On the identification of source term in the heat equation from sparse data

TL;DR

This paper demonstrates that it is possible to uniquely recover a space-time dependent source term in the heat equation using only two measurement points on the boundary, and provides a method for numerical reconstruction despite the problem's ill-posedness.

Contribution

The paper proves uniqueness of source term recovery from minimal boundary data and introduces a numerical approach for reconstruction with low noise levels.

Findings

Unique recovery of source term with only two boundary measurement points.

Effective numerical reconstructions possible under low noise conditions.

Addresses severe ill-posedness with minimal overposed data.

Abstract

We consider the recovery of a source term $f(x,t)=p(x)q(t)$ for the nonhomogeneous heat equation in $\Omega\times (0,\infty)$ where $\Omega$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary $\partial\Omega$ from overposed lateral data on a sparse subset of $\partial\Omega\times(0,\infty)$. Specifically, we shall require a small finite number $N$ of measurement points on $\partial\Omega$ and prove a uniqueness result; namely the recovery of the pair $(p,q)$ within a given class, by a judicious choice of $N=2$ points. Naturally, with this paucity of overposed data, the problem is severely ill-posed. Nevertheless we shall show that provided the data noise level is low, effective numerical reconstructions may be obtained.