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A Variational Finite Element Method for Source Inversion for ConvectiveDiffusive Transport
"... We consider the inverse problem of determining an arbitrary source in a timedependent convectivediffusive 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 pollut ..."
Abstract

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We consider the inverse problem of determining an arbitrary source in a timedependent convectivediffusive 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 illposedness of the problem, we employ Tikhonov and total variation regularization. We present a variational formulation of the first order optimality system, which includes the initialboundary value state problem, the finalboundary value adjoint problem, and the spacetime boundary value source problem. We discretize in the spacetime 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 convectiondominated, the ability of the spacetime inversion formulation to track moving sources, and the optimal convergence rate of the finite element approximation.
Source Inversion for ConvectiveDiffusive Transport �
"... We consider the inverse problem of determining an arbitrary source in a timedependent convectivediffusive 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 pollut ..."
Abstract
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We consider the inverse problem of determining an arbitrary source in a timedependent convectivediffusive 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 illposedness of the problem, we employ Tikhonov and total variation regularization. We present a variational formulation of the first order optimality system, which includes the initialboundary value state problem, the finalboundary value adjoint problem, and the spacetime boundary value source problem. We discretize in the spacetime volume using Galerkin finite