  We consider the quasilinear 1D inverse heat convection problem (IHCP) of
determining the enthalpy-dependent heat fluxes from noisy internal enthalpy
measurements. This problem arises in the Accelerated Cooling (ACC) process of
producing thermomechanically controlled processed (TMCP) heavy plates made of
steel. In order to adjust the complex microstructure of the underlying
material, the Leidenfrost behavior of the hot surfaces with respect to the
application of the cooling fluid has to be studied. Since the heat fluxes
depend on the enthalpy and hence on the solution of the underlying initial
boundary value problem (IBVP), the parameter-to-solution operator, and thus the
forward operator of the inverse problem, can only be defined implicitly. To
guarantee well-defined operators, we study two approaches for showing existence
and uniqueness of solutions of the IBVP. One approach deals with the theory of
pseudomonotone operators and so-called strong solutions in Sobolev-Bochner
spaces. The other theory uses classical solutions in H\"older spaces. Whereas
the first approach yields a solution under milder assumptions, it fails to show
the uniqueness result in contrast to the second approach. Furthermore, we
propose a convenient parametrization approach for the nonlinear heat fluxes in
order to decouple the parameter-to-solution relation and use an iterative
solver based on a Projected Quasi-Newton (PQN) method together with
box-constraints to solve the inverse problem. For numerical experiments, we
derive the necessary gradient information of the objective functional and use
the discrepancy principle as a stopping rule. Numerical tests show that the PQN
method outperforms the Landweber method with respect to computing time and
approximation accuracy.
