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Compositional maps for registration in complex geometries

Tommaso Taddei

TL;DR

A parametric registration procedure for manifolds associated with the solutions to parametric partial differential equations in two-dimensional domains is developed and a class of compositional maps that satisfy the desired requirements and enable non-trivial deformations over curved (non-straight) boundaries of $\Omega$ are proposed.

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 $Ω\subset \mathbb{R}^2$ and the manifold $M=\{ u_μ : μ\in P\}$ associated with the parameter domain $P \subset \mathbb{R}^P$ and the parametric field $μ\mapsto u_μ \in L^2(Ω)$, our approach takes as input a set of snapshots from $M$ and returns a parameter-dependent mapping $Φ: Ω\times P \to Ω$, 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 $Φ=\texttt{N}(\mathbf{a})$ where $\texttt{N}:\mathbb{R}^M \to {\rm Lip}(Ω; \mathbb{R}^2)$ 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 $Ω$; 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.

Compositional maps for registration in complex geometries

TL;DR

A parametric registration procedure for manifolds associated with the solutions to parametric partial differential equations in two-dimensional domains is developed and a class of compositional maps that satisfy the desired requirements and enable non-trivial deformations over curved (non-straight) boundaries of are proposed.

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 and the manifold associated with the parameter domain and the parametric field , our approach takes as input a set of snapshots from and returns a parameter-dependent mapping , 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 where is a suitable linear or nonlinear operator; then, we state the registration problem as an unconstrained optimization statement for the coefficients . We identify minimal requirements for the operator 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 ; 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.
Paper Structure (31 sections, 10 theorems, 64 equations, 13 figures, 1 algorithm)

This paper contains 31 sections, 10 theorems, 64 equations, 13 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Registration in bounded domains
  3. Optimization-based registration
  4. Compositional maps for registration
  5. Affine maps in polytopes
  6. Definition of the penalty function
  7. Mathematical background
  8. Construction of finite-dimensional operators for registration
  9. Approximation properties of affine mappings
  10. Approximation of Lipschitz maps
  11. Nonlinear maps for general domains
  12. Inadequacy of affine maps for domains with curved boundaries
  13. Approximation properties of compositional maps
  14. Multi-layer compositional maps
  15. Discussion
  16. ...and 16 more sections

Key Result

Proposition 2.1

Let $\Omega_{\rm p}$ be a regular bounded polytope. Define the space ${\mathfrak{U}}_0={\mathfrak{U}}_0(\Omega_{\rm p}) = \{ \varphi \in C^1(\overline{\Omega}_{\rm p}; \mathbb{R}^2) \,: \, \varphi \cdot \mathbf{n} |_{\partial \Omega_{\rm p}} = 0 \}$ and consider the vector-valued function $\Phi = \t

Figures (13)

  • Figure 1: compositional maps. The bijection $\Psi$ is designed to recast the registration task from the curved domain $\Omega$ to the polytope $\Omega_{\rm p}$.
  • Figure 2: interpretation of Definition \ref{['def:regular_polytopes']}. (a) regular polytope. (b) irregular polytope.
  • Figure 3: example of Lipschitz map with $\Phi(V)\neq V$. (a) Lipschitz map from the unit square to the unit triangle (cf \ref{['eq:silly_example']}). (b) smooth map in the unit triangle $D$.
  • Figure 4: approximation of large deformations over curved boundaries using a multi-layer ($\ell=3$) compositional map.
  • Figure 5: construction of the polytope $\Omega_{\rm p}$ and the mapping $\Psi$ for a curved mesh with two elements.
  • ...and 8 more figures

Theorems & Definitions (23)

  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 13 more