Identification of an N-valued heterogeneous conductivity profile in an inverse heat conduction problem

In this article we deal with the problem of determining a non-homogeneous $N$-valued heat conductivity profile in a steady-state heat conduction boundary-value problem with mixed Dirichlet-Neumann boundary conditions over a bounded domain in ${ℝ}^n$, from the knowledge of the temperature field over the whole domain. In a previous work we developed a method based on a variational approach of the PDE leading to an optimality equation which is then projected into a finite dimensional space. Discretization of the optimality equation then yields a linear although severely ill-posed equation which is then regularized via appropriate ad-hoc penalizers based upon a-priori information about the conductivities of all materials present. This process results in a generalized Tikhonov-Phillips functional whose global minimizer yields our approximate solution to the inverse problem. In our previous work we showed that this approach yields quite satisfactory results in the cases of two different conductivities. We considered here an appropriate extension of that approach for the $N$ materials case and show a few numerical examples for the case $N=3$ in which the method is able to produce very good reconstructions of the exact solution.