A registration method for model order reduction: data compression and geometry reduction
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We propose a general --- i.e., independent of the underlying equation --- registration method for parameterized Model Order Reduction. Given the spatial domain Ω⊂R^d and a set of snapshots { u^k }_k=1^n_ train over Ω associated with n_ train values of the model parameters μ^1,..., μ^n_ train∈P, the algorithm returns a parameter-dependent bijective mapping Φ: Ω×P→R^d: the mapping is designed to make the mapped manifold { u_μ∘Φ_μ: μ∈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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