Determination of thermal conductivity of inhomogeneous orthotropic materials from temperature measurements

We consider the two-dimensional inverse determination of the thermal conductivity of inhomogeneous orthotropic materials from internal temperature measurements. The inverse problem is general and is classified as a function estimation since no prior information about the functional form of the thermal conductivity is assumed in the inverse calculation. The least-squares functional minimizing naturally the gap between the measured and computed temperature leads to a set of direct, sensitivity and adjoint problems, which have forms of direct well-posed initial boundary value problems for the heat equation, and new formulas for its gradients are derived. The conjugate gradient method employs recursively the solution of these problems at each iteration. Stopping the iterations according to the discrepancy principle criterion yields a stable solution. The employment of the Sobolev -gradient is shown to result in much more robust and accurate numerical reconstructions than when the standard -gradient is used

Refinement indicators for estimating hydrogeologic parameters

We identify simultaneously the hydraulic transmissivity and the storage coefficient in a ground water flow governed by a linear parabolic equation. Both coefficients are assumed to be functions which are piecewise constant in space and constant in time. Therefore the unknowns are the coefficient values as well as the geometry of the zones where these parameters are constant. The identification problem is formulated as the minimization of a misfit least-square function. Using refinement indicators, we refine the parameterization locally and iteratively. We distinguish the cases where the two parameters have the same parameterization or different parameterizations