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 $\Omega \subset \mathbb{R}^2$ and the manifold $M=\{ u_{\mu} : \mu\in P\}$ associated with the parameter domain $P \subset \mathbb{R}^P$ and the parametric field $\mu\mapsto u_{\mu} \in L^2(\Omega)$, our approach takes as input a set of snapshots from $M$ and returns a parameter-dependent mapping $\Phi: \Omega \times P \to \Omega$, 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 $\Phi=\texttt{N}(\mathbf{a})$ where $\texttt{N}:\mathbb{R}^M \to {\rm Lip}(\Omega; \mathbb{R}^2)$ is a suitable linear or nonlinear operator; then, we state the registration problem as an unconstrained optimization statement for the coefficients $\mathbf{a}$. We identify minimal requirements for the operator $\texttt{N}$ 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 (non-straight) boundaries of $\Omega$; 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.

翻译：我们开发并分析了一种参数化配准方法，用于处理二维区域中与参数化偏微分方程解相关的流形。给定区域 $\Omega \subset \mathbb{R}^2$、与参数域 $P \subset \mathbb{R}^P$ 和参数化场 $\mu\mapsto u_{\mu} \in L^2(\Omega)$ 相关的流形 $M=\{ u_{\mu} : \mu\in P\}$，我们的方法以来自 $M$ 的一组快照为输入，并返回一个参数依赖的映射 $\Phi: \Omega \times P \to \Omega$，该映射追踪解场的相干特征（例如激波、剪切层），最终简化模型降阶任务。我们考虑形如 $\Phi=\texttt{N}(\mathbf{a})$ 的映射，其中 $\texttt{N}:\mathbb{R}^M \to {\rm Lip}(\Omega; \mathbb{R}^2)$ 是合适的线性或非线性算子；随后，我们将配准问题表述为关于系数 $\mathbf{a}$ 的无约束优化问题。我们确定了算子 $\texttt{N}$ 需满足的最小要求以确保双射约束的成立；提出了一类满足所需要求并能在 $\Omega$ 的弯曲（非直线）边界上实现非平凡形变的复合映射；针对多面体域对所提出的假设进行了详尽分析，并讨论了一般弯曲域的逼近性质。我们通过一个参数化无粘跨音速可压缩流经涡轮叶片叶栅的数值实验，展示了该方法的诸多特性。