An optimization-based registration approach to geometry reduction
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We develop and assess an optimization-based approach to parametric geometry reduction. Given a family of parametric domains, we aim to determine a parametric diffeomorphism Φ that maps a fixed reference domain Ω into each element of the family, for different values of the parameter; the ultimate goal of our study is to determine an effective tool for parametric projection-based model order reduction of partial differential equations in parametric geometries. For practical problems in engineering, explicit parameterizations of the geometry are likely unavailable: for this reason, our approach takes as inputs a reference mesh of Ω and a point cloud {y_i^ raw}_i=1^Q that belongs to the boundary of the target domain V and returns a bijection Φ that approximately maps Ω in V. We propose a two-step procedure: given the point clouds {x_j}_j=1^N⊂∂Ω and {y_i^ raw}_i=1^Q ⊂∂ V, we first resort to a point-set registration algorithm to determine the displacements { v_j }_j=1^N such that the deformed point cloud {y_j:= x_j+v_j }_j=1^N approximates ∂ V; then, we solve a nonlinear non-convex optimization problem to build a mapping Φ that is bijective from Ω in ℝ^d and (approximately) satisfies Φ(x_j) = y_j for j=1,…,N.We present a rigorous mathematical analysis to justify our approach; we further present thorough numerical experiments to show the effectiveness of the proposed method.
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