A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

Abstract

We propose a general -i.e., independent of the underlying equationregistration method for parameterized Model Order Reduction. Given the spatial domain Ω ⊂ R d and a set of snapshots {u k } n train k=1 over Ω associated with ntrain 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 {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 slowlydecaying 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.

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

“…-Lagrangian approaches [24,33,46,47] rely on linear compression methods to approximate the mapped solutioñ︀ := ∘Φ , where Φ : Ω× → Ω is a suitably-chosen bijection from Ω into itself: the mapping Φ should be chosen to make the mapped solution manifold̃︁ ℳ = {̃︀ : ∈ } more amenable for linear approximations.…”

“…In the framework of linear compression methods, we refer to [11,17,20] for representative examples of non-intrusive techniques, and to the reduced basis literature (e.g., [21,38]) for a thorough discussion about hyper-reduced projection schemes. Non-intrusive techniques can be trivially extended to nonlinear compression methods; on the other hand, the extension of projection-based schemes is more challenging: for Lagrangian approaches, following [33,46], we might perform projection and hyper-reduction in the mapped configuration; for Eulerian approaches, specialized techniques need to be proposed to ensure rapid ROM evaluations (see [26,41]).…”

“…Given the space-time snapshots { = } train =1 ⊂ ℳ, our data compression procedure returns (i) the linear operators : R → and : R → [Lip(Ω)] 2 in (1.1), and (ii) the coefficients { } train =1 ⊂ R and {a } train =1 ⊂ R such that ≈̂︀ ∘ (Φ ) −1 wherê︀ = and Φ = id + a . We develop an adaptive registration algorithm -which is an extension of the approach in [46] -to construct the mappings {Φ } ; on the other hand, we resort to proper orthogonal decomposition (POD, [4,49]) to generate the low-dimensional linear approximation operators , . Since the procedure can be viewed as a generalization of POD, we here refer to our approach as to RePOD (Registered POD): as rigorously showed below, our approach is general, that is it does not depend on the underlying mathematical model.…”

Breaking the Kolmogorov Barrier in Model Reduction of Fluid Flows

“…The Kolmogorov n-width is a concept from approximation theory that determines the linear reducibility of a system [65,66]. Mathematically, it is defined as [66][67][68] where S n is a linear n-dimensional subspace, M is the solution manifold, and Π S n is the orthogonal projector onto S n . In other words, d n (M) quantifies the maximum possible error that might arise from the projection of solution manifold onto the best-possible n-dimensional linear subspace.…”

“…The Kolmogorov n-width is a concept from approximation theory that determines the linear reducibility of a system [65,66]. Mathematically, it is defined as [66][67][68]…”