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213
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 71 (3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
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.
Guaranteed Passive Balancing Transformations for Model Order Reduction
, 2002
"... The major concerns in stateoftheart model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteedpassive models ..."
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Cited by 66 (8 self)
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The major concerns in stateoftheart model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteedpassive models, for systems with special internal structure, using numerically stable and efficient Krylovsubspace iterations. Truncated Balanced Realization (TBR) algorithms, as used to date in the design automation community, can achieve smaller models with better error control, but do not necessarily preserve passivity. In this paper we show how to construct TBRlike methods that guarantee passive reduced models and in addition are applicable to statespace systems with arbitrary internal structure.
Projectionbased approaches for model reduction of weakly nonlinear, timevarying systems
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
"... Abstract—The problem of automated macromodel generation is interesting from the viewpoint of systemlevel design because if small, accurate reducedorder models of system component blocks can be extracted, then much larger portions of a design, or more complicated systems, can be simulated or verifi ..."
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Cited by 59 (1 self)
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Abstract—The problem of automated macromodel generation is interesting from the viewpoint of systemlevel design because if small, accurate reducedorder models of system component blocks can be extracted, then much larger portions of a design, or more complicated systems, can be simulated or verified than if the analysis were to have to proceed at a detailed level. The prospect of generating the reduced model from a detailed analysis of component blocks is attractive because then the influence of secondorder device effects or parasitic components on the overall system performance can be assessed. In this way overly conservative design specifications can be avoided. This paper reports on experiences with extending model reduction techniques to nonlinear systems of differential–algebraic equations, specifically, systems representative of RF circuit components. The discussion proceeds from linear timevarying, to weakly nonlinear, to nonlinear timevarying analysis, relying generally on perturbational techniques to handle deviations from the linear timeinvariant case. The main intent is to explore which perturbational techniques work, which do not, and outline some problems that remain to be solved in developing robust, general nonlinear reduction methods. Index Terms—Circuit noise, circuit simulation, nonlinear systems, reducedorder systems, timevarying circuits. I.
The Sylvester equation and approximate balanced reduction
, 2002
"... The purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems. The resulting approach is completely automatic once an error tolerance is specified and also yields an error bound. This is to be contrasted with existing projection me ..."
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Cited by 53 (4 self)
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The purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems. The resulting approach is completely automatic once an error tolerance is specified and also yields an error bound. This is to be contrasted with existing projection methods, namely PVL (Pad via Lanczos) and rationa Krylov, which do not satisfy these properties. Our approach is based on the computation and approximation of the cross gramtan of the system. The cross gramtan is the solution of a Sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights.
Projection Frameworks for Model Reduction of Weakly . . .
, 2000
"... In this paper we present a generalization of popular linear model reduction methods, such as Lanczos and Arnoldibased algorithms based on rational approximation, to systems whose response to interesting external inputs can be described by a few terms in a functional series expansion such as a Volt ..."
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Cited by 43 (1 self)
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In this paper we present a generalization of popular linear model reduction methods, such as Lanczos and Arnoldibased algorithms based on rational approximation, to systems whose response to interesting external inputs can be described by a few terms in a functional series expansion such as a Volterra series. The approach allows automatic generation of macromodels that include frequencydependent nonlinear effects.
Model reduction of MIMO systems via tangential interpolation
 SIAM J. Matrix Anal. Appl
, 2004
"... Abstract. In this paper, we address the problem of constructing a reduced order system of minimal McMillan degree that satisfies a set of tangential interpolation conditions with respect to the original system under some mild conditions. The resulting reduced order transfer function appears to be ge ..."
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Cited by 34 (6 self)
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Abstract. In this paper, we address the problem of constructing a reduced order system of minimal McMillan degree that satisfies a set of tangential interpolation conditions with respect to the original system under some mild conditions. The resulting reduced order transfer function appears to be generically unique and we present a simple and efficient technique to construct this interpolating reduced order system. This is a generalization of the multipoint Padé technique which is particularly suited to handle multiinput multioutput systems.
Piecewise polynomial nonlinear model reduction
 in Design Automation Conference
"... We present a novel, general approach towards modelorder reduction (MOR) of nonlinear systems that combines good global and local approximation properties. The nonlinear system is first approximated as piecewise polynomials over a number of regions, following which each region is reduced via pol ..."
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Cited by 33 (3 self)
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We present a novel, general approach towards modelorder reduction (MOR) of nonlinear systems that combines good global and local approximation properties. The nonlinear system is first approximated as piecewise polynomials over a number of regions, following which each region is reduced via polynomial modelreduction methods. Our approach, dubbed PWP, generalizes recent piecewise linear approaches and ties them with polynomialbased MOR, thereby combining their advantages. In particular, reduced models obtained by our approach reproduce smallsignal distortion and intermodulation properties well, while at the same time retaining fidelity in largeswing and largesignal analyses, e.g., transient simulations. Thus our reduced models can be used as dropin replacements for timedomain as well as frequencydomain simulations, with small or large excitations. By exploiting sparsity in system polynomial coefficients, we are able to make the polynomial reduction procedure linear in the size of the original system. We provide implementation details and illustrate PWP with an example.
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 28 (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.
Model Reduction of Large Linear Systems via Low Rank System Gramians
, 2000
"... This dissertation concerns the model reduction of large, linear, timeinvariant systems. A new method called the Dominant Gramian Eigenspaces method, which utilizes low rank approximations to the exact system gramians, is proposed for such system. The Cholesky Factor ..."
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Cited by 26 (0 self)
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This dissertation concerns the model reduction of large, linear, timeinvariant systems. A new method called the Dominant Gramian Eigenspaces method, which utilizes low rank approximations to the exact system gramians, is proposed for such system. The Cholesky Factor