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