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We propose a general --- i.e., independent of the underlying equation ---registration method for parameterized Model Order Reduction. Given the spatialdomain $\Omega \subset \mathbb{R}^d$ and a set of snapshots $\{ u^k\}_{k=1}^{n_{\rm train}}$ over $\Omega$ associated with $n_{\rm train}$ valuesof the model parameters $\mu^1,\ldots, \mu^{n_{\rm train}} \in \mathcal{P}$,the algorithm returns a parameter-dependent bijective mapping$\boldsymbol{\Phi}: \Omega \times \mathcal{P} \to \mathbb{R}^d$: the mapping isdesigned to make the mapped manifold $\{ u_{\mu} \circ \boldsymbol{\Phi}_{\mu}:\, \mu \in \mathcal{P} \}$ more suited for linear compression methods. We applythe registration procedure, in combination with a linear compression method, todevise low-dimensional representations of solution manifolds withslowly-decaying Kolmogorov $N$-widths; we also consider the application toproblems in parameterized geometries. We present a theoretical result to showthe mathematical rigor of the registration procedure. We further presentnumerical results for several two-dimensional problems, to empiricallydemonstrate the effectivity of our proposal.

A registration method for model order reduction: data compression and geometry reduction