Identification of a Point Source in the Heat Equation from Sparse Boundary Measurements

TL;DR

This paper addresses the inverse problem of identifying a single heat source in a domain from limited boundary measurements, proving uniqueness under certain conditions and demonstrating the approach with numerical experiments.

Contribution

It establishes unique recovery results for the source location and amplitude in specific domain settings, combining analytical techniques and numerical validation.

Findings

Unique recovery of source location and amplitude in the unit ball.

Recovery of location and compactly supported amplitude in simply connected domains.

Numerical experiments confirming the feasibility of the method.

Abstract

In this work we investigate the inverse problem of recovering one point source in the heat equation from sparse boundary measurement, i.e., the flux data at several points on the boundary. We prove the unique recovery of the location and piecewise constant in time amplitude when the domain is the unit ball in $\mathbb{R}^d$ ($d\geq2$), and the unique recovery of the location and compactly supported amplitude when the domain is simply connected, smooth and bounded in $\mathbb{R}^2$, under mild conditions on the observational points. The proof combines distinct analytical tools, including the representation of the flux data via Laplacian eigenfunctions on the unit ball, a detailed analysis of the properties of the heat and Poisson kernels, as well as methods drawn from complex analysis. Further we present several numerical experiments to illustrate the feasibility of the recovery from sparse boundary data.