Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Riemannian approach for PDE-constrained shape optimization over the diffeomorphism group using outer metrics

Estefania Loayza-Romero, Lidiya Pryymak, Kathrin Welker

Abstract

In this paper, we study the use of outer metrics, in particular Sobolev-type metrics on the diffeomorphism group in the context of PDE-constrained shape optimization. Leveraging the structure of the diffeomorphism group we analyze the connection between the push-forward of a smooth function defined on the diffeomorphism group and the classical shape derivative as an Eulerian semi-derivative. We consider in particular, two predominant examples on PDE-constrained shape optimization. An electric impedance tomography inspired problem, and the optimization of a two-dimensional bridge. These problems are numerically solved using the Riemannian steepest descent method where the descent directions are taken to be the Riemannian gradients associated to various outer metrics. For comparison reasons, we also solve the problem using other previously proposed Riemannian metrics in particular the Steklov-Poincaré metric.

A Riemannian approach for PDE-constrained shape optimization over the diffeomorphism group using outer metrics

Abstract

In this paper, we study the use of outer metrics, in particular Sobolev-type metrics on the diffeomorphism group in the context of PDE-constrained shape optimization. Leveraging the structure of the diffeomorphism group we analyze the connection between the push-forward of a smooth function defined on the diffeomorphism group and the classical shape derivative as an Eulerian semi-derivative. We consider in particular, two predominant examples on PDE-constrained shape optimization. An electric impedance tomography inspired problem, and the optimization of a two-dimensional bridge. These problems are numerically solved using the Riemannian steepest descent method where the descent directions are taken to be the Riemannian gradients associated to various outer metrics. For comparison reasons, we also solve the problem using other previously proposed Riemannian metrics in particular the Steklov-Poincaré metric.

Paper Structure

This paper contains 22 sections, 1 theorem, 30 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Riemannian structure on $\textup{Diff}_c(\mathbb{R}^2)$
  3. Preliminaries on differential geometry
  4. Properties of the diffeomorphism group
  5. Sobolev-type metrics
  6. Gradient-based shape optimization on $(\textup{Diff}_c(\mathbb{R}^2), H^s)$
  7. Basics on shape calculus
  8. Shape calculus
  9. Optimization on Riemannian shape manifolds
  10. Diffeomorphism group as shape space in optimization
  11. Equivalence of shape derivative and push-forward
  12. Shape gradients with respect to the $H^s$ metric
  13. Applications
  14. Interface identification problem
  15. Two-dimensional optimal bridge
  16. ...and 7 more sections

Key Result

Theorem 1

Let $j \in \mathcal{J}$ be a shape functional such that $j(\varphi)=J(u_{\varphi})$ for all $\varphi \in \textup{Diff}_c(\mathbb{R}^2)$ for some $J\colon \mathcal{D} \subset P(\mathbb{R}^2) \rightarrow \mathbb{R}$. Then, for any tangent curve $\gamma\colon \mathbb{R} \rightarrow \textup{Diff}_c(\ma

Figures (10)

  • Figure 1: Tangent vectors on $B_e$, with inner metrics (left) and outer metrics where the vector field deforms the ambient space and induces a deformation of the shape (right).
  • Figure 2: A shape $\varphi \in \textup{Diff}_c(\mathbb{R}^2)$ is identified with its image of the unit circle $u_{\varphi}=\varphi(\Xi)$
  • Figure 3: Data used in the experiment described in \ref{['subsec:Expe1']}.
  • Figure 4: Different shape gradients from the experiment described in \ref{['subsec:Expe1']}.
  • Figure 5: Final iterates of each variant of the method for the experiment described in \ref{['subsec:Expe1']}
  • ...and 5 more figures

Theorems & Definitions (9)

  • Remark 1
  • Definition 1: Shape derivative
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • proof
  • Remark 5
  • Remark 6