We present a general -- i.e., independent of the underlying equation -- registration procedure for parameterized model order reduction. Given the spatial domain \Omega \subset R^2 and the manifold M= \{ u_{\mu} : \mu \in P \} associated with the parameter domain P \subset R^P and the parametric field \mu \mapsto u_{\mu} \in L^2(\Omega), our approach takes as input a set of snapshots \{ u^k \}_{k=1}^{n_{\rm train}} \subset M and returns a parameter-dependent bijective mapping \Phi : \Omega \times P \to R^2: the mapping is designed to make the mapped manifold \{ u_{\mu} \circ \Phi _{\mu}: \, \mu \in P \} more amenable for linear compression methods. In this work, we extend and further analyze the registration approach proposed in [Taddei, SISC, 2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition \{ q \}_{q=1} ^{N_{\rm dd}} of the domain \Omega such that \Omega_1,\ldots,N_{\rm dd} are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain and for a potential flow past a rotating airfoil to demonstrate the effectiveness of our method.
We present a general -- i.e., independent of the underlying equation -- registration procedure for parameterized model order reduction. Given the spatial domain \Omega \subset R^2 and the manifold M= \{ u_{\mu} : \mu \in P \} associated with the parameter domain P \subset R^P and the parametric field \mu \mapsto u_{\mu} \in L^2(\Omega), our approach takes as input a set of snapshots \{ u^k \}_{k=1}^{n_{\rm train}} \subset M and returns a parameter-dependent bijective mapping \Phi : \Omega \times P \to R^2: the mapping is designed to make the mapped manifold \{ u_{\mu} \circ \Phi _{\mu}: \, \mu \in P \} more amenable for linear compression methods. In this work, we extend and further analyze the registration approach proposed in [Taddei, SISC, 2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition \{ q \}_{q=1} ^{N_{\rm dd}} of the domain \Omega such that \Omega_1,\ldots,N_{\rm dd} are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain , a potential flow past a rotating symmetric airfoil, and an inviscid transonic compressible flow past a non-symmetric airfoil, to demonstrate the effectiveness of our method.
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