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

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.