Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces

Falcó, Antonio; Nouy, Anthony

doi:10.1007/s00211-011-0437-5

Cited by 70 publications

(78 citation statements)

References 41 publications

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“…It can be noticed that the size of the reduced basis has a considerable influence on both, the local and the global accuracy, however, the influence of τ is negligible. Note that the convergence of the PGD solution in the ∞-norm is not ensured [31], but the experimental results prove that it converges in this particular case. Figure 8 shows the evolution of the error in the parameter identification with the size of the reduced basis.…”

Section: Sensors Placementmentioning

confidence: 86%

Real time parameter identification and solution reconstruction from experimental data using the Proper Generalized Decomposition

Nadal

Chinesta

Dı́ez

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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Some industrial processes are modelled by parametric partial differential equations. Integrating computational modelling and data assimilation into the control process requires obtaining a solution of the numerical model at the characteristic frequency of the process (realtime). This paper introduces a computational strategy allowing to efficiently exploit measurements of those industrial processes, providing the solution of the model at the required frequency. This is particularly interesting in the framework of control algorithms that rely on a model involving a set of parameters. For instance, the curing process of a composite material is modelled as a thermo-mechanical problem whose corresponding parameters describe the thermal and mechanical 1 behaviours. In this context, the information available (measurements) is used to update the parameters of the model and to produce new values of the control variables (data assimilation). The methodology presented here is devised to ensure the possibility of having a response in real-time of the problem and therefore the capability of integrating it in the control scheme. The Proper Generalized Decomposition is used to describe the solution in the multi-parametric space. The realtime data assimilation requires a further simplification of the solution representation that better fits the data (reconstructed solution) and it provides an implicit parameter identification. Moreover, the analysis of the assimilated data sensibility with respect to the points where the measurements are taken suggests a criterion to locate of the sensors.

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Section: Sensors Placementmentioning

confidence: 86%

Real time parameter identification and solution reconstruction from experimental data using the Proper Generalized Decomposition

Nadal

Chinesta

Dı́ez

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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“…However, it is observed that the decrease in e is not monotonic, with an increase in the 5 th iteration. Although the actual reason of the increase in e has not been investigated here, it can be attributed to the global nature of the PGD based algorithm [22]. Investigating the details about the convergence of PGD based algorithm is a topic of our future research.…”

Section: B Rate Of Convergencementioning

confidence: 98%

“…From Propositions 1 and 2, it is evident that for the dynamical system in (20), the corresponding LQR objective function is given by (22). It is well known that the gain K is can be written as,…”

Section: Linear Quadratic Regulator Using Proper Generalized Decomentioning

confidence: 99%

Reduced order modeling for systems with parametric uncertainty using proper generalized decomposition

Dutta

2015

2015 American Control Conference (ACC)

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Abstract-In this work we have proposed a new technique of model order reduction for linear time invariant (LTI) systems with parametric uncertainty. The model order reduction method is based on proper generalized decomposition (PGD). Using PGD, the underlying state variable is expanded as a sum of separated functions of time and uncertain parameters. At first, the stochastic states of the LTI system is represented using PGD. Then equations to obtain the PGD basis functions are derived. Furthermore a state feedback structure for the control input is assumed where the gain is found by solving a minimum expectation linear quadratic regulator (LQR) problem. An algorithm is then proposed, from which the PGD basis functions and the control input gain are found. The proposed algorithm is then applied to control the angle of attack and pitch rate of a F-16 aircraft having uncertain parameters. It is found that the proposed technique based on PGD could successfully achieve the control objective for the current application.

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“…Concerning the variational approach we refer to the following papers: Espig-Hackbusch-Rohwedder-Schneider [7], Falcó-Nouy [8], Holtz-Rohwedder-Schneider [21], Mohlenkamp [26], Osedelets [27] and others cited in these papers.…”

Section: Variational Approachmentioning

confidence: 99%

Solution of linear systems in high spatial dimensions

Hackbusch

2015

Comput. Visual Sci.

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Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces

Cited by 70 publications

References 41 publications

Real time parameter identification and solution reconstruction from experimental data using the Proper Generalized Decomposition

Real time parameter identification and solution reconstruction from experimental data using the Proper Generalized Decomposition

Reduced order modeling for systems with parametric uncertainty using proper generalized decomposition

Solution of linear systems in high spatial dimensions

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