Errors in reduction methods

Utku, S.; Clemente, José L. M.; Salama, M.

doi:10.1016/0045-7949(85)90170-1

Cited by 19 publications

(14 citation statements)

References 6 publications

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“…Unlike the method presented in [25], our estimates and bounds are valid over the entire time interval considered and not only in a neighborhood of the initial time.…”

mentioning

confidence: 96%

“…Although we study only a particular projection-based model reduction technique (POD) among those considered in [25], the methodology developed here for POD can be easily extended to other types of projection. Compared to the approach taken in [14], our method is applicable to a larger class of problems, our main requirement being that the norm of the POD-based error is small enough for the linearized error equation to be a good enough approximation.…”

mentioning

confidence: 99%

“…To judge the quality of the reduced model, it is important to estimate its error. An algorithm for estimating the error of a class of reduction methods based on projection techniques was presented in [25]. In this approach, the original problem is linearized around the initial time.…”

mentioning

confidence: 99%

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Error Estimation for Reduced‐Order Models of Dynamical Systems

Homescu¹,

Petzold²,

Serban³

2007

SIAM Rev.

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Abstract. The use of reduced order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors, by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most importantly, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude. Key words. model reduction, proper orthogonal decomposition, small sample statistical condition estimation, adjoint method AMS subject classifications. 65L10, 65L991. Introduction. Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [1]. Reduced models are used for efficient simulation [12,24] and control [13,22]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments or "black box" optimization methods [23].The Proper Orthogonal Decomposition (POD) method has been used extensively in a variety of fields including fluid dynamics [18], identification of coherent structures [8,16] , and study of the dynamic wind pressures acting on buildings [11], to name only a few. A detailed description [8] of the POD approach as a reduction method shows that, for a given number of modes, POD is the most efficient choice among all linear decompositions in the sense that it retains, on average, the greatest possible kinetic energy.As soon as one contemplates the use of a reduced model, questions concerning the quality of the approximation become paramount. To judge the quality of the reduced model, it is important to estimate its error. An algorithm for estimating the error of a class of reduction methods based on projection techniques was presented in [25]. In this approach, the original problem is linearized around the initial time. The resulting first-order error estimates are valid for only a small number of time steps (during which the Jacobian matrix can be considered constant). First-order estimates

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“…Unlike the method presented in [25], our estimates and bounds are valid over the entire time interval considered and not only in a neighborhood of the initial time.…”

mentioning

confidence: 96%

mentioning

confidence: 99%

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Error Estimation for Reduced‐Order Models of Dynamical Systems

Homescu¹,

Petzold²,

Serban³

2007

SIAM Rev.

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“…However, the analysis in [44] is based on linearization, and hence, large perturbations may require some knowledge of the solution of the perturbed system. Some related works on error estimations such as in [88,32,58,43] can be found in the extensive review from [44].…”

Section: Error Estimate and Stability Analysis For Pod-galerkin Reducmentioning

confidence: 99%

Nonlinear Model Reduction via Discrete Empirical Interpolation

Chaturantabut¹,

Sorensen²

2010

SIAM J. Sci. Comput.

1,651

1,778

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“…These two methods still bootstrap off of existing simulation data, so they cannot be applied to the case where the simulation is only computed once. Utku et al [1985] proposed an a priori method of estimating the reduced-order error of a non-linear system over time with the goal of evaluating the effectiveness of different bases. While the details differ significantly, our error estimator for dynamic simulation shares the same high-level strategy of linearizing about the last known unreduced state.…”

Section: Related Workmentioning

confidence: 99%

Skipping steps in deformable simulation with online model reduction

Kim¹,

James²

2009

ACM SIGGRAPH Asia 2009 Papers

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Figure 1: We learn fast subspace models on-the-fly to accelerate deformable simulation: (Top) Ground truth frames from a 1000-frame deformable character simulation with 165,941 tetrahedra and Arruda-Boyce material which took 48 hours (174,521 seconds). (Middle) Frames from our online integrator which were generated in only 51 minutes (3,101 seconds) -56.28× faster -by learning a fast 11-dimensional subspace model on-the-fly (Bottom) Visualization of the temporal behavior of the online integrator reveal full steps (in black), reduced steps (in gray), and basis updates (red ticks). A reduced basis is detected quickly, and the first "skip" occurs at timestep 2. Only a handful of full steps are needed, and our reduced solver computes over 98% of the the steps. AbstractFinite element simulations of nonlinear deformable models are computationally costly, routinely taking hours or days to compute the motion of detailed meshes. Dimensional model reduction can make simulations orders of magnitude faster, but is unsuitable for general deformable body simulations because it requires expensive precomputations, and it can suppress motion that lies outside the span of a pre-specified low-rank basis. We present an online model reduction method that does not have these limitations. In lieu of precomputation, we analyze the motion of the full model as the simulation progresses, incrementally building a reduced-order nonlinear model, and detecting when our reduced model is capable of performing the next timestep. For these subspace steps, full-model computation is "skipped" and replaced with a very fast (on the order of milliseconds) reduced order step. We present algorithms for both dynamic and quasistatic simulations, and a "throttle" parameter that allows a user to trade off between faster, approximate previews and slower, more conservative results. For detailed meshes undergoing low-rank motion, we have observed speedups of over an order of magnitude with our method.

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Errors in reduction methods

Cited by 19 publications

References 6 publications

Error Estimation for Reduced‐Order Models of Dynamical Systems

Error Estimation for Reduced‐Order Models of Dynamical Systems

Nonlinear Model Reduction via Discrete Empirical Interpolation

Skipping steps in deformable simulation with online model reduction

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