In this work the authors consider the recovery of the point source in the heat equation. The used data is the sparse boundary measurements. The uniqueness theorem of the inverse problem is given. After that, the numerical reconstruction is considered. We propose a numerical method to reconstruct the location of a Dirac point source by reformulating the inverse problem as a least-squares optimization problem, which is efficiently solved using a gradient descent algorithm. Numerical experiments confirm the accuracy of the proposed method and demonstrate its robustness to noise.

翻译：本文研究热方程中点源的恢复问题。所用数据为稀疏边界测量。文中给出了该反问题的唯一性定理。随后，考虑数值重构问题。我们通过将反问题重构为最小二乘优化问题，提出了一种数值方法来重建狄拉克点源的位置，并采用梯度下降算法进行高效求解。数值实验验证了所提方法的准确性，并证明了其对噪声的鲁棒性。