# On the numerical solution of a time-dependent shape optimization problem for the heat equation

## TL;DR

This paper presents a gradient-based numerical method to identify a time-varying inclusion within a domain governed by the heat equation, using boundary measurements to minimize Neumann data mismatch.

## Contribution

It introduces a novel approach for solving a time-dependent shape optimization problem involving heat equations, with validated numerical results.

## Key findings

- Successful detection of time-dependent inclusions from boundary data
- Validation of the gradient-based optimization method through numerical experiments
- Effective minimization of Neumann data mismatch for shape identification

## Abstract

This article is concerned with the solution of a time-dependent shape identification problem. Specifically we consider the heat equation in a domain, which contains a time-dependent inclusion of zero temperature. The objective is to detect this inclusion from the given temperature and heat flux at the exterior boundary of the domain. To this end, for a given temperature at the exterior boundary, the mismatch of the Neumann data is minimized. This time-dependent shape optimization problem is then solved by a gradient-based optimization method. Numerical results are presented which validate the present approach.