Ailin Qian

Zhiliang Deng, Xiaomei Yang, Ailin Qian

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.

Zhiliang Deng, Ailin Qian, Xiaomei Yang

This paper studies the identification of an unknown number of stationary point sources in a two-dimensional heat equation from boundary flux measurements. Unlike many reconstruction approaches that assume the number of sources to be known in advance, we develop a determinant-based counting method that extracts this number directly from the measured data. In the unit disk, the Laplace-transformed and normalized boundary flux admits a Fourier moment representation whose low-frequency limit has a finite exponential-sum structure. This structure leads to a family of Hankel matrices and associated determinant characteristics. We prove that, under a generic determinant lifting condition, the vanishing order of the Hankel determinant at the zero Laplace frequency changes exactly when the Hankel order exceeds the true number of sources. Consequently, the source number is characterized by the first nonzero contour count of the determinant characteristic through the argument principle. We further establish a Rouché-type stability result showing that the determinant zero count is preserved under sufficiently small perturbations induced by measurement noise, boundary discretization, and time truncation. After the source number is identified, the source locations and strengths are recovered from the low-frequency moment sequence by an annihilating-polynomial and Vandermonde reconstruction procedure. Numerical experiments confirm the predicted count pattern, demonstrate robustness with respect to contour selection, illustrate the role of the Rouché margin under noise and near-degenerate configurations, and validate the subsequent recovery of source locations and strengths.