Compositional maps for registration in complex geometries

TL;DR

This paper introduces a novel parametric registration method for manifolds associated with solutions to PDEs in 2D domains, using compositional maps to improve model reduction by tracking solution features like shocks and shear layers.

Contribution

The paper proposes a class of compositional maps for registration that satisfy bijectivity and handle complex geometries, with thorough analysis and numerical validation.

Findings

Effective tracking of coherent features in parametric PDE solutions.

Handles non-trivial deformations over curved boundaries.

Numerical experiments demonstrate method's accuracy and robustness.

Abstract

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.