Balanced Truncation Model Reduction for a Class of Descriptor Systems with Application to the Oseen Equations

Heinkenschloss, Matthias; Sorensen, Danny C.; Sun, Kai

doi:10.1137/070681910

Cited by 130 publications

(200 citation statements)

References 30 publications

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“…Another assumption on system structure in this paper is that the E(p) matrix is nonsingular for any parameter selection p. However, in several important applications (e.g., incompressible flows and circuit design) one obtains a system of the form (2.1), where E(p) could be singular for some or all selections of p. Such systems with a singular E(p) matrix are called systems of differential-algebraic equations (DAEs). Projection-based model reduction of DAEs has been studied extensively; see, for example, [45,50,118,129,172,195,208,209]. The theoretical discussions of this paper directly extend to this setting.…”

Section: 7mentioning

confidence: 98%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale 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 large-scale 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 the 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 large-scale 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 state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: 7mentioning

confidence: 98%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…However, A 21 = A T 12 can be treated in the current bilinear framework as well, if one extends the arguments used in [14]. By A T 12 x 1 (t) = 0, we have that Π T x 1 (t) = x 1 (t), cf., e.g., [21]. Replacing x 1 (t) by Π T x 1 (t) in (3) and premultiplying by Π, we obtain the following system:…”

Section: Mor For Bilinear Descriptor Systemsmentioning

confidence: 99%

“…As stated in [21], the dynamical system (5) evolves in the n 1 −n 2 dimensional subspace ker (Π). Therefore, with the decomposition of Π,…”

Section: Mor For Bilinear Descriptor Systemsmentioning

confidence: 99%

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An Iterative Model Order Reduction Scheme for a Special Class of Bilinear Descriptor Systems Appearing in Constraint Circuit Simulation

Goyal¹,

Benner²

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. We focus on interpolatory-based model order reduction for a special class of bilinear descriptor systems in the H 2 -optimal framework, appearing in constraint circuit simulations. The straightforward extension of the H 2 -optimality conditions for ODE systems to descriptor systems generically may produce an unbounded error in the H 2 or H ∞ norm, or both. This arises due to the inappropriate use of the polynomial part of the system. To ensure bounded error, one needs to deal with the polynomial part of the systems properly. To do so, we first transform these descriptor systems into equivalent ODE systems by means of oblique projectors, as it is widely done in the literature for linear index-2 ODEs. This enables us to employ bilinear iterative rational Krylov algorithm (B-IRKA) which provides us locally H 2 -optimal reducedorder systems on convergence, if it converges. Unfortunately, the direct implementation of B-IRKA on equivalent ODEs requires the expensive explicit computation of the oblique projectors. Therefore, as one of our contributions, we show how to apply B-IRKA to the equivalent bilinear ODE system without an explicit computation of the projectors. We demonstrate the efficiency of the proposed technique by means of several constraint circuit examples and compare the quality of the reduced-order systems with the ones obtained by using the projection matrices determined by applying IRKA on the corresponding linear part.

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“…Other approaches to an approximation of the I/O map are balanced truncation [20], moment matching [16], and proper orthogonal decomposition [42]. In particular, balanced truncation has been investigated for spatially discretized linearized Navier-Stokes equations (LNSE), see [28,45]. This strategy is motivated by the observation that control often acts locally in time, and therefore the design of a controller based on an approximated linear model still promises a good result.…”

Section: Introductionmentioning

confidence: 99%

Distributed control of linearized Navier‐Stokes equations via discretized input/output maps

Heiland

Mehrmann

2012

Z Angew Math Mech

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The construction of reduced order models for flow control via a direct discretization of the input/output behavior of the system is discussed. The spatially discretized equations are linearized such that an explicit formula for the corresponding input/output map can be used to generate a matrix representation of the input/output map. Estimates for the approximation error are derived and the applicability is illustrated via a numerical example for the control of a driven cavity flow.

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Balanced Truncation Model Reduction for a Class of Descriptor Systems with Application to the Oseen Equations

Cited by 130 publications

References 30 publications

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

An Iterative Model Order Reduction Scheme for a Special Class of Bilinear Descriptor Systems Appearing in Constraint Circuit Simulation

Distributed control of linearized Navier‐Stokes equations via discretized input/output maps

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