Shape optimization problems with random coefficients via the penalty method

TL;DR

This paper addresses shape optimization problems with uncertain diffusion coefficients using a penalization approach, combining finite element, Monte Carlo, and accelerated gradient methods, with convergence analysis and numerical validation.

Contribution

It introduces a penalization technique for shape optimization with random coefficients and integrates multiple numerical methods for efficient solution and analysis.

Findings

Convergence of the proposed penalization method is established.

Numerical experiments demonstrate the effectiveness of the combined approach.

The method efficiently handles uncertainties in diffusion coefficients.

Abstract

For shape optimization problems, governed by elliptic equations with Dirichlet boundary condition and random coefficients, we utilize a penalization technique to get the approximate problem. We consider that uncertainties exists in the diffusion coefficients and minimize objective functions in mean value form. Finite element method, Monte Carlo method and accelerated version of the gradient descent method are applied to solve the corresponding discretized problem. The convergence analysis and numerical results are included.