A Probabilistic Approach to Shape Derivatives
Optimization and Control
2026-01-27 v2 Probability
Abstract
We introduce a novel mesh-free and direct method for computing the shape derivative in PDE-constrained shape optimization problems. Our approach is based on a probabilistic representation of the shape derivative and is applicable for second-order semilinear elliptic PDEs with Dirichlet boundary conditions and a general class of target functions. The probabilistic representation derives from an extension of a boundary sensitivity result for diffusion processes due to Costantini, Gobet and El Karoui [14]. Moreover, we present a simulation methodology based on our results that does not necessarily require a mesh of the relevant domain, and provide Taylor tests to verify its numerical accuracy
Cite
@article{arxiv.2409.15967,
title = {A Probabilistic Approach to Shape Derivatives},
author = {Luka Schlegel and Volker Schulz and Frank T. Seifried and Maximilian Würschmidt},
journal= {arXiv preprint arXiv:2409.15967},
year = {2026}
}