{
  "id": "2409.15967",
  "version": "v1",
  "published": "2024-09-24T10:53:35.000Z",
  "updated": "2024-09-24T10:53:35.000Z",
  "title": "A Probabilistic Approach to Shape Derivatives",
  "authors": [
    "Luka Schlegel",
    "Volker Schulz",
    "Frank T. Seifried",
    "Maximilian Würschmidt"
  ],
  "categories": [
    "math.OC",
    "math.PR"
  ],
  "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",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-24T10:53:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "shape derivative",
      "probabilistic approach",
      "order semilinear elliptic pdes",
      "pde-constrained shape optimization problems",
      "boundary sensitivity result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}