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 $\Omega \subset \mathbb{R}^d$ and the manifold $\mathcal{M}_{u}= \{ u_{\mu} : \mu \in \mathcal{P} \}$ associated with the parameter domain $\mathcal{P} \subset \mathbb{R}^P$ and the parametric field $\mu \mapsto u_{\mu} \in L^2(\Omega)$, the algorithm takes as input a set of snapshots $\{ u^k \}_{k=1}^{n_{\rm train}} \subset \mathcal{M}_{u}$ and returns a parameter-dependent bijective mapping ${\Phi}: \Omega \times \mathcal{P} \to \mathbb{R}^d$: the mapping is designed to make the mapped manifold $\{ u_{\mu} \circ {\Phi}_{\mu}: \, \mu \in \mathcal{P} \}$ 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.