We aim to identify the time-dependent source term in the diffusion equation using boundary measurements. This facilitates tracing back the origins of environmental pollutants. Based on the idea of dynamic complex geometrical optics (CGO) solutions, we analyze a variational formulation of the inverse source problem and prove the uniqueness result. We propose a two-step reconstruction algorithm. Initially, the locations of the point sources are determined, followed by the reconstruction of the Fourier components of the emission concentration functions. Numerical experiments on simulated data are conducted. The results demonstrate that our proposed two-step reconstruction algorithm can reliably reconstruct multiple point sources and accurately reconstruct the emission concentration functions. Additionally, we partition the algorithm into online and offline computations, with the bulk of the work done offline. This paves the way for real-time traceability of pollutants. Our proposed method, applicable in various fields - especially those related to water pollution, can identify the source of a contaminant in the environment, thus serving as a valuable tool in environmental protection.

翻译：本文旨在利用边界测量值识别扩散方程中的时变源项，从而追溯环境污染物源头。基于动态复几何光学（CGO）解的思想，我们分析了逆源问题的变分形式，并证明了唯一性结果。我们提出了一种两步重构算法：首先确定点源位置，进而重构排放浓度函数的傅里叶分量。基于模拟数据的数值实验表明，该两步重构算法能够可靠地重构多个点源，并精确重构排放浓度函数。此外，我们将算法划分为在线计算与离线计算两个阶段，其中大部分工作通过离线完成，这为污染物的实时溯源奠定了基础。本方法适用于多个领域（尤其是水污染相关领域），可识别环境中污染物来源，从而成为环境保护中的有效工具。