A Probabilistic Approach to Shape Derivatives

TL;DR

This paper presents a new mesh-free, probabilistic method for computing shape derivatives in PDE-constrained shape optimization, applicable to semilinear elliptic PDEs, with a simulation approach that avoids meshing.

Contribution

It introduces a novel probabilistic representation of shape derivatives and a mesh-free simulation method for PDE-constrained shape optimization.

Findings

The method accurately computes shape derivatives without meshing.

Simulation results verify the numerical accuracy of the approach.

Applicable to a broad class of target functions and PDEs.

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