Nonlinear reduced-order modeling approaches for discrete contact models

Virginie Ehrlacher (CERMIC, ENPC)
(joint work with Loïc Gouarin, Aline Lefebvre-Lepot and Giulia Sambattaro)

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The aim of this talk is to present some preliminary results about new nonlinear reduced 
order models for parametrized discrete contact models, inspired from the series of works 
[1,2]. Discrete contact models, such as the one introduced in [3], are used in various 
types of applications ranging from crowd motion to granular media. The output of interest 
of these models read as the solutions of time-dependent variational inequalities. 
The Kolmogorov width of the set of solutions of parameter-dependent problems decay very 
slowly in general. Motivated by this observation, we introduce new nonlinear reduced basis 
methods to accelerate the resolution of such problems, augmented with random forest 
regression, in a similar spirit to [2] and discuss possible extensions and improvements.  


[1] Joshua Barnett, Charbel Farhat and Yvon Maday. 
"Neural-network-augmented projection-based model order reduction for mitigating 
the Kolmogorov barrier to reducibility." Journal of Computational Physics 492 (2023): 112420.

[2] Albert Cohen, Charbel Farhat, Yvon Maday et al. 
"Nonlinear compressive reduced basis approximation for PDE’s." 
Comptes Rendus Mécanique, 2023, vol. 351, no S1, p. 357-374.

[3] Bertrand Maury and Juliette Venel. 
"A discrete contact model for crowd motion."
ESAIM: Mathematical Modelling and Numerical Analysis, 2011, vol. 45, no 1, p. 145-168.

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