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64
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.
Asymptotic waveform evaluation via a Lanczos method
 Appl. Math. Lett
, 1994
"... AbstractIn this paper we show that the twosided Lanczos procedure combined with implicit restarts, offers significant advantages over Pad6 approximations used typically for model reduction in circuit simulation. ..."
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Cited by 58 (4 self)
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AbstractIn this paper we show that the twosided Lanczos procedure combined with implicit restarts, offers significant advantages over Pad6 approximations used typically for model reduction in circuit simulation.
Model reduction of state space systems via an Implicitly Restarted Lanczos method
 Numer. Algorithms
, 1996
"... The nonsymmetric Lanczos method has recently received significant attention as a model reduction technique for largescale systems. Unfortunately, the Lanczos method may produce an unstable partial realization for a given, stable system. To remedy this situation, inexpensive implicit restarts are de ..."
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Cited by 56 (8 self)
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The nonsymmetric Lanczos method has recently received significant attention as a model reduction technique for largescale systems. Unfortunately, the Lanczos method may produce an unstable partial realization for a given, stable system. To remedy this situation, inexpensive implicit restarts are developed which can be employed to stabilize the Lanczos generated model.
Reducedorder modeling techniques based on Krylov subspaces and their use in circuit simulation
, 1998
"... ..."
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.
Krylov space methods on statespace control models
 Circuits, Systems, and Signal Processing
, 1994
"... We give an overview of various Lanczos/Krylov space methods and how they are being used for solving certain problems in Control Systems Theory based on statespace models. The matrix methods used are based on Krylov sequences and are closely related to modern iterative methods for standard matrix pr ..."
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Cited by 43 (4 self)
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We give an overview of various Lanczos/Krylov space methods and how they are being used for solving certain problems in Control Systems Theory based on statespace models. The matrix methods used are based on Krylov sequences and are closely related to modern iterative methods for standard matrix problems such as sets of linear equations and eigenvalue calculations. We show how these methods can be applied to problems in Control Theory such as controllability, observability and model reduction. All the methods are based on the use of statespace models, which may be very sparse and of high dimensionality. For example, we show how one may compute an approximate solution to a Lyapunov equation arising from discretetime linear dynamic system with a large sparse system matrix by the use of the Arnoldi Algorithm, and so obtain an approximate Grammian matrix. This has applications in model reduction. The close relation between the matrix Lanczos algorithm and the algebraic structure of linear control systems is also explored. 1
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 35 (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.
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 27 (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.
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 27 (15 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.