Determine the point source of the heat equation with sparse boundary measurements

TL;DR

This paper introduces a new numerical method for accurately locating a point heat source within a domain using sparse boundary measurements, employing a least-squares approach and gradient descent, with proven robustness to noise.

Contribution

It presents a novel numerical reconstruction technique for the inverse heat source problem based on sparse boundary data, reformulating it as a least-squares optimization solved efficiently.

Findings

The method accurately reconstructs the point source location.

The approach is robust to measurement noise.

Numerical experiments validate the effectiveness of the method.

Abstract

In this work the authors consider the recovery of the point source in the heat equation. The used data is the sparse boundary measurements. The uniqueness theorem of the inverse problem is given. After that, the numerical reconstruction is considered. We propose a numerical method to reconstruct the location of a Dirac point source by reformulating the inverse problem as a least-squares optimization problem, which is efficiently solved using a gradient descent algorithm. Numerical experiments confirm the accuracy of the proposed method and demonstrate its robustness to noise.