Hyper-reduction for parametrized transport dominated problems via online-adaptive reduced meshes

We propose an efficient residual minimization technique for the nonlinear model-order reduction of parameterized hyperbolic partial differential equations. Our nonlinear approximation space is a span of snapshots evaluated on a shifted spatial domain, and we compute our reduced approximation via residual minimization. To speed-up the residual minimization, we compute and minimize the residual on a (preferably small) subset of the mesh, the so-called reduced mesh. Due to the nonlinearity of our approximation space we show that, similar to the solution, the residual also exhibits transport-type behaviour. To account for this behaviour, we introduce online-adaptivity in the reduced mesh by "moving" it along the spatial domain with parameter dependent shifts. We also present an extension of our method to spatial transforms different from shifting. Numerical experiments showcase the effectiveness of our method and the inaccuracies resulting from a non-adaptive reduced mesh.