A variational finite element method for source inversion for convective–diffusive transport

• Published in: Finite elements in analysis and design 2003-05, Vol.39 (8), p.683-705
• Main Authors: Akçelik, Volkan, Biros, George, Ghattas, Omar, Long, Kevin R., Waanders, Bart van Bloemen
• Format: Article
• Language: English
• Subjects: Computational techniques; Exact sciences and technology; Finite-element and galerkin methods; Mathematical methods in physics; Numerical approximation and analysis; Ordinary and partial differential equations, boundary value problems; Physics; Statistical physics, thermodynamics, and nonlinear dynamical systems; Transport processes
• ISSN: 0168-874X; 1872-6925
• DOI: 10.1016/S0168-874X(03)00054-4

We consider the inverse problem of determining an arbitrary source in a time-dependent convective–diffusive transport equation, given a velocity field and pointwise measurements of the concentration. Applications that give rise to such problems include determination of groundwater or airborne pollutant sources from measurements of concentrations, and identification of sources of chemical or biological attacks. To address ill-posedness of the problem, we employ Tikhonov and total variation regularization. We present a variational formulation of the first-order optimality system, which includes the initial-boundary value state problem, the final-boundary value adjoint problem, and the space–time boundary value source problem. We discretize in the space–time volume using Galerkin finite elements. Several examples demonstrate the influence of the density of the sensor array, the effectiveness of total variation regularization for discontinuous sources, the invertibility of the source as the transport becomes increasingly convection-dominated, the ability of the space–time inversion formulation to track moving sources, and the optimal convergence rate of the finite element approximation.