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

Abstract

We propose a general --- i.e., independent of the underlying equation ---\nregistration method for parameterized Model Order Reduction. Given the spatial\ndomain $\\Omega \\subset \\mathbb{R}^d$ and a set of snapshots $\\{ u^k\n\\}_{k=1}^{n_{\\rm train}}$ over $\\Omega$ associated with $n_{\\rm train}$ values\nof the model parameters $\\mu^1,\\ldots, \\mu^{n_{\\rm train}} \\in \\mathcal{P}$,\nthe algorithm returns a parameter-dependent bijective mapping\n$\\boldsymbol{\\Phi}: \\Omega \\times \\mathcal{P} \\to \\mathbb{R}^d$: the mapping is\ndesigned to make the mapped manifold $\\{ u_{\\mu} \\circ \\boldsymbol{\\Phi}_{\\mu}:\n\\, \\mu \\in \\mathcal{P} \\}$ more suited for linear compression methods. We apply\nthe registration procedure, in combination with a linear compression method, to\ndevise low-dimensional representations of solution manifolds with\nslowly-decaying Kolmogorov $N$-widths; we also consider the application to\nproblems in parameterized geometries. We present a theoretical result to show\nthe mathematical rigor of the registration procedure. We further present\nnumerical results for several two-dimensional problems, to empirically\ndemonstrate the effectivity of our proposal.\n