Recovering the functional form of nonlinear heat transfer by means of thermal imaging

I consider a thin metallic plate whose top side is inaccessible and in contact with an aggressive environment (a corroding fluid, hard particles hitting the boundary, ...). On first approximation, heat exchange between metal and fluid follows linear Newtons cooling law at least as long as the inaccessible side is not damaged. I assume that deviations from Newton's law are modelled by means of a nonlinear perturbative term h. On the other hand, I am able to heat the conductor and take temperature maps of the accessible side (Active Infrared Thermography). My goal is to recover the nonlinear perturbation of the exchange law on the inaccessible side. The problem is stated as an inverse ill-posed problem for the heat equation with nonlinear boundary conditions. I prove that the nonlinear term is identified by one Cauchy data set and produce approximated solutions by means of optimization. In conclusion, I try to drive mathematical modelling as far as possible: although I use advanced results in mathematical analysis (theory of nonlinear boundary value problems, domain derivative), I apply them immediately to technical problems.