A nonlinear reduced-order model for parametrized variational inequalities: application to crowd motion

Abstract

In this work we adapt recent nonlinear model reduction approaches to predict the solutions of time-dependent parametrized variational inequalities. In our present context, we make a specific focus on discrete contact problems. A prototypical example is the agentbased model proposed in Maury, Venel, M2NA 2011 to describe crowd motion in the presence of obstacles. In this model, in a discrete time setting, the set of velocities of each agent in the crowd is the solution at each time step to a constrained least-squares optimization statement. The parametric variations of the problem (associated with the geometric configuration of the domain where the agents evolve) have a very strong impact on the variability of the solution, both in terms of positions of the agents and of contact forces between them, the latter being given by the Lagrange multipliers associated to non-interpenetration constraints. More precisely, the Kolmogorov n-width of the solution set is very slowly decaying. Motivated by this observation, we investigate new developments and combinations of the reduced-basis method and supervised machine-learning techniques to provide more accurate approximations of the primal and dual solutions, inspired from the recent works (Cohen et al., Comptes Rendus. Mécanique 2023). The proposed nonlinear compressive strategy then reads as a postprocessing step of the solution of a standard reduced basis reduced-order model, which yields an improvement of the accuracy of the approximation by an order of magnitude for a negligible extra computational cost. We see this work as a preliminary step before the investigation of more efficient nonlinear reduced order modeling approaches for this type of problems.

Key words. Multi-particles systems; Discrete contact problems; Reduced Basis method; Supervised Machine Lerning.