Parametric vector flows for registration fields in bounded domains with applications to nonlinear interpolation of shock-dominated flows

We present a registration procedure for parametric model order reduction (MOR) in two- and three-dimensional bounded domains. In the MOR framework, registration methods exploit solution snapshots to identify a parametric coordinate transformation that improves the approximation of the solution set through linear subspaces. For each training parameter, optimization-based (or variational) registration methods minimize a target function that measures the alignment of the coherent structures of interest (e.g., shocks, shear layers, cracks) for different parameter values, over a family of bijections of the computational domain $\Omega$. We consider diffeomorphisms $\Phi$ that are vector flows of given velocity fields $v$ with vanishing normal component on $\partial \Omega$; we rely on a sensor to extract appropriate point clouds from the solution snapshots and we develop an expectation-maximization procedure to simultaneously solve the point cloud matching problem and to determine the velocity $v$ (and thus the bijection $\Phi$); finally, we combine our registration method with the nonlinear interpolation technique of [Iollo, Taddei, J. Comput. Phys., 2022] to perform accurate interpolations of fluid dynamic fields in the presence of shocks. Numerical results for a two-dimensional inviscid transonic flow past a NACA airfoil and a three-dimensional viscous transonic flow past an ONERA M6 wing illustrate the many elements of the methodology and demonstrate the effectiveness of nonlinear interpolation for shock-dominated fields.