Traceability of Water Pollution: An Inversion Scheme Via Dynamic CGO Solutions

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.