A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

We propose a general—i.e., independent of the underlying equation—registration method for parameterized model order reduction. Given the spatial domain $Ømega \subset \mathbbR^d$ and the manifold $\mathcalM_u= { u_\mu : \mu ın \mathcalP }$ associated with the parameter domain $\mathcalP \subset \mathbbR^P$ and the parametric field $\mu \mapsto u_\mu ın L^2(Ømega)$, the algorithm takes as input a set of snapshots ${ u^k }_k=1^n_\rm train \subset \mathcalM_u$ and returns a parameter-dependent bijective mapping $\Phi: Ømega \times \mathcalP \to \mathbbR^d$: the mapping is designed to make the mapped manifold ${ u_\mu \circ \Phi_\mu: \, \mu ın \mathcalP }$ more suited for linear compression methods. We apply the registration procedure, in combination with a linear compression method, to devise low-dimensional representations of solution manifolds with slowly decaying Kolmogorov $N$-widths; we also consider the application to problems in parameterized geometries. We present a theoretical result to show the mathematical rigor of the registration procedure. We further present numerical results for several two-dimensional problems, to empirically demonstrate the effectivity of our proposal.