A separated representation of an error indicator for the mesh refinement process under the proper generalized decomposition framework

Nadal, E.; Leygue, Adrien; Chinesta, Francisco; Béringhier, Marianne; Ródenas, Juan José; Fuenmayor, F. J.

doi:10.1007/s00466-014-1097-y

Cited by 13 publications

(13 citation statements)

References 33 publications

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“…As mentioned above, the speed of convergence to the reference FEM solution is noticeably slower than the case without dropper slackening. The 27 computed modes of the separated solution can be compressed into a few modes using (28) with almost the same accuracy, as shown in Fig. 7.…”

Section: Example 1: Academic Examplementioning

confidence: 99%

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Parametric model for the simulation of the railway catenary system static equilibrium problem

Gregori

Tur

Nadal

et al. 2016

Finite Elements in Analysis and Design

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a b s t r a c tDynamic simulations of pantograph-catenary interaction are nowadays essential for improving the performance of railway locomotives, by achieving better current collection at higher speeds and lower wear of the collecting parts. The first step in performing these simulations is to compute the static equilibrium of the overhead line. The initial dropper lengths play an important role in hanging the contact wire at an appropriate height. From a classical point of view, if one wants to obtain the static equilibrium configuration of the system for different combinations of dropper lengths, one static problem must be solved for each combination of lengths, which involves a prohibitive computational cost. In this paper we propose a parametric model of the catenary, including the undeformed dropper lengths as extra-coordinates of the problem. This multidimensional problem is efficiently solved by means of the Proper Generalized Decomposition (PGD) technique, avoiding the curse of dimensionality issue. The capabilities and performance of the proposed method are shown by numerical examples.

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Section: Example 1: Academic Examplementioning

confidence: 99%

“…Therefore, a post-compression should be envisaged in order to express the solution in a more compact form (see [28,16]). If rn is the PGD solution of the original problem withn modes, the postcompression is carried out by solving the following problem:…”

Section: Pgd Formulationmentioning

confidence: 99%

Parametric model for the simulation of the railway catenary system static equilibrium problem

Gregori

Tur

Nadal

et al. 2016

Finite Elements in Analysis and Design

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“…Different sources of errors are present in the solution provided by PGD (see for example [6][7][8]. If u is the analytical solution of the BVP and u H,M is the solution of PGD characterized by a mesh size H and a number of terms M, the PGD error is then defined by e := u − u H,M .…”

Section: A Priori Estimates For Fementioning

confidence: 99%

“…As usual, u sep is inserted in the weak form (8). In this case, the diffusivity function k(x, y) is also replaced by its separable approximation k sep , see Eq.…”

Section: Space-separated Pgd Algorithmmentioning

confidence: 99%

Effect of the separated approximation of input data in the accuracy of the resulting PGD solution

Zlotnik

Dı́ez

González

et al. 2015

Adv. Model. and Simul. in Eng. Sci.

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The proper generalized decomposition (PGD) requires separability of the input data (e.g. physical properties, source term, boundary conditions, initial state). In many cases the input data is not expressed in a separated form and it has to be replaced by some separable approximation. These approximations constitute a new error source that, in some cases, may dominate the standard ones (discretization, truncation…) and control the final accuracy of the PGD solution. In this work the relation between errors in the separated input data and the errors induced in the PGD solution is discussed. Error estimators proposed for homogenized problems and oscillation terms are adapted to asses the behaviour of the PGD errors resulting from approximated input data. The PGD is stable with respect to error in the separated data, with no critical amplification of the perturbations. Interestingly, we identified a high sensitiveness of the resulting accuracy on the selection of the sampling grid used to compute the separated data. The separation has to be performed on the basis of values sampled at integration points: sampling at the nodes defining the functional interpolation results in an important loss of accuracy. For the case of a Poisson problem separated in the spatial coordinates (a complex diffusivity function requires a separable approximation), the final PGD error is linear with the truncation error of the separated data. This relation is used to estimate the number of terms required in the separated data, that has to be in good agreement with the truncation error accepted in the PGD truncation (tolerance for the stoping criteria in the enrichment procedure). A sensible choice for the prescribed accuracy of the PGD solution has to be kept within the limits set by the errors in the separated input data.

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“…In contrast to the error estimator we introduce in this paper, the estimators presented in [34,35,38,12] also take into account the discretization error; for the a posteriori error estimator suggested in this paper this is the subject of a forthcoming work. Moreover,in [34,35,38,12] the authors consider (parametrized) coercive elliptic and parabolic problems; it seems that the techniques just described have not been used yet to address problems that are merely inf-sup stable.Error estimators based on local error indicators are proposed to drive mesh adaptation within the construction of the PGD approximation in [39]. In [7], an ideal minimal residual formulation is proposed in order to build optimal PGD approximations.…”

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confidence: 99%

Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

Smetana

Zahm

2020

Numerical Meth Engineering

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This paper introduces a novel error estimator for the Proper Generalized Decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: It builds on concentration inequalities of Gaussian maps and an adjoint problem with random right-hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user-defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time-harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high-dimensional parameter space.in general only as costly as the computation of the PGD approximation or even significantly less expensive, which makes our error estimator strategy attractive from a computational viewpoint. To build this estimator, we first estimate the norm of the error with a Monte Carlo estimator using Gaussian random vectors whose covariance is chosen according to the desired error measure, e.g., user-defined norms or quantity of interest. Then, we introduce a dual problem with random right-hand side the solution of which allows us to rewrite the error estimator in terms of the residual of the original equation. Approximating the random dual problems with the PGD yields a fast-to-evaluate error estimator that has low marginal cost.Next, we relate our error estimator to other stopping criteria that have been proposed for the PGD solution. In [2] the authors suggest letting an error estimator in some linear QoI steer the enrichment of the PGD approximation. In order to estimate the error they employ an adjoint problem with the linear QoI as a right-hand side and approximate this adjoint problem with the PGD as well. It is important to note that, as observed in [2], this type of error estimator seems to require in general a more accurate solution of the dual than the primal problem. In contrast, the framework we present in this paper often allows using a dual PGD approximation with significantly less terms than the primal approximation. An adaptive strategy for the PGD approximation of the dual problem for the error estimator proposed in [2] was suggested in [23] in the context of computational homogenization. In [1] the approach proposed in [2] is extended to problems in nonlinear solids by considering a suitable linearization of the problem.An error estimator for the error between the exact solution of a parabolic PDE and the PGD approximation in a suitable norm is derived in [34] based on the constitutive relation error and the Prager-Synge theorem. The discretization error is thus also assessed, and the computational costs of the error estimator depend on the number of elements in the underlying spatial mesh. Relying on the same techniques an error estimator for the error between the exact solution of a parabolic PDE and a...

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A separated representation of an error indicator for the mesh refinement process under the proper generalized decomposition framework

Cited by 13 publications

References 33 publications

Parametric model for the simulation of the railway catenary system static equilibrium problem

Parametric model for the simulation of the railway catenary system static equilibrium problem

Effect of the separated approximation of input data in the accuracy of the resulting PGD solution

Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

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