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 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.

Theoretical and numerical studies of inverse source problem for the linear parabolic equation with sparse boundary measurements

“…The proof depended on the explicit representation of the eigensystem of the Laplacian ∆ on the two-dimensional unit disc. The conclusions in [19,23] confirm that in the heat equation, if the domain has a smooth shape, then it is possible to recover the source from sparse boundary data. Sequentially, there is a natural question that can we solve the similar inverse source problem in the parabolic equation with a general domain, in which the explicit representation of the eigensystem is not applicable.…”

“…To save cost, absolutely we want the observed area to be as small as possible. In [19,23], the authors considered the heat equation on the two-dimensional unit disc, and proved the uniqueness theorem under the boundary flux data from two chosen points on the boundary. The proof depended on the explicit representation of the eigensystem of the Laplacian ∆ on the two-dimensional unit disc.…”

Logarithmic stable recovery of the source and the initial state of time fractional diffusion equations

“…), by virtue of (33). Thus, there exists a positive constant C, depending only on α, such that the following estimate α+1) , t ∈ [1, +∞), holds uniformly in k ∈ N. As a consequence we have ≤ S 0 (•)u 0 L r (R+;H Finally, (43) follows readily from this and the inequality M ≥ 1 arising from (40).…”

“…While several authors have already addressed the inverse problem of retrieving the space-varying part of the source term in a fractional diffusion equation, see e.g., [12,34], only uniqueness results are available in the mathematical literature, see e.g., [16,19,22,25,33] (see also [20] for some related inverse problem) with the exception of the recent stability result of [9] stated with specific norms. This is not surprising since the classical methods used to build a stability estimate for the source of parabolic (α = 1) or hyperbolic (α = 2) systems, see e.g., [10,14,24,3], do not apply in a straightforward way to time-fractional diffusion equations.…”

Restoring the discontinuous heat equation source using sparse boundary data and dynamic sensors

“…We shall delve into scenarios involving two distinct unknown sources, each characterized by disparate geometries and dimensions pertaining to the unknown parameters. Across both sets of problems, the task involves measuring boundary observation flux at two distinct points along the boundary-an endeavor that has proven to be particularly challenging for conventional implementation methodologies [31,32]. Employing a comparative approach alongside the fixed observation points method, our analysis reveals a consistent trend: the methods we propose adeptly capture the source.…”