Enhanced goal‐oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems

Serafin, Kevin; Magnain, Benoît; Florentin, Éric; Mariné, Núria Parés; Dı́ez, Pedro

doi:10.1002/nme.5363

Cited by 8 publications

(13 citation statements)

References 31 publications

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“…In order to be computationally efficient, the final dimension of the reduced basis must remain small with respect to the dimension of the initial problem. It was evidenced in previous work [25] that the proposed methodology may yield a very significant decrease of computation times. Table 3 sums up the dimension of the reduced basis at the end of the simulations carried out for each set of parameters a, α and ε 0 .…”

Section: Reduced Basis Evolutionmentioning

confidence: 82%

“…Equation. (25) is then left multiplied byK(θ k ) in order to yield a symmetric matrix so that the system may be solved :…”

Section: New Time Formulation Of the Discretized Problemmentioning

confidence: 99%

“…The impact of the approximation presented in Eq. (39) has previously been assessed [25] and compared with a finer, but more costly, approximation. It was found that for small variations of the variability, the obtained results are identical.…”

Section: Adaptive Strategy Using Error Assessmentmentioning

confidence: 99%

“…In addition, the sequence of draws is fully arbitrary. In previous work [25], several sequences are assessed and provide similar results. In its current implementation, the proposed methodology retains all admissible solutions within the reduced basis without any mathematical modification or optimization.…”

Section: Proposed Algorithmmentioning

confidence: 99%

“…It builds up on the idea of using a reduced basis combined with a goal oriented error assessment criterion which controls the content of the reduced basis in order to maximize its numerical efficacy. The present investigation is an extension of the work initiated in [24,25] to transient thermal processes [26,27]. As a matter of fact, the time dependence of the investigated problem makes it more arduous to solve.…”

Section: Introductionmentioning

confidence: 96%

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Adaptive reduced basis strategy dedicated to the solution of nonstationary stochastic thermal problems

Magnain

Batailly

Florentin

2017

Computers & Structures

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Adaptive reduced basis strategy dedicated to the solution of nonstationary stochastic thermal problems. Computers and Structures, Elsevier, 2017, 182, pp.Abstract This contribution addresses the modelling and the stochastic analysis of transient thermal processes by means of the finite element method. It focuses on the theoretical presentation as well as the application of an efficient reduced basis strategy that advantageously lowers the dimension of the investigated system. The modal content of the reduced basis is driven by the goal oriented error assessment of a user-defined quantity of interest. The first section of the article presents the stochastic system of interest: key aspects of a stochastic analysis are recalled along with the employed spatial discretization. The newly developed adaptive reduced basis strategy is then detailed in the second section before extensive numerical investigations are carried out in order to validate it in the last section of the article. A numerical benchmark allowing for the confrontation of the proposed strategy with usual Monte-Carlo simulations highlights the benefits of the method that allows for a precise control of the maximum admissible error on the quantity of interest. RésuméCet article porte sur la modélisation et l'analyse stochastique de phénomènes thermiques transitoires par la méthode deséléments finis. Les développements effectués portent sur la présentation théorique et l'application d'une méthodologie efficace reposant sur l'utilisation de bases réduites et permettant ainsi une diminution de la dimension du systèmeétudié. Le contenu modal de la base réduite est reliéà l'évaluation d'erreur avec objectif d'une quantité d'intérêt choisie par l'utilisateur. La première section de l'article porte sur le système stochastiqueétudié, et permet un rappel des notions fondamentales associéesà une analyse stochastique ainsi que des bases de la procédure de discrétisation spatiale. Par la suite, la deuxième section présente en détail la méthodologie développée. Enfin, dans une dernière section, plusieurs calculs permettent de valider cette méthodologie. Un point de comparaison numérique entre la méthodologie proposée et des simulations de Monte-Carlo mettent eń evidence les avantages de la méthode proposée notamment au niveau de la précision de l'erreur maximale admissible sur la quantité d'intérêt.

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Section: Reduced Basis Evolutionmentioning

confidence: 82%

“…Equation. (25) is then left multiplied byK(θ k ) in order to yield a symmetric matrix so that the system may be solved :…”

Section: New Time Formulation Of the Discretized Problemmentioning

confidence: 99%

Section: Adaptive Strategy Using Error Assessmentmentioning

confidence: 99%

Section: Proposed Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

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Adaptive reduced basis strategy dedicated to the solution of nonstationary stochastic thermal problems

Magnain

Batailly

Florentin

2017

Computers & Structures

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Nonlinear dimensionality reduction for parametric problems: A kernel proper orthogonal decomposition

Dı́ez

Muixí

Zlotnik

et al. 2021

Numerical Meth Engineering

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Reduced-order models are essential tools to deal with parametric problems in the context of optimization, uncertainty quantification, or control and inverse problems. The set of parametric solutions lies in a low-dimensional manifold (with dimension equal to the number of independent parameters) embedded in a large-dimensional space (dimension equal to the number of degrees of freedom of the full-order discrete model). A posteriori model reduction is based on constructing a basis from a family of snapshots (solutions of the full-order model computed offline), and then use this new basis to solve the subsequent instances online. Proper orthogonal decomposition (POD) reduces the problem into a linear subspace of lower dimension, eliminating redundancies in the family of snapshots. The strategy proposed here is to use a nonlinear dimensionality reduction technique, namely, the kernel principal component analysis (kPCA), in order to find a nonlinear manifold, with an expected much lower dimension, and to solve the problem in this low-dimensional manifold. Guided by this paradigm, the methodology devised here introduces different novel ideas, namely, 1) characterizing the nonlinear manifold using local tangent spaces, where the reduced-order problem is linear and based on the neighboring snapshots, 2) the approximation space is enriched with the cross-products of the snapshots, introducing a quadratic description, 3) the kernel for kPCA is defined ad hoc, based on physical considerations, and 4) the iterations in the reduced-dimensional space are performed using an algorithm based on a Delaunay tessellation of the cloud of snapshots in the reduced space. The resulting computational strategy is performing outstandingly in the numerical tests, alleviating many of the problems associated with POD and improving the numerical accuracy.

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Projection methods for stochastic dynamic systems: A frequency domain approach

Pryse

Kundu

Adhikari

2018

Computer Methods in Applied Mechanics and Engineering

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A set of novel hybrid projection approaches are proposed for approximating the response of stochastic partial differential equations which describe structural dynamic systems. An optimal basis for the response of a stochastic system has been computed from the eigen modes of the parametrized structural dynamic system. The hybrid projection methods are obtained by applying appropriate approximations and by reducing the modal basis. These methods have been further improved by an implementation of a sample based Galerkin error minimization approach. In total four methods are presented and compared for numerical accuracy and efficiency by analysing the bending of a Euler-Bernoulli cantilever beam.

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Enhanced goal‐oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems

Cited by 8 publications

References 31 publications

Adaptive reduced basis strategy dedicated to the solution of nonstationary stochastic thermal problems

Adaptive reduced basis strategy dedicated to the solution of nonstationary stochastic thermal problems

Nonlinear dimensionality reduction for parametric problems: A kernel proper orthogonal decomposition

Projection methods for stochastic dynamic systems: A frequency domain approach

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