Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation

TL;DR

This paper explores the application of reproducing kernel Hilbert spaces with radial kernels to PDE-constrained shape optimization, introducing a smoothing parameter for shape adjustment and demonstrating efficiency through numerical experiments.

Contribution

It introduces a novel approach using kernel reproducing Hilbert spaces with radial kernels for shape derivatives in PDE-constrained optimization, including a smoothing parameter for shape control.

Findings

Radial kernels provide efficient shape gradient formulas.

The smoothing parameter allows shape smoothness adjustment.

Numerical experiments confirm theoretical advantages.

Abstract

In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE constrained shape optimisation problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so called smoothing parameter that allows a smoothness adjustment of the shape during the optimisation process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments.