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We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been based on the approximation of the solution set by linear spaces on Hilbert or Banach spaces. This approach can be expected to be successful only when the Kolmogorov width of the set decays fast. While this is the case on certain parabolic or elliptic problems, most transport-dominated problems are expected to present a slow decaying width and require to study nonlinear approximation methods. In this work, we propose to address the reduction problem from the perspective of general metric spaces with a suitably defined notion of distance. We develop and compare two different approaches, one based on barycenters and another one using tangent spaces when the metric space has an additional Riemannian structure. As a consequence of working in metric spaces, both approaches are automatically nonlinear. We give theoretical and numerical evidence of their efficiency to reduce complexity for one-dimensional conservative PDEs where the underlying metric space can be chosen to be the $L^2$-Wasserstein space.

Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces