Space Mapping for PDE Constrained Shape Optimization

TL;DR

This paper introduces novel space mapping techniques for PDE-constrained shape optimization, combining accuracy and efficiency by leveraging Riemannian metrics and numerical discretization, demonstrating significant performance improvements.

Contribution

It proposes new space mapping methods in a Riemannian setting for PDE-constrained shape optimization, with detailed discretization and implementation strategies.

Findings

Methods show high efficiency on model problems

Numerical results validate the approach's effectiveness

Significant reduction in computational cost

Abstract

The space mapping technique is used to efficiently solve complex optimization problems. It combines the accuracy of fine model simulations with the speed of coarse model optimizations to approximate the solution of the fine model optimization problem. In this paper, we propose novel space mapping methods for solving shape optimization problems constrained by partial differential equations (PDEs). We present the methods in a Riemannian setting based on Steklov-Poincar\'e-type metrics and discuss their numerical discretization and implementation. We investigate the numerical performance of the space mapping methods on several model problems. Our numerical results highlight the methods' great efficiency for solving complex shape optimization problems.