  In this work, we investigate the estimation of the transient mold-slab heat
flux in continuous casting molds given some thermocouples measurements in the
mold plates. Mathematically, we can see this problem as the estimation of a
Neumann boundary condition given pointwise state observations in the interior
of the domain. We formulate it in a deterministic inverse problem setting.
After introducing the industrial problem, we present the mold thermal model and
related assumptions. Then, we formulate the boundary heat flux estimation
problem in a deterministic inverse problem setting using a sequential approach
according to the sequentiality of the temperature measurements. We consider
different formulations of the inverse problem. For each one, we develop novel
direct methodologies exploiting a space parameterization of the heat flux and
the linearity of the mold model. We construct these methods to be divided into
a computationally expensive offline phase that can be computed before the
process starts, and a cheaper online phase to be performed during the casting
process. To conclude, we test the performance of the proposed methods in two
benchmark cases.
