  Here we report on the characterization of one of the large-volume
LaBr$_{3}$:Ce detectors for the ELIGANT project at ELI-NP. The main focus of
this work is the response function for high-energy $\gamma$ rays of such
detectors. In particular, we compare a selection of unfolding methods to
resolve small structures in $\gamma$-ray spectra with high-energies. Three
methods have been compared using $\gamma$-ray spectra with energies up to 12
MeV obtained in an experiment at the 3 MV Tandetron\texttrademark\ facility at
IFIN-HH. The results show that the iterative unfolding approach gives the best
qualitative reproduction of the emitted $\gamma$-ray spectrum. Furthermore, the
correlation fluctuations in high-energy regime from the iterative method are
two orders of magnitude smaller than when using the matrix inversion approach
with second derivative regularization. In addition, the iterative method is
computationally faster as it does not contain large matrix inversions. The
matrix inversion method does, however, give more consistent results over the
full energy range and in the low-statistics limit. Our conclusion is that the
performance of the iterative approach makes it well suitable for semi-online
analysis of experimental data. These results will be important, both for
experiments with the ELIGANT setup, and for on-line diagnostics of the energy
spread of the $\gamma$-ray beam which is under implementation at ELI-NP.
