Sparse Source Identification in Transient Advection-Diffusion Problems with a Primal-Dual-Active-Point Strategy

TL;DR

This paper introduces a novel sparse source identification method for transient advection-diffusion problems using a primal-dual-active-point algorithm, enabling rapid and reliable prediction of airborne contaminants with limited sensor data.

Contribution

It develops a problem-specific PDAP algorithm for efficiently solving sparse inverse problems in contaminant source identification, outperforming existing regularization techniques.

Findings

Effective in 2D and 3D test cases

Handles both instantaneous and continuous sources

Works well in complex real-world geometries

Abstract

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, we formulate an inverse problem governed by an advection-diffusion equation. Given the problem's underdetermined nature, we further employ a variational regularization ansatz and model the unknown contaminant sources as distribution over the spatial domain. To efficiently solve the arising inverse problem, we employ a problem-specific variant of the Primal-Dual-Active-Point (PDAP) algorithm which efficiently 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 $L^2$-regularization. Its effectiveness is further illustrated in complex domains with real-world building geometries imported from OpenStreetMap.