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 a set of snapshots ${ u^k }_{k=1}^{n_{rm train}}$ over $Omega$ associated with $n_{rm train}$ values of 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 is designed to make the mapped manifold ${ u_{mu} circ boldsymbol{Phi}_{mu}: , mu in mathcal{P} }$ 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.
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