Elliptic Partial Differential Equations are very common mathematical models used to describe various physical processes. A new approach of parameter identification is described here. The problem is regarded as a problem of optimization of an objective function. The variable is the parameter and the objective function to minimize is a measurement of the difference between the simulated state with the chosen parameters, and the observed state variable. The gradient of the objective function will be calculated by the adjoint state method.
Parametrization is based on zonation. We solved the following problem: how to estimate the parameter based on a constant function per parts on a partition which is also unknown. Using parametrization by zonation and refinement indicators, we succeeded in finding an algorithm which detects the interface between two zones and the parameter values using two types of indicators only (horizontal and vertical). It has been tested with success on several cases. This algorithm can be generalized on parameterizations of higher size.