A release of hazardous materials poses a significant threat to individuals and communities. Such releases can occur through industrial accidents, such as leaks or spills, or through deliberate acts of sabotage or terrorism. This work presents a mathematical model to enable rapid prediction of airborne contaminant transport based on scarce sensor measurements. The method is designed for applications in critical infrastructure protection (CIP), such as evacuation planning following contaminant release. In such scenarios, timely and reliable decision-making is essential, despite limited observation data. To identify contaminant sources (and compute the future contaminant dispersion with this information), we formulate an inverse problem governed by an advection–diffusion equation. Given the problem’s underdetermined nature, we further employ a sparsity-promoting regularization and model the unknown contaminant sources as distribution over the spatial domain. This approach allows us to incorporate prior knowledge about the source structure and the measurement noise. To efficiently solve the source identification problem, we employ a problem-specific variant of the Primal-Dual-Active-Point (PDAP) algorithm. This algorithm approximates sparse minimizers of the inverse problem by alternating between greedy location updates and source intensity optimization. The approach is demonstrated on two- and three-dimensional test cases involving both instantaneous and continuous contaminant sources and outperforms state-of-the-art techniques with L2-regularization. Its effectiveness is further illustrated in complex domains with real-world building imprints imported from OpenStreetMap. Finally, our approach is developed towards a digital twin framework for rapid predictions of contaminant dispersion to support informed decision making in emergency situations.