An inverse problem from 2D ground-water modelling

The paper is devoted to the inverse problem of identifying the coefficient in the main term of an elliptic differential equation describing the filtration of ground water. Experience suggests that the gradient of the piezometric head, i.e. Darcy's velocity, may have discontinuities and the transmissivity coefficient is a piecewise constant function. To solve this problem we use a modification of a direct method of Vainikko. Starting with a weak formulation of the problem, a suitable discretization is obtained by the method of least error. If necessary, this method can be combined with Tikhonov's regularization. The main difficulty consists of generating distributed state observations from measurements of the ground-water level. For this step we propose an optimized data preparation procedure using additional information such as knowledge of the sought parameter values at some points and lower and upper bounds for the parameter. Numerical tests confirm the expected local behaviour of the method, i.e. locally sufficiently many measurements provide locally satisfactory results. Two numerical examples, one with simulated data and the other with real life data, are given.