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Mathematical aspects of registration methods in bounded domains

Angelo Iollo, Jon Labatut, Pierre Mounoud, Tommaso Taddei

TL;DR

This work analyzes registration methods in bounded Lipschitz domains for parametric MOR by comparing vector flows (VFs) and compositional maps (CMs). It develops existence and gradient results, showing how to enforce bijectivity and compute sensitivities via direct and adjoint formulations, while establishing that VFs are dense in ${\rm Diff}_0(\overline{\Omega})$ and CM expressivity is geometry-dependent. The paper also introduces modal (reduced-order) strategies to lower problem dimensionality and provides numerical insights on LS89 test cases, highlighting the trade-offs between mathematical rigor and computational efficiency. Overall, it offers a rigorous framework and practical guidance for selecting basis representations and optimization approaches in parametric registration on bounded domains.

Abstract

Registration methods in bounded domains have received significant attention in the model reduction literature, as a valuable tool for nonlinear approximation. The aim of this work is to provide a concise yet complete overview of relevant results for registration methods in $n$-dimensional domains, from the perspective of parametric model reduction. We present a thorough analysis of two classes of methods, vector flows and compositional maps: we discuss the enforcement of the bijectivity constraint and we comment on the approximation properties of the two methods, for Lipschitz $n$-dimensional domains.

Mathematical aspects of registration methods in bounded domains

TL;DR

This work analyzes registration methods in bounded Lipschitz domains for parametric MOR by comparing vector flows (VFs) and compositional maps (CMs). It develops existence and gradient results, showing how to enforce bijectivity and compute sensitivities via direct and adjoint formulations, while establishing that VFs are dense in and CM expressivity is geometry-dependent. The paper also introduces modal (reduced-order) strategies to lower problem dimensionality and provides numerical insights on LS89 test cases, highlighting the trade-offs between mathematical rigor and computational efficiency. Overall, it offers a rigorous framework and practical guidance for selecting basis representations and optimization approaches in parametric registration on bounded domains.

Abstract

Registration methods in bounded domains have received significant attention in the model reduction literature, as a valuable tool for nonlinear approximation. The aim of this work is to provide a concise yet complete overview of relevant results for registration methods in -dimensional domains, from the perspective of parametric model reduction. We present a thorough analysis of two classes of methods, vector flows and compositional maps: we discuss the enforcement of the bijectivity constraint and we comment on the approximation properties of the two methods, for Lipschitz -dimensional domains.
Paper Structure (22 sections, 22 theorems, 77 equations, 3 figures)

This paper contains 22 sections, 22 theorems, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Formulation
  3. Vector flows for registration
  4. Compositional maps for registration
  5. Representative target functions
  6. Mathematical analysis
  7. Existence of minimizers
  8. Application of gradient descent procedures
  9. Bijectivity and approximation properties of vector flows
  10. Bijectivity and approximation properties of compositional maps in polytopes
  11. Discussion
  12. Choice of the truncated basis
  13. Modal expansions for registration
  14. Numerical results: choice of the modal expansion for CMs
  15. Proofs of Propositions \ref{['th:bijectivity']} and \ref{['th:approximation_DB']}
  16. ...and 7 more sections

Key Result

Proposition 3.1

Suppose that $\texttt{N}$ and $\mathfrak{f}_{\rm pen}$ satisfy eq:desiderata_mapping_space. Suppose also that (i) $E:= \mathbf{a} \mapsto \mathfrak{f}^{\rm tg}(\texttt{N}(\mathbf{a}))$ is Lipschitz continuous and bounded from below, and (ii) ${\lim_{\| \mathbf{a} \|_2 \to \infty}}$$\mathfrak{f}_{\rm

Figures (3)

  • Figure 1: interpretation of Definitions \ref{['def:polyhedron']} and \ref{['def:polyhedral_domain']}. (a) example of polyhedral domain. (b) example of polyhedron that is not Lipschitz.
  • Figure 2: Inviscid flow past an array of LS89 turbine blades. (a) polytope $\Omega_{\rm p}$ and FE mesh; (b) curved domain $\Omega$.
  • Figure 3: Inviscid flow past an array of LS89 turbine blades. (a) relative projection error $E_m^{\rm proj}$. (b) worst-case objective $E_m^{\rm obj}$ and comparison with FE solution.

Theorems & Definitions (45)

  • Definition 2.1
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • ...and 35 more