# 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.

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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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2016 10 Aug 2016 07:27 10 Aug 2016 07:27 SpringerOpen 2213-7467 https://doi.org/10.1186/s40323-015-0055-3
Permanent URL: https://doi.org/10.5167/uzh-124442

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