A Riemannian Approach for PDE-Constrained Shape Optimization Using Outer Metrics скачать в хорошем качестве

A Riemannian Approach for PDE-Constrained Shape Optimization Using Outer Metrics

05-02-2026 Oxford-RAL Computational Mathematics and Applications Seminar Speaker: Estefania Loayza Romero (Strathclyde University) Title: A Riemannian Approach for PDE-Constrained Shape Optimization Using Outer Metrics Abstract: In PDE-constrained shape optimisation, shapes are traditionally viewed as elements of a Riemannian manifold, specifically as embeddings of the unit circle into the plane, modulo reparameterizations. The standard approach employs the Steklov-Poincaré metric to compute gradients for Riemannian optimisation methods. A significant limitation of current methods is the absence of explicit expressions for the geodesic equations associated with this metric. Consequently, algorithms have relied on retractions (often equivalent to the perturbation of identity method in shape optimisation) rather than true geodesic paths. Previous research suggests that incorporating geodesic equations, or better approximations thereof, can substantially enhance algorithmic performance. This talk presents numerical evidence demonstrating that using outer metrics, defined on the space of diffeomorphisms with known geodesic expressions, improves Riemannian gradient-based optimisation by significantly reducing the number of required iterations and preserving mesh quality throughout the optimisation process.