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 Ω ⊂ Rᵈ and the manifold Mᵤ= \ uμ : μ ∈ P \ associated with the parameter domain P ⊂ RP and the parametric field μ ↦ uμ ∈ L²(Ω), the algorithm takes as input a set of snapshots \ uᵏ \ₖ₌₁^nₜᵣₐᵢₙ ⊂ Mᵤ and returns a parameter-dependent bijective mapping Φ: Ω × P → Rᵈ: the mapping is designed to make the mapped manifold \ uμ ∘ Φμ: \, 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.