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Robust model reduction by $\mathit{L^{1}}$-norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems


Abgrall, Rémi; Amsallem, David; Crisovan, Roxana (2016). Robust model reduction by $\mathit{L^{1}}$-norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems. Advanced Modeling and Simulation in Engineering Sciences, 3(1):online.

Abstract

We propose a novel model reduction approach for the approximation of non linear hyperbolic equations in the scalar and the system cases. The approach relies on an offline computation of a dictionary of solutions together with an online $\mathit{L^{1}}$- norm minimization of the residual. It is shown why this is a natural framework for hyperbolic problems and tested on nonlinear problems such as Burgers’ equation and the one-dimensional Euler equations involving shocks and discontinuities. Efficient algorithms are presented for the computation of the L1-norm minimizer, both in the cases of linear and nonlinear residuals. Results indicate that the method has the potential of being accurate when involving only very few modes, generating physically acceptable, oscillation-free, solutions.

Abstract

We propose a novel model reduction approach for the approximation of non linear hyperbolic equations in the scalar and the system cases. The approach relies on an offline computation of a dictionary of solutions together with an online $\mathit{L^{1}}$- norm minimization of the residual. It is shown why this is a natural framework for hyperbolic problems and tested on nonlinear problems such as Burgers’ equation and the one-dimensional Euler equations involving shocks and discontinuities. Efficient algorithms are presented for the computation of the L1-norm minimizer, both in the cases of linear and nonlinear residuals. Results indicate that the method has the potential of being accurate when involving only very few modes, generating physically acceptable, oscillation-free, solutions.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2016
Deposited On:10 Aug 2016 07:27
Last Modified:12 Jan 2017 11:07
Publisher:SpringerOpen
ISSN:2213-7467
Publisher DOI:https://doi.org/10.1186/s40323-015-0055-3

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