Shape Optimisation with $W^{1,\infty}$: A connection between the steepest descent and Optimal Transport
Optimization and Control
2023-01-20 v1 Numerical Analysis
Numerical Analysis
Abstract
In this work, we discuss the task of finding a direction of optimal descent for problems in Shape Optimisation and its relation to the dual problem in Optimal Transport. This link was first observed in a previous work which sought minimisers of a shape derivative over the space of Lipschitz functions which may be closely related to the -Laplacian. We provide some results of shape optimisation using this novel Lipschitz approach, highlighting the difference between the Lipschitz and semi-norms. After this, we provide an overview of the necessary results from Optimal transport in order to make a direct link to the Shape optimisation of star-shaped domains. Demonstrative numerical experiments are provided.
Cite
@article{arxiv.2301.07994,
title = {Shape Optimisation with $W^{1,\infty}$: A connection between the steepest descent and Optimal Transport},
author = {Philip J. Herbert},
journal= {arXiv preprint arXiv:2301.07994},
year = {2023}
}