Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method

Benner, Peter; Kürschner, Patrick; Saak, Jens

doi:10.1007/s11075-012-9569-7

Cited by 66 publications

(83 citation statements)

References 32 publications

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“…For effective methods to solve large-scale Lyapunov equations, see, e.g., [46,47,152,219] for ADI-type methods, [117,191,199,206] for the Smith method and its variants, [78,135,203,210] for Krylov-based methods, or the recent survey [48] and the references therein.…”

Section: Balanced Truncationmentioning

confidence: 99%

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: Balanced Truncationmentioning

confidence: 99%

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

2015

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“…As in [13,27,2,28] the occurring large-scale generalized Lyapunov equations were solved approximately with the SO-LR-ADI iterations. A new result showing the low-rank structure of the Lyapunov residual in low-rank ADI methods was given.…”

Section: Discussionmentioning

confidence: 99%

“…However, in the latter case it is assumed that µ k , µ k+1 := µ k holds, and Z is augmented by 2m new real columns. This construction is possible due to a result in [28] which allows the computation of real low-rank factors even if complex shift parameters are used. The main result there is that the increment with respect to µ k+1 can be constructed by…”

Section: Handling the Second Order Structure And Complex Shift Paramementioning

confidence: 99%

An improved numerical method for balanced truncation for symmetric second-order systems

Benner

Kürschner

Saak

2013

Mathematical and Computer Modelling of Dynamical Systems

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We consider balanced truncation model order reduction for symmetric second order systems. The occurring large-scale generalized and structured Lyapunov equations are solved with a specially adapted low-rank ADI type method. Stopping criteria for this iteration are investigated and a new result concerning the Lyapunov residual within the low-rank ADI method is established. We also propose a goal oriented stopping criterion which tries to incorporate the balanced truncation approach already during the ADI iteration. The model reduction approach using the ADI method with different stopping criteria is evaluated on several test systems.

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“…In [11,Algorithm 4] it is shown how the LRCF-ADI method can be rewritten to achieve this goal. However, another approach given in [10] generates real low-rank factors as well, but appears to be more efficient in term of the computational complexity although temporarily still complex arithmetic operations are employed. In the remainder we assume we have generated real LRCFs using the latter approach.…”

Section: Algorithm 2 Generalized Low-rank Cholesky Factor Adi Iteratimentioning

confidence: 99%

“…In the presence of complex shift parameters, it is again possible to rewrite Algorithm 3 into a real form along the lines of [11,Algorithm 4] or to apply the strategy proposed in [10] in order to generate real LRCFs. b) Structure Preserving Balanced Truncation for Second Order Systems.…”

Section: Algorithm 2 Generalized Low-rank Cholesky Factor Adi Iteratimentioning

confidence: 99%

Model reduction of an elastic crankshaft for elastic multibody simulations

et al. 2012

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System analysis and optimization of combustion engines and engine components are increasingly supported by digital simulations. In the simulation process of combustion engines multi physics simulations are used. As an example, in the simulation of a crank drive the mechanical subsystem is coupled to a hydrodynamic subsystem. As far as the modeling of the mechanical subsystems is concerned, elastic multibody systems are frequently used. During the simulation many equations must be solved simultaneously, the hydrodynamic equations as well as the equations of motion of each body in the elastic multibody system. Since the discretization of the elastic bodies, e.g with the help of the finite element method, introduces a large number of elastic degrees of freedom, an efficient simulation of the system becomes difficult. The linear model reduction of the elastic degrees of freedom is a key step for using flexible bodies in multibody systems and turning simulations more efficient from a computational point of view. In recent years a variety of new reduction methods alongside the traditional techniques were developed in applied mathematics. Some of these methods are introduced and compared for reducing the equations of motion of an elastic multibody system. The special focus of this work is on balanced truncation model order reduction which is a singular value based reduction technique using the Gramian matrices of the system. We investigate a version of this method that is adapted to the structure of a special class of second order dynamical systems which is important for the particular application discussed here. The main computational task in balanced truncation is the solution or large-scale Lyapunov equations for which we apply a modified variant of the low-rank ADI method. The simulation of a crank drive with a flexible crankshaft is taken as technically relevant example. The results are compared to other methods like Krylov approaches or modal reduction.

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Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method

Cited by 66 publications

References 32 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 improved numerical method for balanced truncation for symmetric second-order systems

Model reduction of an elastic crankshaft for elastic multibody simulations

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