Traceability of Water Pollution: An Inversion Scheme Via Dynamic Complex Geometrical Optics Solutions

TL;DR

This paper presents a novel two-step inversion scheme using dynamic complex geometrical optics solutions to accurately identify and reconstruct water pollution sources from boundary measurements, enabling real-time environmental monitoring.

Contribution

It introduces a new variational formulation and a two-step reconstruction algorithm for source identification in diffusion equations, with proven uniqueness and practical numerical validation.

Findings

Successfully reconstructs multiple point sources

Accurately reconstructs emission concentration functions

Enables real-time pollutant source tracing

Abstract

We investigate the identification of the time-dependent source term in the diffusion equation using boundary measurements. This facilitates tracing back the origins of environmental pollutants. Employing the concept of dynamic complex geometrical optics (CGO) solutions, a variational formulation of the inverse source problem is analyzed, leading to a proof of uniqueness result. Our proposed two-step reconstruction algorithm first determines the point source locations and subsequently reconstructs the Fourier components of the emission concentration functions. Numerical experiments on simulated data are conducted. The results demonstrate that the proposed two-step reconstruction algorithm can reliably reconstruct multiple point sources and accurately reconstruct the emission concentration functions. Additionally, by partitioning the algorithm into online and offline computations, and concentrating computational demand offline, real-time pollutant traceability becomes feasible. This 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.