedorder models from realistic 3D structures has received less attention. In this paper we describe a Krylovsubspace based method for deriving reducedorder mode ..."
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Cited by 52 (10 self)
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Reducedorder modeling techniques are now commonly used to efficiently simulate circuits combined with interconnect, but generating reducedorder models from realistic 3D structures has received less attention. In this paper we describe a Krylovsubspace based method for deriving reducedorder models directly from the 3D magnetoquasistatic analysis program FastHenry. This new approach is no more expensive than computing an impedance matrix at a single frequency.
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 52 (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 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.
A Rational Lanczos Algorithm for Model Reduction II: Interpolation Point Selection
 Numerical Algorithms
, 1998
"... In part I of this work [10], a rational Lanczos algorithm was developed which led to rational interpolants of dynamical systems. In this sequel, the important implementational issue of interpolation point selection is analyzed in detail. A residual expression is derived for the rational Lanczos al ..."
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Cited by 45 (0 self)
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In part I of this work [10], a rational Lanczos algorithm was developed which led to rational interpolants of dynamical systems. In this sequel, the important implementational issue of interpolation point selection is analyzed in detail. A residual expression is derived for the rational Lanczos algorithm and is used to govern the placement and type of the interpolation points. Algorithms are developed and applied to a problem arising from circuit interconnect modeling. AMS classification: Primary 65F15; Secondary 65G05. Key Words : State space systems, rational Lanczos algorithm, preconditioning, rational interpolation, model reduction. 1 Introduction A variety of Lanczosbased methods are now available for acquiring a reducedorder model for a stable, linear, timeinvariant system. Many of these Lanczosbased methods interpolate the value and consecutive derivatives of the frequency response of the original system at one or more points, see [10] and references therein. Yet by...
Algorithms for Model Reduction of Large Dynamical Systems
, 1999
"... Three algorithms for the model reduction of largescale, continuoustime, timeinvariant, linear, dynamical systems with a sparse or structured transition matrix and a small number of inputs and outputs are described. They rely on low rank approximations to the controllability and observability Gram ..."
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Cited by 43 (1 self)
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Three algorithms for the model reduction of largescale, continuoustime, timeinvariant, linear, dynamical systems with a sparse or structured transition matrix and a small number of inputs and outputs are described. They rely on low rank approximations to the controllability and observability Gramians, which can eciently be computed by ADI based iterative low rank methods. The rst two model reduction methods are closely related to the wellknown square root method and Schur method, which are balanced truncation techniques. The third method is a heuristic, balancingfree technique. The performance of the model reduction algorithms is studied in numerical experiments.
Approximation of largescale dynamical systems: An overview
, 2001
"... In this paper we review the state of affairs in the area of approximation of largescale systems. We distinguish among three basic categories, namely the SVDbased, the Krylovbased and the SVDKrylovbased approximation methods. The first two were developed independently of each other and have dist ..."
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Cited by 43 (2 self)
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In this paper we review the state of affairs in the area of approximation of largescale systems. We distinguish among three basic categories, namely the SVDbased, the Krylovbased and the SVDKrylovbased 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 SVDbased 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
Model Reduction Methods Based on Krylov Subspaces
 Acta Numerica
, 2003
"... This paper reviews the main ideas of reducedorder modeling techniques based on Krylov subspaces and describes some applications of reducedorder modeling in circuit simulation ..."
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Cited by 36 (6 self)
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This paper reviews the main ideas of reducedorder modeling techniques based on Krylov subspaces and describes some applications of reducedorder modeling in circuit simulation
Error estimation of the Pad'e approximation of transfer functions via the Lanczos process
 Trans. Numer. Anal
, 1998
"... 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 matr ..."
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Cited by 25 (8 self)
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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 multipleinputmultipleoutput circuit system via a block Lanczos process is also given.