# Computational Aspects for Interface Identification Problems with Stochastic Modelling

## TL;DR

This paper addresses interface identification problems under uncertainty by formulating a shape optimization constrained by stochastic PDEs, introducing a generalized stochastic gradient method, and demonstrating its effectiveness through numerical experiments.

## Contribution

It introduces a novel stochastic gradient approach for shape optimization with uncertain PDE constraints, extending classical methods to stochastic shape spaces.

## Key findings

- The proposed algorithm effectively solves stochastic interface identification problems.
- Numerical experiments confirm the method's convergence and robustness.
- The approach handles uncertainty in coefficients and inputs in PDE-based shape optimization.

## Abstract

In this paper, a shape optimization problem constrained by a random elliptic partial differential equation with a pure Neumann boundary is presented. The model is motivated by applications in interface identification, where we assume coefficients and inputs are subject to uncertainty. The problem is posed as a minimization of the expectation of a random objective functional depending on the uncertain parameters. A numerical method for iteratively solving the problem is presented, which is a generalization of the classical stochastic gradient method in shape spaces. Moreover, we perform numerical experiments, which demonstrate the effectiveness of the algorithm.