A Moment--Hankel Rank Method for Identifying the Number of Point Sources in the Heat Equation
This work provides a theoretically grounded approach for source number identification in inverse heat conduction problems, which is important for applications like thermal monitoring, but the method is domain-specific (unit disk) and incremental.
The paper develops a low-frequency moment-Hankel rank method to identify the number of time-independent point sources in the heat equation from boundary flux data, achieving exact rank recovery under noiseless conditions and stable recovery under noise, with numerical validation.
We develop a low-frequency moment--Hankel rank method for identifying the number of time-independent point sources in the heat equation from boundary flux data. The method is formulated in the unit disk, where the Laplace-transformed and normalized boundary flux admits an explicit Fourier moment representation. By taking the low-frequency limit, we obtain a finite exponential-sum moment sequence in which the nodes encode the source locations, the weights encode the source strengths, and the number of terms equals the number of point sources. The associated Hankel matrix admits a Vandermonde factorization, and its rank is exactly the source number under the natural assumptions that the source locations are distinct and the source strengths are nonzero. We also analyze the effect of discrete and noisy boundary data. A uniform moment perturbation bound is propagated to the empirical Hankel matrix, and Weyl's singular-value perturbation inequality yields a sufficient condition for stable numerical rank recovery in terms of the smallest nonzero singular value of the ideal Hankel matrix. Numerical experiments confirm the exact rank pattern in the noiseless case, validate the stability threshold under moment noise, and illustrate the loss of resolution for close or weak sources. After the source number is identified, the same moment sequence can be used for location and strength recovery through an annihilating-polynomial and Vandermonde reconstruction procedure.