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 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：我们针对二维区域中参数化偏微分方程解对应的流形，建立并分析了一种参数化配准方法。给定区域 $\Omega \subset \mathbb{R}^2$ 以及由参数域 $P \subset \mathbb{R}^P$ 和参数场 $\mu\mapsto u_{\mu} \in L^2(\Omega)$ 关联的流形 $M=\{ u_{\mu} : \mu\in P\}$，本方法以 $M$ 的一组快照为输入，返回一个依赖于参数的映射 $\Phi: \Omega \times P \to \Omega$，该映射能追踪解场的相干特征（如激波、剪切层），从而简化模型降阶任务。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；随后将配准问题表述为关于系数 $\mathbf{a}$ 的无约束优化问题。我们识别了算子 $\texttt{N}$ 满足双射约束所需的最低条件；提出了一类满足上述要求且能在 $\Omega$ 的弯曲边界上实现非平凡形变的组合映射；对多面体域上的假设形式进行了深入分析，并讨论了适用于一般弯曲域的逼近性质。我们以参数化无粘跨声速可压缩流绕过涡轮叶片叶栅的数值实验，展示了该方法的多种特性。