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Balanced model reduction via the proper orthogonal decomposition
 AIAA Journal
, 2002
"... A new method for performing a balanced reduction of a highorder linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshots is used to obtain lowrank, reducedrange approximationsto the system control ..."
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Cited by 45 (5 self)
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A new method for performing a balanced reduction of a highorder linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshots is used to obtain lowrank, reducedrange approximationsto the system controllability and observability grammiansin either the time or frequency domain.The approximationsare then used to obtain a balanced reducedorder model. The method is particularly effective when a small number of outputs is of interest. It is demonstrated for a linearized highorder system that models unsteady motion of a twodimensional airfoil. Computation of the exact grammians would be impractical for such a large system. For this problem, very accurate reducedorder models are obtained that capture the required dynamics with just three states. The new models exhibit far superior performance than those derived using a conventionalproper orthogonaldecomposition. Although further development is necessary, the concept also extends to nonlinear systems. W Nomenclature h = airfoil plunge displacement K = proper orthogonal decomposition (POD) kernel m = number of POD snapshots n = number of states in computational � uid dynamics (CFD) model nr = number of states in reducedordermodel R = correlation matrix T = matrix whose columns contain the balancing transformation vectors u; U = vector containing inputs for models, time and frequency domain Wc = controllabilitygrammian Wco = grammian product Wo = observability grammian x; X = aerodynamic state vector for CFD model, time and frequency domain xr = aerodynamic state vector for reducedordermodel y; Y = vector containing outputs of CFD model, time and frequency domain yr = vector containing outputs of reducedordermodel z = dual state vector for CFD model i = ith Hankel singular value = basis vector! = forcing frequency
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
Optimizing dominant time constant in RC circuits
, 1996
"... We propose to use the dominant time constant of a resistorcapacitor (RC) circuit as a measure of the signal propagation delay through the circuit. We show that the dominant time constant is a quasiconvex function of the conductances and capacitances, and use this property to cast several interestin ..."
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Cited by 16 (8 self)
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We propose to use the dominant time constant of a resistorcapacitor (RC) circuit as a measure of the signal propagation delay through the circuit. We show that the dominant time constant is a quasiconvex function of the conductances and capacitances, and use this property to cast several interesting design problems as convex optimization problems, specifically, semidefinite programs (SDPs). For example, assuming that the conductances and capacitances are affine functions of the design parameters (which is a common model in transistor or interconnect wire sizing), one can minimize the power consumption or the area subject to an upper bound on the dominant time constant, or compute the optimal tradeoff surface between power, dominant time constant, and area. We will also note that, to a certain extent, convex optimization can be used to design the topology of the interconnect wires. This approach has two advantages over methods based on Elmore delay optimization. First, it handles a far wider class of circuits, e.g., those with nongrounded capacitors. Second, it always results in convex optimization problems for which very efficient interiorpoint methods have recently been developed. We illustrate the method, and extensions, with several examples involving optimal wire and transistor sizing.
Missing point estimation in models described by proper orthogonal decomposition
, 2007
"... This paper presents a new method of Missing Point Estimation (MPE) to derive efficient reducedorder models for largescale parametervarying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projectionbased model reduction framework is used w ..."
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Cited by 11 (4 self)
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This paper presents a new method of Missing Point Estimation (MPE) to derive efficient reducedorder models for largescale parametervarying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projectionbased model reduction framework is used where projection spaces are inferred from proper orthogonal decompositions of datadependent correlation operators. The key contribution of the MPE method is to perform online computations efficiently by computing Galerkin projections over a restricted subset of the spatial domain. Quantitative criteria for optimally selecting such a spatial subset are proposed and the resulting optimization problem is solved using an efficient heuristic method. The effectiveness of the MPE method is demonstrated by applying it to a nonlinear computational fluid dynamic model of an industrial glass furnace. For this example, the Galerkin projection can be computed using only 25 % of the spatial grid points without compromising the accuracy of the reduced model.
Model Reduction of LargeScale Systems Rational Krylov Versus Balancing Techniques
"... . In this paper, we describe some recent developments in the use of projection methods to produce a reducedorder model for a linear timeinvariant dynamical system which approximates its frequency response. We give an overview of the family of Rational Krylov methods and compare them with "nearopti ..."
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Cited by 4 (1 self)
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. In this paper, we describe some recent developments in the use of projection methods to produce a reducedorder model for a linear timeinvariant dynamical system which approximates its frequency response. We give an overview of the family of Rational Krylov methods and compare them with "nearoptimal" approximation methods based on balancing transformations. 1.
Parametric ReducedOrder Models for Probabilistic Analysis of Unsteady Aerodynamic Applications
"... Methodology is presented to derive reducedorder models for largescale parametric applications in unsteady aerodynamics. The specific case considered in this paper is a computational fluid dynamic (CFD) model with parametric dependence that arises from geometric shape variations. The first key cont ..."
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Cited by 2 (0 self)
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Methodology is presented to derive reducedorder models for largescale parametric applications in unsteady aerodynamics. The specific case considered in this paper is a computational fluid dynamic (CFD) model with parametric dependence that arises from geometric shape variations. The first key contribution of the methodology is the derivation of a linearized model that permits the effects of geometry variations to be represented with an explicit affine function. The second key contribution is an adaptive sampling method that utilizes an optimization formulation to derive a reduced basis that spans the space of geometric input parameters. The method is applied to derive efficient reducedorder models for probabilistic analysis of the effects of blade geometry variation for a twodimensional model problem governed by the Euler equations. Reducedorder models that achieve three orders of magnitude reduction in the number of states are shown to accurately reproduce CFD Monte Carlo simulation results at a fraction of the computational cost. I.
Krylov Subspace Methods and Applications to System and Control Problems
"... In this report the Krylov subspace methods are reviewed and some applications in linear system theory and modern control theory are introduced. A modification to the Arnoldibased method to solve the Lyapunov matrix equation is also proposed. ..."
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Cited by 1 (0 self)
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In this report the Krylov subspace methods are reviewed and some applications in linear system theory and modern control theory are introduced. A modification to the Arnoldibased method to solve the Lyapunov matrix equation is also proposed.
Hessianbased model reduction for largescale systems with initialcondition inputs
, 2008
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Approach to Inverse Problems
, 2008
"... In many settings, distributed sensors provide dynamic measurements over a specified time horizon that can be used to reconstruct information such as parameters, states or initial conditions. This estimation task can be posed formally as an inverse problem: given a model and a set of measurements, es ..."
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In many settings, distributed sensors provide dynamic measurements over a specified time horizon that can be used to reconstruct information such as parameters, states or initial conditions. This estimation task can be posed formally as an inverse problem: given a model and a set of measurements, estimate the parameters of interest. We consider the specific problem of computing in realtime the prediction of a contamination event, based on measurements obtained by mobile sensors. The spread of the contamination is modeled by the convection diffusion equation. A Bayesian approach to the inverse problem yields an estimate of the probability density function of the initial contaminant concentration, which can then be propagated through the forward model to determine the predicted contaminant field at some future time and its associated uncertainty distribution. Sensor steering is effected by formulating and solving an optimization problem that seeks the sensor locations that minimize the uncertainty in this prediction. An important aspect of this Dynamic Sensor Steering Algorithm is the ability to