In continuous casting machinery, the molten metal solidifies in a mold. In order to control the casting process, a proper knowledge of the heat flux between the mold and the metal in real time is crucial. The heat flux in a portion of the boundary of the mold can be estimated using thermocouples measurements inside the mold and solving an inverse problem. We use a least square approach for this inverse problem minimizing the difference between the temperature measured in the thermocouples and the one computed with the estimated heat flux. Different methodologies for the solution of this problem are discussed. These methodologies are generally computational expensive. Then to solve the problem in real time, we discuss the use of model order reduction techniques.