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We are interested on an inverse problem of distributed parameter estimation in a partial differential equation (PDE) from measures of the PDE's solution. The considered parameter is supposed to be a piecewise constant function. Identifying the parameterization consists on identifying both values of the parameter and shapes of zones where the parameter is constant. We develop a posteriori error estimators for the considered inverse problem and we present a new algorithm, result of the combination between  adaptive parameterization technique , leading to overcome the underdetermination problem and, an adaptive mesh technique, guided by  a posteriori error estimators , providing more precise results.

Incorporating a posteriori error estimators in an adaptive parametrization algorithm