Model reduction, centering, and the Karhunen-Loeve expansion

Glavaski, S.; Marsden, Jerrold E.; Murray, Richard M.

doi:10.1109/cdc.1998.758639

Cited by 22 publications

(19 citation statements)

References 4 publications

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“…Applications of POD include image processing [22], data compression, signal analysis [2], modeling and control of chemical reaction systems [12,25,26], turbulence models [14], coherent structures in fluids [14], control of fluids [10], electrical power grids [21,20,18], and wind engineering to name a few.…”

Section: Background On Proper Orthogonal Decompositionmentioning

confidence: 99%

A New Look at Proper Orthogonal Decomposition

Rathinam¹,

Petzold²

2003

SIAM J. Numer. Anal.

393

290

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Abstract. We investigate some basic properties of the proper orthogonal decomposition (POD) method as it is applied to data compression and model reduction of finite dimensional nonlinear systems. First we provide an analysis of the errors involved in solving a nonlinear ODE initial value problem using a POD reduced order model. Then we study the effects of small perturbations in the ensemble of data from which the POD reduced order model is constructed on the reduced order model. We explain why in some applications this sensitivity is a concern while in others it is not. We also provide an analysis of computational complexity of solving an ODE initial value problem and study the computational savings obtained by using a POD reduced order model. We provide several examples to illustrate our theoretical results.

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Section: Background On Proper Orthogonal Decompositionmentioning

confidence: 99%

A New Look at Proper Orthogonal Decomposition

Rathinam¹,

Petzold²

2003

SIAM J. Numer. Anal.

393

290

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“…Experience has shown that if a system has symmetry, then using travelling Karhunen-Loève bases can dramatically cut down the dimension of the reduced system; see, for example, [8,36] and references therein. It is likewise well understood that discrete symmetries are also important to take into account when building bases.…”

Section: Improvementsmentioning

confidence: 99%

Structure-preserving model reduction for mechanical systems

Lall¹,

Krysl²,

Marsden³

2003

Physica D: Nonlinear Phenomena

142

112

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This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen-Loève expansion, balancing, and wavelet decompositions. The model reduction procedure is implemented for three-dimensional finite-element models of elasticity, and we show that using the standard Newmark implicit integrator, significant savings are obtained in the computational costs of simulation. In particular simulation of the reduced model scales linearly in the number of degrees of freedom, and parallelizes well.

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“…For model reduction of dynamical systems POD may be used on data points obtained from system trajectories obtained via experiments, numerical simulations, or analytical derivations. For more information see Rathinam and Petzold [13], Holmes, Lumley, and Berkooz [5], Moore [10], Lall, Marsden, and Glavaski [6], Glavaski, Marsden, and Murray [3], and references therein.…”

Section: Podmentioning

confidence: 99%

“…Berkooz [5]), cascading failures in power grids (Parrilo et al [11]), and control of compressors (Glavaski, Marsden, and Murray [3]), etc., and selective modal analysis has been developed by researchers in the electrical power field (Perèz-Arriaga et al [12]), to name a few.…”

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confidence: 99%

Dynamic Iteration Using Reduced Order Models: A Method for Simulation of Large Scale Modular Systems

Rathinam¹,

Petzold²

2002

SIAM J. Numer. Anal.

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Abstract. We describe a new iterative method, dynamic iteration using reduced order models (DIRM), for simulation of large scale modular systems using reduced order models that preserve the interconnection structure. This method may be compared to the waveform relaxation technique; however, unlike DIRM, waveform relaxation does not take advantage of model reduction techniques. The DIRM method involves simulating in turn each subsystem connected to model reduced versions of the other subsystems. The data from this simulation is then used to update the reduced model for that particular subsystem. We provide analytical results on convergence and accuracy of the DIRM method as well as numerical examples that demonstrate the success of DIRM and verify the analysis.

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Model reduction, centering, and the Karhunen-Loeve expansion

Cited by 22 publications

References 4 publications

A New Look at Proper Orthogonal Decomposition

A New Look at Proper Orthogonal Decomposition

Structure-preserving model reduction for mechanical systems

Dynamic Iteration Using Reduced Order Models: A Method for Simulation of Large Scale Modular Systems

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