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Nonlinear programming provides a framework for this inverse problem, which is formulated as a least squares minimization subject to the differential and algebraic constraints of the network water quality model. Unknown time-dependent injection terms are introduced at every network node, and the solution of the nonlinear program provides an estimate of the time and location of contamination sources. The network water quality model contains partial differential pipe equations that are a function of both time and displacement. A naive discretization of this system in
time and space produces a large-scale, nonlinear math programming problem that is unreasonably large for current optimization tools. To overcome this difficulty, an origin tracking algorithm based on the Lagrangian technique of Liou and Kroon is
developed to reformulate the partial differential pipe expressions into a set of algebraic time delay constraints, removing the need to spatially discretize along the length of the pipes.