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Conference Papers Year : 2023

Wasserstein model reduction approach for parametrized flow problems in porous media

Abstract

The aim of this work is to build a reduced order model for parametrized porous media equations. The main challenge of this type of problems is that the Kolmogorov width of the solution manifold typically decays quite slowly and thus makes usual linear model order reduction methods inappropriate. In this work, we investigate an adaptation of the methodology proposed in [Ehrlacher et al., Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces , ESAIM: Mathematical Modelling and Numerical Analysis (2020)], based on the use of Wasserstein barycenters [Agueh & Carlier, Barycenters in the Wasserstein Space , SIAM Journal on Mathematical Analysis (2011)], to the case of non-conservative problems. Numerical examples in one-dimensional test cases illustrate the advantages and limitations of this approach and suggest further research directions that we intend to explore in the future.
Le but de ce travail est de construire un modèle réduit pour des problèmes d’écoulements en milieux poreux paramétrés. La difficulté principale de ce type de problèmes est que la distance de Kolmogorov de l’ensemble de solutions décroît lentement, rendant ainsi les méthodes de réduction de modèles linéaires usuelles inefficaces. Ici, nous proposons une adaptation de la méthodologie proposée dans [Ehrlacher et al., Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces , ESAIM: Mathematical Modelling and Numerical Analysis (2020)], utilisant des barycentres de Wasserstein [Agueh & Carlier, Barycenters in the Wasserstein Space , SIAM Journal on Mathematical Analysis (2011)], au cas de problèmes non conservatifs. Des tests numériques en dimension 1 permettent d’illustrer les avantages et les limitations de cette approche et d’identifier des pistes de recherche que nous souhaiterons aborder dans un futur travail.

Dates and versions

hal-04390131 , version 1 (12-01-2024)

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Beatrice Battisti, Tobias Blickhan, Guillaume Enchery, Virginie Ehrlacher, Damiano Lombardi, et al.. Wasserstein model reduction approach for parametrized flow problems in porous media. CEMRACS 2021 - Data Assimilation and Reduced Modeling for High Dimensional Problems, Jul 2021, Luminy, France. pp.28-47, ⟨10.1051/proc/202373028⟩. ⟨hal-04390131⟩
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