Optimal Shape for Elliptic Problems with Random Perturbations

TL;DR

This paper investigates the optimal shape of elliptic PDEs under random perturbations in the source term, proposing a relaxed optimization framework and demonstrating numerical results to compare with deterministic cases.

Contribution

It introduces a new shape optimization approach accounting for stochastic perturbations in the source term of elliptic problems.

Findings

The optimal shape depends significantly on the random perturbations.

Numerical examples illustrate differences from deterministic shape optimization.

The framework effectively incorporates uncertainty into shape design.

Abstract

In this paper we analyze the relaxed form of a shape optimization problem with state equation $\{{array}{ll} -div \big(a(x)Du\big)=f\qquad\hbox{in}D \hbox{boundary conditions on}\partial D. {array}.$ The new fact is that the term $f$ is only known up to a random perturbation $\xi(x,\omega)$. The goal is to find an optimal coefficient $a(x)$, fulfilling the usual constraints $\alpha\le a\le\beta$ and $\displaystyle\int_D a(x) dx\le m$, which minimizes a cost function of the form $$\int_\Omega\int_Dj\big(x,\omega,u_a(x,\omega)\big) dx dP(\omega).$$ Some numerical examples are shown in the last section, to stress the difference with respect to the case with no perturbation.