Nonlinear Conjugate Gradient Methods for PDE Constrained Shape Optimization Based on Steklov-Poincar\'e-Type Metrics

TL;DR

This paper introduces nonlinear conjugate gradient methods using Steklov-Poincaré metrics for PDE-constrained shape optimization, demonstrating their efficiency and competitiveness through numerical comparisons with existing methods.

Contribution

It develops and analyzes a new class of nonlinear conjugate gradient algorithms tailored for shape optimization constrained by PDEs, incorporating Steklov-Poincaré-type metrics.

Findings

The proposed methods perform well in numerical experiments.

They are more efficient than gradient descent and BFGS in benchmark problems.

The methods are a valuable addition to existing shape optimization algorithms.

Abstract

Shape optimization based on shape calculus has received a lot of attention in recent years, particularly regarding the development, analysis, and modification of efficient optimization algorithms. In this paper we propose and investigate nonlinear conjugate gradient methods based on Steklov-Poincar\'e-type metrics for the solution of shape optimization problems constrained by partial differential equations. We embed these methods into a general algorithmic framework for gradient-based shape optimization methods and discuss the numerical discretization of the algorithms. We numerically compare the proposed nonlinear conjugate gradient methods to the already established gradient descent and limited memory BFGS methods for shape optimization on several benchmark problems. The results show that the proposed nonlinear conjugate gradient methods perform well in practice and that they are an efficient and attractive addition to already established gradient-based shape optimization algorithms.