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

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.

💡 Research Summary

The paper proposes a novel framework for PDE‑constrained shape optimization that works directly on the diffeomorphism group (\mathrm{Diff}{c}(\mathbb{R}^{2})) equipped with Sobolev‑type outer metrics. By representing a shape as the image of a reference domain under a compactly supported diffeomorphism, the authors replace the classical shape space of embedded curves (B{e}) with the quotient (\mathrm{Diff}{c}(\mathbb{R}^{2})/\mathrm{Diff}{c}(\mathbb{R}^{2},\Xi)), where (\Xi) is the unit circle. This representation allows the ambient space (\mathbb{R}^{2}) itself to be deformed, which is a key difference from inner‑metric approaches that only move the shape boundary.

The core mathematical contribution is the rigorous connection between the push‑forward of a shape functional defined on the diffeomorphism group and the classical Eulerian shape derivative. The authors show that the Riemannian gradient of the lifted functional (\tilde J(\phi)=J(\phi(\Omega))) with respect to a Sobolev metric (H^{s}) coincides, after push‑forward, with the standard shape derivative (DJ(\Omega)