## Section: New Results

### Automatic registration for model reduction

As part of the ongoing team effort on ROMs, we work on the development of automatic registration procedures for model reduction. In computer vision and pattern recognition, registration refers to the process of finding a spatial transformation that aligns two datasets; in our work, registration refers to the process of finding a parametric transformation that improves the linear compressibility of a given parametric manifold. For advection-dominated problems, registration is motivated by the inadequacy of linear approximation spaces due to the presence of parameter-dependent boundary layers and travelling waves.

In [48], we proposed and analysed a computational procedure for stationary PDEs and investigated performance for two-dimensional model problems. In Figure 15, we show slices of the parametric solution for three different parameters before (cf. Left) and after (cf. Right) registration: we observe that the registration procedure is able to dramatically reduce the sensitivity of the solution to the parameter value $\mu $. In Figure 16, we show the behaviour of the normalised POD eigenvalues (cf. Left) and of the relative ${L}^{2}$ error of the corresponding POD-Galerkin ROM (cf. Right) for an advection-reaction problem: also in this case, the approach is able to improve the approximation properties of linear approximation spaces and ultimately simplify the reduction task.

We aim to extend the approach to a broad class of non-linear steady and unsteady PDEs: in October 2019, we funded a 16-month postdoc to work on the reduction of hyperbolic systems of PDEs. We are also collaborating with EDF (departments PERICLES and LNHE) to extend the approach to the Saint-Venant (shallow water) equations.