Departmental Papers (MEAM)

Document Type

Journal Article

Date of this Version

May 2003

Comments

Postprint version. Published in Finite Elements in Analysis and Design, Volume 39, Issue 8, May 2003, pages 683-705.
Publisher URL: http://dx.doi.org/10.1016/S0168-874X(03)00054-4

NOTE: At the time of publication, author George Biros was affiliated with New York University. Currently (January 2006), he is a faculty member in the Department of Mechanical Engineering and Applied Mechanics at the University of Pennsylvania.

Abstract

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.

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Date Posted: 25 January 2006

This document has been peer reviewed.