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

08/06/2019

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by   William Rundell, et al.

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We consider the recovery of a source term f(x,t)=p(x)q(t) for the nonhomogeneous heat equation in Ω× (0,∞) where Ω is a bounded domain in R^2 with smooth boundary ∂Ω from overposed lateral data on a sparse subset of ∂Ω×(0,∞). Specifically, we shall require a small finite number N of measurement points on ∂Ω 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.