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A Probabilistic Approach to Shape Derivatives

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

TL;DR

This work presents a mesh-free, direct method for computing shape derivatives in PDE-constrained shape optimization by leveraging a probabilistic representation of the shape derivative. The authors derive a Feynman–Kac type expression for the Eulerian shape derivative of the PDE solution $u$ and a boundary-focused representation for the derivative of shape functionals $\Phi$, enabling Monte Carlo evaluation without meshing the domain. The framework avoids Lagrangian multipliers and adjoint equations, offering a potentially more efficient alternative to classical mesh-sensitivity approaches. Numerical verification on a 2D benchmark demonstrates agreement with traditional FE-based derivatives via Taylor tests, highlighting the method's practical viability and accessibility through mesh-free simulations.

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

A Probabilistic Approach to Shape Derivatives

TL;DR

This work presents a mesh-free, direct method for computing shape derivatives in PDE-constrained shape optimization by leveraging a probabilistic representation of the shape derivative. The authors derive a Feynman–Kac type expression for the Eulerian shape derivative of the PDE solution and a boundary-focused representation for the derivative of shape functionals , enabling Monte Carlo evaluation without meshing the domain. The framework avoids Lagrangian multipliers and adjoint equations, offering a potentially more efficient alternative to classical mesh-sensitivity approaches. Numerical verification on a 2D benchmark demonstrates agreement with traditional FE-based derivatives via Taylor tests, highlighting the method's practical viability and accessibility through mesh-free simulations.

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
Paper Structure (8 sections, 20 theorems, 196 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 8 sections, 20 theorems, 196 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discussion of Main Results
  3. Probabilistic Representation of Shape Derivatives
  4. Proof of Theorem \ref{['theorem:ProbabilisticRepresentationShapeDerivative']}
  5. Simulation Methodology
  6. Numerical Verification
  7. Stochastic Gronwall Inequality with Stopping Times
  8. Supplements

Key Result

Proposition 3.1

The PDE eqnSemilinearPDE admits a unique solution $u\in\mathcal{C}^{2,\gamma}(\overline{\Omega})$. $\diamond$

Figures (1)

  • Figure 1: Taylor Test Results -- Perturbations are introduced in Table \ref{['tab:TestResults']}. For readability the legends simply indicate the type of shape derivative used to compute the error $\mathcal{E}(\mathbb{D}\Phi;\varepsilon)$.

Theorems & Definitions (47)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Theorem 3.2: Probabilistic Representation of Shape Derivative
  • proof
  • Remark 3.3
  • Example 3.4: $L^1$-derivative of exit times
  • Proposition 3.5
  • ...and 37 more