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

Abstract

We propose an efficient residual minimization technique for the nonlinear model-order reduction of parameterized hyperbolic partial differential equations. Our nonlinear approximation space is spanned by snapshots functions over spatial transformations, and we compute our reduced approximation via residual minimization. To speedup the residual minimization, we compute and minimize the residual on a (preferably small) subset of the mesh, the so-called reduced mesh. We show that, similar to the solution, the residual also exhibits transport-type behaviour. To account for this behaviour, we introduce adaptivity in the reduced mesh by “moving” it along the spatial domain depending on the parameter value. Numerical experiments showcase the effectiveness of our method and the inaccuracies resulting from a non-adaptive reduced mesh.