Compositional maps for registration in complex geometries
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We develop and analyze a parametric registration procedure for manifolds associated with the solutions to parametric partial differential equations in two-dimensional domains. Given the domain Ω⊂ℝ^2 and the manifold M={ u_μ : μ∈ P} associated with the parameter domain P ⊂ℝ^P and the parametric field μ↦ u_μ∈ L^2(Ω), our approach takes as input a set of snapshots from M and returns a parameter-dependent mapping Φ: Ω× P →Ω, which tracks coherent features (e.g., shocks, shear layers) of the solution field and ultimately simplifies the task of model reduction. We consider mappings of the form Φ=(𝐚) where :ℝ^M → Lip(Ω; ℝ^2) is a suitable linear or nonlinear operator; then, we state the registration problem as an unconstrained optimization statement for the coefficients 𝐚. We identify minimal requirements for the operator to ensure the satisfaction of the bijectivity constraint; we propose a class of compositional maps that satisfy the desired requirements and enable non-trivial deformations over curved boundaries of Ω; we develop a thorough analysis of the proposed ansatz for polytopal domains and we discuss the approximation properties for general curved domains. We perform numerical experiments for a parametric inviscid transonic compressible flow past a cascade of turbine blades to illustrate the many features of the method.
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