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

Authors:
Volkan Akçelik;George Biros;Omar Ghattas;Kevin R. Long;Bart van Bloemen Waanders
Affiliations:
Mechanics, Algorithms, and Computing Laboratory, Departments of Biomedical and Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA;Courant Institute for Mathematical Sciences, New York University, New York, NY;Mechanics, Algorithms, and Computing Laboratory, Departments of Biomedical and Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA;Sandia National Laboratories, Livermore, CA;Sandia National Laboratories, Albuquerque, NM
Venue:
Finite Elements in Analysis and Design - Special issue: 14th Robert J. Melosh competition
Year:
2003

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Abstract

We consider the inverse problem of determining an arbitrary source in a time-dependent corvective-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.