Measure-based Approaches for Model Order Reduction

The objective of this thesis is to solve non-linear hyperbolic conservation laws by measure-based approaches. The first contribution is a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre's hierarchy. Finally, several post-treatments can be performed, including reconstructing the graph of the solution. The second contribution is a model order reduction method to solve scalar hyperbolic PDEs with a moment approach, inspired of a previous work from Ehrlacher et al. (2020). In this method, we see the snapshots as measures linked to the solutions, and define the approximation space as the set of Wasserstein barycenters of these snapshots. The offline and online phases are then based on a projection onto the approximation space, which is obtained through an optimization phase which consists in minimizing deviation from some moment information on the solution.