Estimating two time-varying reaction coefficients in a water quality model from inexact initial and boundary data

An inverse problem of identifying two time-varying reaction coefficients in a system of one-dimensional advection-reaction equations is investigated. The system is used for modeling the transportation of pollutants in rivers or streams. The coefficients to be identified represent the deoxygenation and reaeration rates. All the initial and boundary data in the model are assumed to contain noise. Lipschitz-type stability estimates of the coefficients are obtained. The proofs of the stability estimates are based on a Carleman estimate for a first-order transport operator. For numerical computation, the coefficient identification problem is reformulated as an optimization problem using the least-squares method coupled with the adjoint equation method for computing the gradient of the objective functional. Error estimates are derived and numerical examples are provided for demonstrating the performance of the proposed algorithm.