A Riemannian View on Shape Optimization

TL;DR

This paper introduces a Riemannian framework for shape optimization, enabling the analysis of Newton methods with properties like symmetry and quadratic convergence, which are advantageous over traditional steepest descent approaches.

Contribution

It defines a Riemannian shape Hessian to facilitate the analysis and development of shape-Newton methods, offering new theoretical insights.

Findings

Riemannian shape Hessian exhibits symmetry

Shape-Newton methods achieve quadratic convergence

Framework enhances understanding of shape optimization algorithms

Abstract

Shape optimization based on the shape calculus is numerically mostly performed by means of steepest descent methods. This paper provides a novel framework to analyze shape-Newton optimization methods by exploiting a Riemannian perspective. A Riemannian shape Hessian is defined yielding often sought properties like symmetry and quadratic convergence for Newton optimization methods.