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Model Reduction of LargeScale Dynamical Systems
"... Abstract. Simulation and control are two critical elements of Dynamic DataDriven Application Systems (DDDAS). Simulation of dynamical systems such as weather phenomena, when augmented with realtime data, can yield precise forecasts. In other applications such as structural control, the presence of ..."
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Abstract. Simulation and control are two critical elements of Dynamic DataDriven Application Systems (DDDAS). Simulation of dynamical systems such as weather phenomena, when augmented with realtime data, can yield precise forecasts. In other applications such as structural control, the presence of realtime data relating to system state can enable robust active control. In each case, there is an ever increasing need for improved accuracy, which leads to models of higher complexity. The basic motivation for system approximation is the need, in many instances, for a simplified model of a dynamical system, which captures the main features of the original complex model. This need arises from limited computational capability, accuracy of measured data, and storage capacity. The simplified model may then be used in place of the original complex model, either for simulation and prediction, oractive control. As sensor networks and embedded processors proliferate our environment, technologies for such approximations and realtime control emerge as the next major technical challenge. This paper outlines the state of the art and outstanding challenges in the development of efficient and robust methods for producing reduced order models of large statespace systems. 1
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 13 (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
Gijzen, Preconditioned multishift BiCG for H2optimal model reduction
, 2012
"... Abstract. We propose the use of a multishift biconjugate gradient method (BiCG) in combination with a suitable chosen polynomial preconditioning, to efficiently solve the two sets of multiple shifted linear systems arising at each iteration of the iterative rational Krylov algorithm (IRKA, [Gugerki ..."
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Abstract. We propose the use of a multishift biconjugate gradient method (BiCG) in combination with a suitable chosen polynomial preconditioning, to efficiently solve the two sets of multiple shifted linear systems arising at each iteration of the iterative rational Krylov algorithm (IRKA, [Gugerkin, Antoulas, and Beattie, 2008]) for H2optimal model reduction of linear systems. The idea is to construct in advance bases for the two preconditioned Krylov subspaces (one for the matrix and one for its adjoint). These bases are then reused inside the model reduction methods for the other shifts, by exploiting the shiftinvariant property of Krylov subspaces. The polynomial preconditioner is chosen to maintain this shiftinvariant property. This means that the shifted systems can be solved without additional matrixvector products. The performance of our proposed implementation is illustrated through numerical experiments. Key words. Model order reduction, IRKA, shifted linear systems, polynomial preconditioning, BiCG
Nonlinear parametric inversion using interpolatory model reduction. arXiv preprint arXiv:1311.0922
, 2013
"... Abstract. Nonlinear parametric inverse problems appear in several prominent applications; one such application is Diffuse Optical Tomography (DOT) in medical image reconstruction. Such inverse problems present huge computational challenges, mostly due to the need for solving a sequence of largesca ..."
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Abstract. Nonlinear parametric inverse problems appear in several prominent applications; one such application is Diffuse Optical Tomography (DOT) in medical image reconstruction. Such inverse problems present huge computational challenges, mostly due to the need for solving a sequence of largescale discretized, parametrized, partial differential equations (PDEs) in the forward model. In this paper, we show how interpolatory parametric model reduction can significantly reduce the cost of the inversion process in DOT by drastically reducing the computational cost of solving the forward problems. The key observation is that function evaluations for the underlying optimization problem may be viewed as transfer function evaluations along the imaginary axis; a similar observation holds for Jacobian evaluations as well. This motivates the use of systemtheoretic model order reduction methods. We discuss the construction and use of interpolatory parametric reduced models as surrogates for the full forward model. Within the DOT setting, these surrogate models can approximate both the cost functional and the associated Jacobian with very little loss of accuracy while significantly reducing the cost of the overall inversion process. Four numerical examples illustrate the efficiency of the proposed approach. Although we focus on DOT in this paper, we believe that our approach is applicable much more generally. Key words. DOT, PaLS, model reduction, rational interpolation. AMS subject classifications. 65F10, 65N22, 93A15, 93C05.
Model Order Reduction by ParameterVarying Oblique Projection
"... Abstract — A method to reduce the dynamic order of linear parametervarying (LPV) systems in grid representation is developed in this paper. It approximates balancing and truncation by an oblique projection onto a dominant subspace. The approach is novel in its use of a parametervarying kernel to ..."
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Abstract — A method to reduce the dynamic order of linear parametervarying (LPV) systems in grid representation is developed in this paper. It approximates balancing and truncation by an oblique projection onto a dominant subspace. The approach is novel in its use of a parametervarying kernel to define the direction of this projection. Parametervarying state transformations in general lead to parameter rate dependence in the model. The proposed projection avoids this dependence and maintains a consistent state space basis for the reducedorder system. The method is compared with LPV balancing and truncation for a nonlinear massspringdamper system. It is shown to yield similar accuracy while the required computation time is reduced by a factor of almost 100,000. I.
Model Reduction by Rational Interpolation
"... Abstract. The last two decades have seen major progress in interpolatory methods for model reduction of largescale dynamical systems have. The ability to produce optimal (at least locally) interpolatory reduced models at a modest cost for linear and bilinear systems, extensions to reducing paramet ..."
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Abstract. The last two decades have seen major progress in interpolatory methods for model reduction of largescale dynamical systems have. The ability to produce optimal (at least locally) interpolatory reduced models at a modest cost for linear and bilinear systems, extensions to reducing parametric systems, and the ability to produce reducedmodels directly from input/output measurements are some examples of these new developments. This chapter will give a survey of interpolatory model reduction methods including a detailed analysis of basic principles together with a presentation of the more recent developments. Discussion will be supported by numerical examples.
applications
"... Model order reduction techniques are known to work reliably for finiteelementtype simulations of MEMS devices. These techniques can tremendously shorten computational times for transient and harmonic analysis. However, standard model reduction techniques cannot be applied if the equation system in ..."
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Model order reduction techniques are known to work reliably for finiteelementtype simulations of MEMS devices. These techniques can tremendously shorten computational times for transient and harmonic analysis. However, standard model reduction techniques cannot be applied if the equation system incorporates timevarying matrices or parameters that are to be preserved for the reduced model. However, design cycles often involve parameter modification, which should remain possible also in the reduced model. In this paper we demonstrate a novel parametrization method to numerically construct highly accurate parametric ODE systems based on a small number of systems with different parameter settings. This method is demonstrated to parameterize the geometry of a model of a microgyroscope, where the relative error introduced by the parametrization lies in the region of 10 −9. We also present novel semiautomatic order reduction methods that can preserve scalar parameters or functions during the reduction process. The first approach is based on a multivariate Padétype expansion. The second approach is a coupling of the balanced truncation method for model order reduction of (deterministic) linear, timeinvariant systems with interpolation. The approach is quite flexible in allowing the use of numerous interpolation techniques like polynomial, Hermite, rational, sinc, and spline interpolation. As technical examples we investigate a micro anemometer as well as the gyroscope. Speedup factors of 20 to 80 could be achieved, whilst retaining up to 6 parameters, and keeping typical relative errors below 1%.
Model Order Reduction for DifferentialAlgebraic Equations: a Survey
, 2015
"... In this paper, we discuss the model order reduction problem for descriptor systems, that is, systems with dynamics described by differentialalgebraic equations. We focus on linear descriptor systems as a broad variety of methods for these exist, while model order reduction for nonlinear descriptor ..."
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In this paper, we discuss the model order reduction problem for descriptor systems, that is, systems with dynamics described by differentialalgebraic equations. We focus on linear descriptor systems as a broad variety of methods for these exist, while model order reduction for nonlinear descriptor systems has not received sufficient attention up to now. Model order reduction for linear statespace systems has been a topic of research for about 50 years at this writing, and by now can be considered as a mature field. The extension to linear descriptor systems usually requires extra treatment of the constraints imposed by the algebraic part of the system. For almost all methods, this causes some technical difficulties, and these have only been thoroughly addressed in the last decade. We will focus on these developments in particular for the popular methods related to balanced truncation and rational interpolation. We will review efforts in extending these approaches to descriptor systems, and also add the extension of the socalled stochastic balanced truncation method to descriptor systems
A SURVEY OF PROJECTIONBASED MODEL REDUCTION METHODS FOR PARAMETRIC DYNAMICAL SYSTEMS ∗
"... Abstract. 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 often leads to unmanageable demands on computational resources. Model reduction aims to reduce this ..."
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Abstract. 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 often 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 state of the art 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 equations. The goal of parametric model reduction is to generate low cost but accurate models that characterize system response for different values of the parameters. This paper surveys stateoftheart meth
Dynamics of Complex Technical Systems
, 2015
"... www.mpimagdeburg.mpg.de/preprints In this article we investigate model order reduction of largescale systems using frequencylimited balanced truncation, which restricts the well known balanced truncation framework to prescribed frequency regions. The main emphasis is put on the efficient numerica ..."
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www.mpimagdeburg.mpg.de/preprints In this article we investigate model order reduction of largescale systems using frequencylimited balanced truncation, which restricts the well known balanced truncation framework to prescribed frequency regions. The main emphasis is put on the efficient numerical realization of this model reduction approach. We discuss numerical methods to take care of the involved matrixvalued functions. The occurring largescale Lyapunov equations are solved for lowrank approximations for which we also establish results regarding the eigenvalues of their solutions. These results, and also numerical experiments, will show that eigenvalues of the Lyapunov solutions in frequencylimited balanced truncation are often smaller than those in standard balanced truncation. Moreover, we show in further numerical examples that frequencylimited balanced truncation generates reduced order models which are significantly more accurate in the considered frequency region. 1