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

Marco Mattuschka, Daniel Walter, Max von Danwitz, Alexander Popp

TL;DR

This paper tackles identifying localized airborne contaminant sources from scarce measurements by formulating a convex inverse problem that lifts source terms to Radon measures and promotes sparsity. It combines a variational regularization framework with a Primal-Dual-Active-Point (PDAP) algorithm that greedily selects source locations and optimizes intensities, requiring only a small number of forward and adjoint PDE solves per iteration. The authors demonstrate strong performance across 2D and 3D geometries, including complex campus and building geometries, and show superiority over traditional $L^2$-regularization, even with limited sensors. The approach is computationally attractive, scalable to large domains, and readily extendable to uncertainty quantification and experimental design for critical infrastructure protection scenarios.

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.

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

TL;DR

This paper tackles identifying localized airborne contaminant sources from scarce measurements by formulating a convex inverse problem that lifts source terms to Radon measures and promotes sparsity. It combines a variational regularization framework with a Primal-Dual-Active-Point (PDAP) algorithm that greedily selects source locations and optimizes intensities, requiring only a small number of forward and adjoint PDE solves per iteration. The authors demonstrate strong performance across 2D and 3D geometries, including complex campus and building geometries, and show superiority over traditional -regularization, even with limited sensors. The approach is computationally attractive, scalable to large domains, and readily extendable to uncertainty quantification and experimental design for critical infrastructure protection scenarios.

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 -regularization. Its effectiveness is further illustrated in complex domains with real-world building geometries imported from OpenStreetMap.

Paper Structure

This paper contains 18 sections, 1 theorem, 28 equations, 17 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Forward problem, parameter-to-observable and misfit-to-adjoint map
  4. A sparse inversion framework for contaminant release
  5. Modelling of contaminant sources
  6. Quadratic regularization in source space
  7. Primal-Dual-Active-Point Algorithm for Source Identification
  8. Finite Element Discretization of Forward and Adjoint Problem
  9. Numerical Results
  10. Benchmark geometry representing two buildings
  11. Source identification in three-dimensional flow field
  12. Source identification on complex campus domain with wind in inflow-outflow configuration
  13. Closed form initial condition -- comparison with --approach
  14. Implicit source definition with elliptic PDE
  15. Instantaneous point sources modeled as Dirac distributions
  16. ...and 3 more sections

Key Result

Theorem 1

Problem eq:sparseObjective admits at least one minimizing pair $(\bar{\mu_{\text{I}}}, \bar{\mu_{\text{C}}}) \in \mathcal{M}^+_N(\bar{\Omega})^2$, i.e., as well as $\bar{N}_{\text{I}}+\bar{N}_{\text{C}} \leq N_{\text{obs}}$. Moreover, the following equivalence holds:

Figures (17)

  • Figure 1: Three-dimensional gas source identification. Manufactured initial condition (a) and simulation at $t=3.0s$ (b), reproduced by algorithmic reconstruction of initial contaminant source (c) and simulation at $t=3.0s$ (d) based on sparse measurements. Positions of noisy point-evaluations (modeled sensors) are marked by green spheres.
  • Figure 2: Illustration of commonly used benchmark problem Villa.2021. Ground truth initial condition $m_{\text{I}}$ (left), and ground truth concentration field at $t=5\s$ (solution of \ref{['eq:forward_equation']}), as well as measurement positions marked by nine green squares (right)
  • Figure 3: Adjoint transport problem. Example right-hand side of \ref{['eq:adjoint_equation']}, i.e., a sensor misfit of "$1$" applied at $({t^{\text{obs}}_i,x_i^\text{obs}}) = (5\s, [ 0.2\m ,0.25\m])$ (left), solution field of \ref{['eq:adjoint_equation']} evaluated at $t=0$.
  • Figure 4: Summary of the inverse problem with sparse regularization
  • Figure 5: Mathematical description of contaminant release. Closed form expression $\mathcal{S}^1([0.35, 0.7],0.26,\cdot)$, elliptic PDE solution $\mathcal{S}^2([0.8, 0.2],1.0,0.001,\cdot)$, and very localized Dirac distribution ${\mathcal{S}^3([0.5, 0.5],\cdot)}$.
  • ...and 12 more figures

Theorems & Definitions (1)

  • Corollary 1