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130
Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations
, 2009
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Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations
 SIAM Journal on Scientific Computing
"... Abstract. We discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete Oseen equations with timeindependent advection and the linearized Navier–Stokes equations, linearized around a steady state. The purpose of this paper is twof ..."
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Cited by 26 (2 self)
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Abstract. We discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete Oseen equations with timeindependent advection and the linearized Navier–Stokes equations, linearized around a steady state. The purpose of this paper is twofold. First, we show how to apply standard balanced truncation model reduction techniques, which apply to dynamical systems given by ordinary differential equations, to this class of descriptor systems. This is accomplished by eliminating the algebraic equation using a projection. The second objective of this paper is to demonstrate how the important class of ADI/Smithtype methods for the approximate computation of reduced order models using balanced truncation can be applied without explicitly computing the aforementioned projection. Instead, we utilize the solution of saddle point problems. We demonstrate the effectiveness of the technique in the computation of reduced order models for semidiscrete Oseen equations.
InputOutput Analysis and Control Design Applied to a Linear Model of Spatially Developing Flows
 Appl. Mech. Rev
"... A framework for the inputoutput analysis, model reduction and control design of spatially developing shear flows is presented using the Blasius boundarylayer flow as an example. An inputoutput formulation of the governing equations yields a flexible formulation for treating stability problems a ..."
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Cited by 15 (5 self)
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A framework for the inputoutput analysis, model reduction and control design of spatially developing shear flows is presented using the Blasius boundarylayer flow as an example. An inputoutput formulation of the governing equations yields a flexible formulation for treating stability problems and for developing control strategies that optimize given objectives. Model reduction plays an important role in this process since the dynamical systems that describe most flows are discretized partial differential equations with a very large number of degrees of freedom. Moreover, as system theoretical tools, such as controllability, observability and balancing has become computationally tractable for largescale systems, a systematic approach to model reduction is presented. I.
Model order reduction for nonlinear dynamical systems based on trajectory piecewiselinear approximations
 LINEAR ALGEBRA AND ITS APPLICATIONS
, 2005
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Fourier Series for Accurate, Stable, ReducedOrder Models for Linear CFD Applications.
"... A new method, Fourier model reduction (FMR), for obtaining stable, accurate, loworder models of very large linear systems is presented. The technique draws on traditional control and dynamical system concepts and utilizes them in a way which is computationally very efficient. Discretetime Fourier ..."
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Cited by 15 (1 self)
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A new method, Fourier model reduction (FMR), for obtaining stable, accurate, loworder models of very large linear systems is presented. The technique draws on traditional control and dynamical system concepts and utilizes them in a way which is computationally very efficient. Discretetime Fourier coefficients of the large system transfer function are calculated and used to construct the Hankel matrix of an intermediate system with guaranteed stability. Explicit balanced truncation formulae are then applied to obtain the final reducedorder model, whose size is determined by the Hankel singular values of the intermediate system. In this paper, the method is applied to two computational fluid dynamic systems, which model unsteady motion of a twodimensional subsonic airfoil and unsteady flow in a supersonic diffuser. In both cases, the new method is found to work extremely well. Results are compared to models developed using the proper orthogonal decomposition and Arnoldi method. In comparison with these widely used techniques, the new method is computationally more efficient, guarantees the stability of the reducedorder model, uses both input and output information, and is valid over a wide range of frequencies.
An efficient reducedorder modeling approach for nonlinear parametrized partial differential equations
"... For general nonlinear parametrized partial differential equations (PDEs), the standard Galerkin projection is no longer efficient to generate reducedorder models. This is because the evaluation of the integrals involving the nonlinear terms has a high computational complexity and cannot be preco ..."
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For general nonlinear parametrized partial differential equations (PDEs), the standard Galerkin projection is no longer efficient to generate reducedorder models. This is because the evaluation of the integrals involving the nonlinear terms has a high computational complexity and cannot be precomputed. This situation also occurs for linear equations when the parametric dependence is nonaffine. In this paper, we propose an efficient approach to generate reducedorder models for largescale systems derived from PDEs, which may involve nonlinear terms and nonaffine parametric dependence. The main idea is to replace the nonlinear and nonaffine terms with a coefficientfunction approximation consisting of a linear combination of precomputed basis functions with parameterdependent coefficients. The coefficients are determined efficiently by an inexpensive and stable interpolation at some precomputed points. The efficiency and accuracy of this method are demonstrated on several test cases, which show significant computational savings relative to the standard Galerkin projection reducedorder approach. Copyright q
A piecewiselinear momentmatching approach to parameterized modelorder reduction for highly nonlinear systems
 IEEE Trans. CAD of Integrated Circuits and Systems
, 2007
"... Abstract—This paper presents a parameterized reduction technique for highly nonlinear systems. In our approach, we first approximate the nonlinear system with a convex combination of parameterized linear models created by linearizing the nonlinear system at points along training trajectories. Each ..."
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Abstract—This paper presents a parameterized reduction technique for highly nonlinear systems. In our approach, we first approximate the nonlinear system with a convex combination of parameterized linear models created by linearizing the nonlinear system at points along training trajectories. Each of these linear models is then projected using a momentmatching scheme into a loworder subspace, resulting in a parameterized reducedorder nonlinear system. Several options for selecting the linear models and constructing the projection matrix are presented and analyzed. In addition, we propose a training scheme which automatically selects parameterspace training points by approximating parameter sensitivities. Results and comparisons are presented for three examples which contain distributed strong nonlinearities: a diode transmission line, a microelectromechanical switch, and a pulsenarrowing nonlinear transmission line. In most cases, we are able to accurately capture the parameter dependence over the parameter ranges of ±50 % from the nominal values and to achieve an average simulation speedup of about 10×. Index Terms—Modelorder reduction (MOR), nonlinear systems, parameterized reducedorder models (PROMs).
A Survey of Model Reduction Methods for Parametric Systems
, 2013
"... Numerical simulation of largescale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent largescale nature of the models leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational bu ..."
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Cited by 12 (4 self)
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Numerical simulation of largescale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent largescale nature of the models leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original largescale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey stateoftheart in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and largescale systems of parameterized ordinary differential
An Optimization Framework for GoalOriented, ModelBased Reduction of LargeScale Systems
 IN PROC. 44TH IEEE CONF. ON DECISION AND CONTROL, SEVILLE
, 2005
"... Optimizationready reducedorder models should target a particular output functional, span an applicable range of dynamic and parametric inputs, and respect the underlying governing equations of the system. To achieve this goal, we present an approach for determining a projection basis that uses a g ..."
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Cited by 10 (2 self)
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Optimizationready reducedorder models should target a particular output functional, span an applicable range of dynamic and parametric inputs, and respect the underlying governing equations of the system. To achieve this goal, we present an approach for determining a projection basis that uses a goaloriented, modelbased optimization framework. The mathematical framework permits consideration of general dynamical systems with general parametric variations. The methodology is applicable to both linear and nonlinear systems and to systems with many input parameters. This paper focuses on an initial presentation and demonstration of the methodology on a simple linear model problem of the twodimensional, timedependent heat equation with a small number of inputs. For this example, the reduced models determined by the new approach provide considerable improvement over those derived using the proper orthogonal decomposition.