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arXiv · 2409.15967

A Probabilistic Approach to Shape Derivatives

Luka Schlegel·Volker Schulz·Frank T. Seifried·Maximilian Würschmidt
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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

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Luka Schlegel, Volker Schulz, Frank T. Seifried, Maximilian Würschmidt. 2024-09-24. A Probabilistic Approach to Shape Derivatives. https://arxiv.org/abs/2409.15967

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